Statistics Worksheet

QUESTION 11. Test of hypothesis Ho: µ ≥ 20 against H1: µ < 20 leads to: Two-tailed test. None of these. Right-tailed test. Left-tailed test. 2 points QUESTION 2 1. For a t-test based on a sample of 18 observations with the test statistic t-stat = -1.75, what is the associated p-value? p-value=0.019. p-value=0.156. p-value=0.425. No enough information. 3 points QUESTION 3 1. In a recent speech to students, the Dean of the College of Business claims that the average grade of students in the college is no longer 75. To conduct a test of the dean's claim, a randomly selected sample of 64 students revealed an average grade of 76 with a standard deviation of 5. Is there evidence that the average grade is no longer 75? Use α = 0.05. H0: µ = 75 vs H1: µ ≠ 75. z-stat = 1.6. p-value = 0.11. Do not reject H0 and thers is not enough evidence to support the dean's claim. H0: µ = 75 vs H1: µ ≠ 75. z-stat = 1.6. p-value = 0.11. Do not reject H0 and there is enough evidence to support the dean's claim. H0: µ = 75 vs H1: µ ≠ 75 t-stat = 1.6. p-value = 0.115. Do not reject H0 and thers is enough evidence to support the dean's claim. H0: µ = 75 vs H1: µ ≠ 75. t-stat = 1.6. p-value = 0.115. Do not reject H0 and there is not enough evidence to support the dean's claim. 4 points QUESTION 4 1. A company claims that the mean weight of coffee in its 16ounce bags is 16.05 ounces. Consumer advocates group constructed the following 95% confidence interval (16.09 , 16.12). Based on this interval, do you think that the company's claim is plausible at the 5% significance level? 16.05 lies in the interval (16.09 , 16.12), so we do not reject the manufacturer's claim. 16.05 does not lie in the interval (16.0 , 16.2), so we reject the manufacturer's claim. 16.05 does not lie in the interval (16.09 , 16.12), so we reject the company's claim. Not enough information to answer the question. 3 points QUESTION 5 1. A cereal company claims that the mean weight of Brand A cereals is less than 16 oz. The p-value is found to be 0.014. If the test had been two-tailed, the p-value would have been: 0.014. 0.028. 0.052. 0.007. 3 points QUESTION 6 1. The probability of rejecting null hypothesis when it is true is is β a Type II error is α a Type I error 3 points QUESTION 7 1. A survey claims that the average cost of a hotel room in Abu Dhabi is at most AED 799. A random sample of 40 hotel rooms has an average cost of AED 835 and a sample standard deviation of AED 90. What is the corresponding pvalue of the test? 4 points QUESTION 8 1. Oasis Company bottles soft drink cans using an automatic filling machine. When the process is running properly, the mean fill is 450 ml per can. Today, the company selected a random sample of 50 cans yielding a mean of 451.75 ml and a standard deviation of 7.5 ml. The company would like to test whether the filling process is working properly. At which significance level α, will the null hypothesis be rejected? 5% 3% 1% 10% 3 points QUESTION 9 1. It was found that in a study of air-bag effectiveness 1000 crashes of midsize cars equipped with air bags, 65 of the crashes resulted in hospitalization of the drivers. We want to test the claim that the air-bag hospitalization rate is more that the 5.5% rate for crashes of midsize cars equipped with automatic safetly belts. Calculate the statistic value of the to three decimal places. test. Round and the pyour answer 4 points QUESTION 10 1. If you do not rejected H0 at the 0.05 level of significance, then you must reject it at the 0.01 level. you must reject it at the 0.03 level. you must reject it at the 0.1 level. you may reject or not reject it at the 0.1 level. 3 points QUESTION 11 1. The average score on Midterm Exam over the past years was 80. With a new teaching curriculum introduced, it is believed that this score will change. A random sample of 20 students is selected and the mean score was found to be 75 with a standard deviation of 15. Assume the scores are normally distributed, what is the corresponding p-value of the test? 4 points QUESTION 12 1. In which of the following situations, the one proportion z-test will not be applicable? n=55, x=5 and p0=0.1 n=125, x=7 and p0=0.07 n=1025, x=51 and p0=0.06 n=55, x=5 and p0=0.09 4 points QUESTION 13 1. A grocery store owner claims that the mean amount spent per checkout is more than AED 1850. We test the hypothesis H0: μ ≤ 1850 versus H1: μ > 1850. The null hypothesis
is not rejected. State the appropriate conclusion.
The mean checkout amount is more than AED 1850.
There is not enough evidence to conclude that the mean
checkout amount is more AED 1850.
The mean checkout amount is at least AED 1850.
There is enough evidence to conclude that the mean
checkout amount is greater than AED 1850.
3 points
QUESTION 14
1.
The health of employees is monitored by periodically weighing
them. A sample of 55 employees has a mean weight of 185 lb
and a standard deviation of 40.5 lb. At a significance level 5%,
we want to test the claim that the population mean of all such
employees weights is less than 200 lb. Assume the population
is normaly distributed, which test would be appropriate for this
problem?
One-sample t-test.
Either one-sample z-test or one-sample t-test.
Neither one-sample z-test nor one-sample t-test.
One-sample z-test.
3 points
QUESTION 15
1.
Find the p-value for a two tailed test, if t-stat = – 1.32 and the
sample size n=13.
2 points
QUESTION 16
1.
State the null and the alternative hypotheses for
the following situation:
It is claimed that the average age of the customers of a car
insurance company does not exceed 38 years.
H0:
μ < 38 versus H1: μ ≥ 38 H0: μ ≥ 38 versus H1: μ < 38 H0: μ = 38 versus H1: μ ≤ 38 H0: μ ≤ 38 versus H1: μ > 38
3 points
QUESTION 17
1.
A magazine article cliams that the mean time of evening longdistance calls is 15.2 minutes per call. A researcher wished to
test the claim and selected a random sample of 49 calls. She
found that the mean time was 15.9 minutes and standard
deviation was 5 minutes. Using a 0.05 level of significance, is
there sufficient evidence to support the claim?
p-value=0.327, there is not enough evidence to support the
claim.
p-value=0.164, there is enough evidence to support the
claim.
p-value=0.327, there is enough evidence to support the
claim.
p-value=0.164, there is not enough evidence to support the
claim.
4 points
QUESTION 18
1.
The inspector at a manufacturing company claims that more
than 4% of the produced smoke detectors are defective. A
random sample of 150 smoke detectors is selected of which 12
are defective. Find the p-value of the test. (Round your
answer to three decimal places).
4 points
QUESTION 19
1.
Marketers believe that less 67% of adults in UAE own an
iPhone. The appropriate null and alternative hypotheses are
H0: p = 0.67 vs Ha: p ≠ 0.67.
H0: p > 0.67 vs Ha: p ≤ 0.67.
H0: p ≤ 0.67 vs Ha: p > 0.67.
H0: p ≥ 0.67 vs Ha: p < 0.67. 3 points QUESTION 20 1. State the null and the alternative hypotheses for the following situation: It is claimed that the mean lifetime of car engines of a particular type is at least 220000 miles. H0: μ ≤ 220000 versus H1: μ > 220000
H0:
μ < 220000 versus H1: μ ≥ 220000 H0: μ ≥ 220000 versus H1: μ < 220000 H0: μ = 220000 versus H1: μ > 220000
3 points
QUESTION 21
1.
An environmental researcher wants to know whether the mean
amount of sulfur dioxide in the air in UAE cities is more than 1.19
parts per billion (ppb). She tested the hypotheses H0: μ
≤1.19 versus Ha: μ>1.19 using a significance level α = 0.01. The
p-value of the test is 0.013. If the true value of μ is 1.20, the
conclusion results in
Both Type I and Type II errors.
Type II error.
Type I error.
Correct decision.
3 points
QUESTION 22
1.
A production line operates with a mean filling weight of 16
ounces per container. A quality control inspector samples
seven items
15.86 16.18 16.23 15.98
16.05 15.95 16. 29
and wants to determine whether or not the filling weight has to
be adjusted. Suppose that the filling weight is normally
distributed.
(a) Compute the mean
and the standard
deviation
of the dataset. Round your answers
to three decimal places.
(b) Calculate the test statistic of the test
answer to three decimal places.
(c) What is the p-value of the test
. Round your
?
6 points
QUESTION 23
1.
Find the p-value for a right tailed test, if z-stat = 1.75.
2 points
QUESTION 24
1.
The health of employees is monitored by periodically weighing
them. A sample of 61 employees has a mean weight of 184 lb
and a standard deviation of 42 lb. At a significance level 5%,
we want to test the claim that the population mean of all such
employees weights is less than 190 lb. Calculate the test
statistic of the test. If needed, round your answer
to two decimal places.
4 points
QUESTION 25
1.
The HR manager of an organization is bargaining to get a
new dental insurance plan. He claims that the yearly family
dental expenses of its employees does not exceed AED
20000. A random sample of 49 employees reveals that the
average family dental expenses is AED 20100 and the standard
deviation is AED 600. At 1% significance level, can we support
the manager’s claim?
(1) State the hypotheses, (2) Check the assumptions of the test
and compute the test statistic (Round your answer
to three decimal place), (3) Find the p-value (Round your
answer to three decimal place), and (4) Make a decision and
state the conclusion.
6 points
QUESTION 26
1.
It was found that in a study of air-bag effectiveness 821 crashes
of midsize cars equipped with air bags, 52 of the crashes
resulted in hospitalization of the drivers. We want to test the
claim that the air-bag hospitalization rate is lower that the 7.8%
rate for crashes of midsize cars equipped with automatic safetly
belts. At 0.01 significance level we conclude that
there is sufficient evidence to support the claim that for
crashes of midsize cars, the air-bag hospitalization rate is
less than the 7.8% rate for automatic safetly belts.
there is no sufficient evidence to support the claim that for
crashes of midsize cars, the air-bag hospitalization rate is
less than the 7.8% rate for automatic safetly belts.
there is sufficient evidence to support the claim that for
crashes of midsize cars, the air-bag hospitalization rate is
at least 7.8% rate for automatic safetly belts.
there is sufficient evidence to support the claim that for
crashes of midsize cars, the air-bag hospitalization rate is
at most 7.8% rate for automatic safetly belts.
4 points
QUESTION 27
1.
Auto insurance companies are beginnning to consider raising
rates for those who use mobile phones while driving. A
consumer group claims that at least 19% of drivers use mobile
phones. The insurance industry conducts a study and finds that
among 500 randomly selected drivers, 80 use mobile phones.
At 0.05 level of significance, test the group’s claim.
(1) State the hypotheses. (2) Check the assumptions of the test
and compute the test statistic (Round your answer
to three decimal place). (3) Find the p-value. (Round your
answer to three decimal place). (4) Make a decision and state
the conclusion.
6 points
QUESTION 28
1.
A random sample of 1000 college applications showed that
769 (of those applications were submitted online. Compute
the test statistic used to test the claim that among all college
applications the percentage submitted online is 75%. Round
your answer to three decimal places.
MTH213: Business Statistics
Chapter 9: Fundamentals of Hypothesis-Testing
Methodology: One-Sample Tests
Zayed University
College of Natural and Health Sciences
Fall 2022
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Fall 2022
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Outline
1
Introduction to Hypothesis Testing
2
Hypothesis Testing for Population Mean
One-Sample z test
One-Sample t test
3
Hypothesis Testing for Population Proportion
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Outline
Objectives
In this chapter, you learn:
The formal hypothesis testing process.
To conduct formal hypothesis tests concerning the population
mean.
To conduct formal hypothesis tests concerning the population proportion.
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Introduction to Hypothesis Testing
Table of Contents
1
Introduction to Hypothesis Testing
2
Hypothesis Testing for Population Mean
One-Sample z test
One-Sample t test
3
Hypothesis Testing for Population Proportion
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Introduction to Hypothesis Testing
Introduction
Statistical inference deals with making statements about population (or model) parameters. The two main areas of statistical
inference are estimation and hypothesis testing.
Hypothesis Testing is used to assess the evidence provided by the
data in favor of some claim about the population. We perform
a test of hypothesis only when we are making a decision about a
population parameter based on the value of the sample statistic.
Statistics is about making decisions:
1
2
3
4
5
Is the normal body temperature actually 37◦ C?
Do male and female ZU students differ in their environmental
awareness?
Does the keto diet cause diabetes?
Which technical support company should we deal with?
Is the average connection speed 100 Mbps, as claimed by the
internet service provider?
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Introduction to Hypothesis Testing
À Hypotheses
A statistical hypothesis is a claim (assertion) about a population
parameter (or parameters).
All hypothesis-testing procedures require generating two hypotheses where the purpose of hypothesis testing is to decide which of
these two hypotheses is more likely to be correct.
Null hypothesis (H0 ):
I A null hypothesis is a claim (or statement) about a population
parameter that is assumed to be true until it is declared false.
I The test is designed to assess the evidence against H0 .
I The null hypothesis will be rejected only if the sample data provide substantial contradictory evidence.
Alternative hypothesis (Ha or H1 ):
I The alternative hypothesis is a claim about a population parameter that will be true if the null hypothesis is false.
I The test is designed to assess the evidence that supports H1 .
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Introduction to Hypothesis Testing
Formulating Hypotheses
The context of the situation is very important in determining how
the hypotheses should be stated.
In some cases it is easier to identify the alternative hypothesis first
while in other cases the null is easier.
Alternative hypothesis as a research hypothesis:
I Many applications of hypothesis testing involve an attempt to
gather evidence in support of a research hypothesis.
I In such cases, it is often best to begin with the alternative hypothesis and make it the conclusion that the researcher hopes to
support.
Null hypothesis as an assumption to be challenged:
I We might begin with a belief or assumption that a statement
about the value of a population parameter is true.
I In such cases, it is helpful to develop the null hypothesis first.
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Introduction to Hypothesis Testing
Hypothesis Tests
Typically, H0 contains some form of equality (≤, ≥ or =), while
H1 contains strict inequality (>, < or 6=). Tests are labeled based on the sign in H1 : 1 Right-tailed test: H1 contains > (greater than).
Results if the problem says increases, improves, better, result is
higher, etc.
2
Left-tailed test:
H1 contains < (less than). Results if the problem says decreases, reduces, worse than, result is lower, etc. 3 Two-tailed test: H1 contains 6= (Not equal to). Results if the problem says different from, no longer the same, changes, etc. Department of Mathematics and Statistics MTH213: Business Statistics Fall 2022 8 / 58 Introduction to Hypothesis Testing Hypotheses The relationship between possible verbal statements about the parameter µ and the corresponding null and alternative hypotheses. Verbal Statement H0 Verbal Statement H1 The mean is ... greater than or equal to µ0 . Hypotheses ( at least µ0 . H1 : µ < µ0 not less than µ0 . less than or equal to µ0 . ( at most µ0 . µ0 . exactly µ0 . Department of Mathematics and Statistics H0 : µ ≤ µ0 H1 : µ > µ0
not more than µ0 .
equal to µ0 .
H0 : µ ≥ µ0
(
H0 : µ = µ0
H1 : µ 6= µ0
MTH213: Business Statistics
The mean is …
less than µ0 .
below µ0 .
fewer than µ0 .
greater than µ0 .
above µ0 .
more than µ0 .
not equal to µ0 .
different from µ0 .
not µ0 .
Fall 2022
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Introduction to Hypothesis Testing
Hypotheses: Examples
Goodyear’s tire will last, on average, longer than 60,000 km.
H0 : µ ≤ 60, 000
vs.
H1 : µ > 60, 000
No more than 6% of the sold laptops are returned for repair during
the warranty period.
H0 : p ≤ 0.06
vs.
H1 : p > 0.06
Average waiting time for COVID-19 test is less than 30 minutes
H0 : µ ≥ 30
vs.
H1 : µ < 30 The average normal body temperature actually is not 37◦ C. H0 : µ = 37 vs. H1 : µ 6= 37 A bank manager claims that 85% of its customers are served within 10 minutes. H0 : p = 0.85 Department of Mathematics and Statistics vs. MTH213: Business Statistics H1 : p 6= 0.85 Fall 2022 10 / 58 Introduction to Hypothesis Testing Exercises State the null and alternative hypotheses for the following tests: a A company advertises that the mean life of its laptops exceeds 10 years. b A consumer analyst reports that the mean life of a certain type of automobile battery is not 74 months. c The average age of customers of an electronics store is 25 years. The company wants to know if customers who use their website are younger on average. d A recent newspaper report claims that more than 67% of workers in UAE say they are more productive working from home. e Last year, 45% of the employees were willing to pay for on-site day care. The company wants to know if that has changed. f The owner of a local pizza restaurant has determined that home delivery will be successful only if the average time spent on a delivery does not exceed 25 minutes. Department of Mathematics and Statistics MTH213: Business Statistics Fall 2022 11 / 58 Introduction to Hypothesis Testing Á Test Statistic Testing hypothesis is based on a test statistic; a quantity calculated from the sample data and has some known distribution if H0 is true. Test statistics are used to discriminate between H0 and H1 . When we verify a hypothesis about some parameter (say µ), the test statistic is usually obtained by a suitable transformation of its estimator (say x). We find a suitable test statistic and compute it from the data. For example, if we are interested in testing H0 : µ = µ0 assuming that the underlying distribution of the population is normal, then the test statistic would be x − µ0 √ z= σ/ n which follows a standard normal distribution. Department of Mathematics and Statistics MTH213: Business Statistics Fall 2022 12 / 58 Introduction to Hypothesis Testing  Decision Rule The next step in hypothesis testing is assessing the strength of the evidence against the null hypothesis by defining a decision rule that prescribes when H0 is to be rejected. Basically, H0 is rejected when the test statistic takes a value which is deemed very unlikely if H0 were true. In practice, there are two approaches for making a statistical decision: 1 2 Rejection region (Region of rejection) P-value (Probability value) Both approaches will ensure the same conclusion and either one will work. However, the p-value approach is more commonly used and provided in published literature. Additionally, it gives us some idea of the strength of the evidence against the null hypothesis. Department of Mathematics and Statistics MTH213: Business Statistics Fall 2022 13 / 58 testing methodology provides clear definitions for evaluating differences. Furthermore, it enables Introduction to Hypothesis Testing you to quantify the decision-making process by computing the probability of getting a certain sample result if the null hypothesis is true. You calculate this probability by determining the sampling distribution for the sample statistic of interest (e.g., the sample mean) and then computing the particular test statistic based on the given sample result. Because the sampling distribution for the test statistic often follows a well-known statistical distribution, such as the standardized normalvalue distribution t distribution, you canbetween use these distributions to help deter- (ReThe critical is orthe boundary the two regions mine whether the null hypothesis is true. Rejection Region p tic folsampling jection region and non-rejection region). The rejection region isand theNonrejection set, or region, of unlikely values of the Regions of Rejection The sampling distribution of the is divided two regions, a region test statistic. Hence, if test thestatistic value of theintotest statistic fallsof in the rejection (sometimes called the critical region) and a region of nonrejection (see rejection region, the null hypothesis is rejected. Figure 9.1). If the test statistic falls into the region of nonrejection, you do not reject the null hypothesis. In the Oxford Cereals scenario, you conclude that there isto insufficient The size of the rejection region is directly related the risks inevidence that the population mean fill is different from 368 grams. If the test statistic falls into the rejection region, you reject the nullevidence hypothesis. Into this make case, you decisions conclude that the volved in using only sample about a population mean is not 368 grams. population parameter. .1 ction on in ing Region of Rejection Department of Mathematics and Statistics Critical Value Region of Nonrejection Critical Value MTH213: Business Statistics Region of Rejection Fall 2022 14 / 58 Introduction to Hypothesis Testing Chapter 9 Rejection Region One-Sample Hypothesis Tests 345 rejection region. You may visualize the level of significance (α) as an area in the tail(s) of a distribution (e.g., normal) far enough from the center that it represents an unlikely outcome if our null hypothesis is true. We will calculate a test statistic that measures the difference between the sample statistic and the hypothesized parameter. A test statistic that falls in the shaded region will cause rejection of H0, as illustrated in Figure 9.2. The area of the nonrejection region (white area) is 1 2 α. The risk, or the level of significance (α) represents the size of the rejection region, which also depends on the test type (twotailed, right-tailed or left-tailed). The decision maker specifies the FIGURE 9.2 value αμ for the test (common choices are 0.10, 0.05, or 0.01). Tests for Hof :μ5 0 0 Left-Tailed Test Reject α Do not reject 12α Critical value Two-Tailed Test Reject Right-Tailed Test Reject Do not reject 12α α/2 Critical values Do not reject 12α α/2 Critical values Reject α Critical value The rejection region does not convey the full information conCritical Value tained in the databetween regarding the against H0 . It tells The critical value is the boundary the two regions (rejectevidence H , do not reject H ). The decision rule states what the critical value of the test statistic would have to be in order to nothing about the location of the value of the test reject H at the chosen level of significance (α). For example, if we arecomputed dealing with a normal sampling distribution for a mean, we might reject H if the sample mean x differs from μ by statistic in the rejection region. more than 1.96 times the standard error of the mean (outside the 95 percent confidence inter0 0 − 0 valMathematics for μ). Department of and Statistics 0 MTH213: Business Statistics 0 Fall 2022 15 / 58  Left-tailed test: The P-value equals the probability of observing a value of the test inHypothesis Fig. 9.7(a). to |z 0 |, as illustrated Introduction to Testing z that is as small as or smaller than the value actually observed, which is The statistic P-value the area under the standard normal curve that lies to the left of z , as illustrated 0 in Fig. 9.7(b). P-value  Right-tailed test: The P-value equals the probability of observing a value of the test statistic zisthat is asprobability, large as or larger than H the0 ,value actually observed, whichofis the the The P-value the under of observing a value as illustrated in area under the standard normal curve lies to extreme the right of(in z 0 , the test statistic that is as extreme as, that or more direction Fig. 9.7(c). of the alternative hypothesis) than the observed value. RE 9.7 P-value st if the tailed, t tailed P-value P-value −|z| 0 |z| z (a) Two tailed z 0 z (b) Left tailed 0 z z (c) Right tailed A P-value is a conditional probability which can be written as: P-value = P (observed statistic value [or even more extreme]|H0 ) E 9.8 Determining the P-Value for a One-Mean z-Test Department of Mathematics and Statistics MTH213: Business Statistics Fall 2022 16 / 58 Introduction to Hypothesis Testing The P-value The p-value captures the strength of the evidence against H0 ; the smaller the p-value, the stronger the evidence against H0 . The p-value, sometimes called the probability of chance, can be thought of as the probability that the results of a statistical experiment are due only to chance. The lower the p-value, the greater the likelihood of obtaining the same (or very similar) results in a repetition of the statistical experiment. Thus, a low p-value is a good indication that your results are not due to random chance alone. Important remarks: I The p-value is not the probability that the null hypothesis is true. I Even if p-value is small the observed failure of H0 may not be practically important (Statistical and practical significance). Department of Mathematics and Statistics MTH213: Business Statistics Fall 2022 17 / 58 Introduction to Hypothesis Testing à Decision To assess the evidence against H0 , we compare the p-value with a fixed value, called the significance level (α). If the p-value is less than the significance level, we reject the null hypothesis. Typically, hypotheses are tested at significance levels as small as 0.01, 0.05, or 0.10, although there are exceptions. Testing at a low level of significance means that only a large amount of evidence can force rejection of H0 . When we carry out the test we assume H0 is true. Hence the test will result in one of two decisions: I Reject H0 : Hence we have sufficient evidence to conclude that the alternative hypothesis is true. Such a test is said to be significant. I Fail to reject H0 : Hence we do not have sufficient evidence to conclude that the alternative hypothesis is true. Such a test is said to be insignificant. Department of Mathematics and Statistics MTH213: Business Statistics Fall 2022 18 / 58 If the to p-value is between Introduction Hypothesis Testing .05 and .10, we say that there is weak evidence to indicate Decision that the alternative hypothesis is true. When the p-value is greater than 5%,we say that the result is not statistically significant. When the p-value exceeds .10, we say that there is little to no evidence to infer that the alternative hypothesis is true. The decision rule thethese p-value Figure 11.6using summarizes terms. approach in all tests is  p-value ≤ α ⇒ Reject H0 FIGUREp-value 11.6 Describing > α p-Values
⇒ Do not reject H0
Overwhelming
Evidence
(Highly Significant)
0
.01
Strong Evidence
(Significant)
Weak Evidence
(Not Significant)
.05
Little to no Evidence
(Not Significant)
.10
1.0
If the test rejects the hypothesis, all we can state is that the data
provides11-2f
sufficient
evidence
H0 .Region
It may
either happen
The p-Value
andagainst
Rejection
Methods
because HIf0 weissonot
true, or because our sample is too extreme.
choose, we can use the p-value to make the same type of decisions we make in
rejection region
rejection region
requires the
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The latter,thehowever,
canmethod.
onlyThe
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withmethod
probability
α.decision
If the
to select a significance level from which the rejection region is constructed. We then
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to reject
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not reject the nullithypothesis.
Anotherthat
way ofthe
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with
the
selected
value
of
the
significance
level. If
obtained from the data is NOT sufficient to reject it.
the p-value is less than α, we judge the p-value to be small enough to reject the null
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Fall 2022
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Introduction to Hypothesis Testing
General Testing Procedure: Summary
Testing Procedure
1
Hypotheses:
From the problem context, define H0 and H1 .
2
Test Statistic:
Summarize the data, verify the assumptions, and compute the
value of the test statistic.
3
P-value:
Use the test statistic and its corresponding sampling distribution
along with the test type to compute the P-value.
4
Decision and Interpretation of the results:
I Decision: If p-value ≤ α, we reject H0 (significant at level α).
If p-value > α, we do not reject H0 .
I Conclusion: Give a simple explanation of your conclusions in the
context of the original problem.
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Introduction to Hypothesis Testing
Errors in Hypothesis Testing
When testing hypotheses, we realize that all we see is a random
sample. Therefore, because of sampling variability, our decision
to reject or not to reject H0 may still be wrong.
Decision
Do not reject H0
Reject H0
True state
H0 is true
H0 is false
Correct decision
Type II error
Type I error
Correct decision
The probability of Type I Error (rejecting H0 when it is true)
is α, the significance level.
The probability of Type II Error (not rejecting H0 when it is
false) is called β.
The quantity 1 − β is called the power of a test and represents
the probability of rejecting H0 when it is, in fact, false.
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Introduction to Hypothesis Testing
Errors in Hypothesis Testing: Example
A company manufactures ball bearings for computer fans. The average
diameter of a certain type of ball bearing should be 80 mm. To check
that theaverage diameter is correct, the company wants to test.
H0 : µ = 80 mm (manufacturer’s specification)
H1 : µ 6= 80 mm (cause for adjusting process)
1
A type I error is caused when sample evidence indicates that we should
reject H0 when, in fact, the average diameter of the ball bearings being
produced is 80 mm. A type I error will cause a needless adjustment
and delay of the manufacturing process.
2
A type II error occurs if the sample evidence leads us not to reject
H0 when, in fact, the average diameter of the ball bearing is either
too large or too small to meet specifications. Such an error would
mean that the production process would not be adjusted even though
it really needed to be adjusted. This could possibly result in a large
production of ball bearings that do not meet specifications.
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Hypothesis Testing for Population Mean
Table of Contents
1
Introduction to Hypothesis Testing
2
Hypothesis Testing for Population Mean
One-Sample z test
One-Sample t test
3
Hypothesis Testing for Population Proportion
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Hypothesis Testing for Population Mean
Hypothesis Testing on the Mean
We hypothesize that the population mean (µ) equals some value µ0 .
A random sample of n is to be selected, and the sample mean, x, and
sample standard deviation, s, are determined.
1
Case I: Large sample and Normality is NOT assumed
I We use one-sample z-test.
2
Case II: Normal population:
I We use one-sample t-test based on t distribution with degrees of
freedom df = n − 1.
I When the sample size is small (Less than 30), normality must be
tested before applying this procedure.
Population
Distribution
Normal
Non-normal
Department of Mathematics and Statistics
Sample size
Large (n ≥ 30) Small (n < 30) t−test z−test Not applicable MTH213: Business Statistics Fall 2022 24 / 58 Hypothesis Testing for Population Mean One-Sample z test Case I: One-Sample z test One-Sample z test 1 Hypotheses: H0 : µ ≤ µ0 vs. H1 : µ > µ0 (Right-tailed test)
H0 : µ ≥ µ0 vs. H1 : µ < µ0 (Left-tailed test) H0 : µ = µ0 vs. H1 : µ 6= µ0 (Two-tailed test) 2 Test statistic: (Large n and normality NOT assumed) zST AT = 3 P-value Department of Mathematics and Statistics Test type Right-tailed Left-tailed Two-tailed x − µ0 √ s/ n p-value P (Z > zST AT )
P (Z < zST AT ) 2P (Z > |zST AT |)
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Hypothesis Testing for Population Mean
One-Sample z test
Finding p−value: Online Normal Calculator
StatDistribution.com
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Hypothesis Testing for Population Mean
One-Sample z test
Example (1)
The number of concurrent users for some internet service provider
has always averaged 5000. After an equipment upgrade, the average
number of users at 100 randomly selected moments of time is 5200
with a standard deviation of 800. Does it indicate, at a 5% level of
significance, that the mean number of concurrent users has increased?
1
Hypotheses:
H0 : µ ≤ 5000 versus H1 : µ > 5000
2
Test statistic: We have n = 100, x = 5200, s = 800 and α =
0.05. We have a large sample and normality of number of concurrent users is not assumed so we can use the z test.
x − µ0
5200 − 5000
√ =
√
zST AT =
= 2.5
s/ n
800/ 100
Department of Mathematics and Statistics
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Hypothesis Testing for Population Mean
One-Sample z test
Example (1)
3
P-value:
p − value = P (Z > 2.5) = 0.006
4
Conclusion:
Since p-value=0.006 < 0.05, we reject H0 and conclude that the mean number of concurrent users has increased. Enter either the p-value (represented by the blue area on the graph) or the test statistic (the coordinate along the horizontal axis) below to have the other value computed. Normal distribution Other distributions: Student's t • Chi-square • F p-value: 0.006 z-value: 2.5 mean: 0 std. dev: 1 two tails right tail left tail mean to z 2-sided mean to z p = 0.006 -3 -2 -1 0 1 2 3 z = 2.5 Link to calculation: http://www.statdistributions.com/normal?z=2.5&tail=2 Department of Mathematics and Statistics MTH213: Business Statistics Fall 2022 28 / 58 Hypothesis Testing for Population Mean One-Sample z test Example (2) Oasis Company bottles soft drinks using an automatic filling machine. When the process is running properly, the mean fill is 450 ml per can. Each day, the company selects a random sample of 36 cans and measures the volume in each can. They then test to determine whether the filling process is working properly. At 5% significance level, what conclusion should the company reach if the sample mean is 450.75 ml and the standard deviation is 7.5 ml? 1 Hypotheses: H0 : µ = 450 versus H1 : µ 6= 450 2 Test statistic: We have n = 36, x = 450.75, s = 7.5 and α = 0.05. The sample size is sufficiently large and normality of the volume is not assumed, so we can use the z test. x − µ0 450.75 − 450 √ = √ zST AT = = 0.6 s/ n 7.5/ 36 Department of Mathematics and Statistics MTH213: Business Statistics Fall 2022 29 / 58 Hypothesis Testing for Population Mean One-Sample z test Example (2) 3 P-value: p − value = 2P (Z > |0.6|) = 0.549
4
Conclusion:
Since p-value=0.549 > 0.05, we do not reject H0 and conclude
that the filling process is working properly.
Enter either the p-value (represented by the blue area on the graph) or the test statistic (the coordinate
along the horizontal axis) below to have the other value computed.
Normal distribution
Other distributions: Student’s t • Chi-square • F
p-value: 0.549
z-value: 0.6
p = 0.549
mean: 0
std. dev: 1
two tails
right tail
left tail
mean to z
2-sided mean to z
-3
-2
-1
0
1
2
3
z = 0.6
Link to calculation: http://www.statdistributions.com/normal?z=0.6
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Hypothesis Testing for Population Mean
One-Sample t test
Case II: One-Sample t test
One-Sample t test
1
Hypotheses:
H0 : µ ≤ µ0 vs. H1 : µ > µ0 (Right-tailed test)
H0 : µ ≥ µ0 vs. H1 : µ < µ0 (Left-tailed test) H0 : µ = µ0 vs. H1 : µ 6= µ0 (Two-tailed test) 2 3 Test statistic: (Normality is assumed) x − µ0 √ tST AT = s/ n follows a t-distribution with df = n − 1. P-value: Test type Right-tailed Left-tailed Two-tailed Department of Mathematics and Statistics p-value P (T > tST AT )
P (T < tST AT ) 2P (T > |tST AT |)
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Fall 2022
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Hypothesis Testing for Population Mean
One-Sample t test
Finding p−value: Online t Calculator
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Fall 2022
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Hypothesis Testing for Population Mean
One-Sample t test
Normality Assumption
Assumptions:
The t-test and the t-interval require that:
1
2
3
The population standard deviation, σ, is unknown.
The data is a random sample.
The sampled population is normally distributed.
If the sample size is small (i.e. n < 30) then we need to check whether the population is normally distributed, or at least not extremely non-normal. However, the t procedures are fairly robust to departures from normality; i.e., even if the population distribution(s) is/are nonnormal, we can still use the t procedures and get approximate results. We can check the normality assumption graphically using histogram, boxplot or normal probability plot. Department of Mathematics and Statistics MTH213: Business Statistics Fall 2022 33 / 58 Hypothesis Testing for Population Mean One-Sample t test Example (1) A traffic department office claims that the mean wait time is less than 14 minutes. A random sample of 10 people has a mean wait time of 13 minutes with a standard deviation of 3.5 minutes. At 5% significance level, test the office’s claim. Assume that the wait time is normally distributed. 1 Hypotheses: H0 : µ ≥ 14 versus H1 : µ < 14 2 Test statistic: We have n = 10, df = 9, x = 13 and s = 3.5. Also, we have a small sample with unknown σ but we know that the distribution of the call times is normal so we can use the t test. x − µ0 13 − 14 √ = √ = −0.904 tST AT = s/ n 3.5/ 10 Department of Mathematics and Statistics MTH213: Business Statistics Fall 2022 34 / 58 Hypothesis Testing for Population Mean One-Sample t test Example (1) 3 P-value: p − value = P (T < −0.904) = 0.195 4 Conclusion: Since p-value> 0.05, we do not reject H0 . There is
not enough evidence to support the office’s claim that the mean
wait time is less than 14 minutes.
Enter either the p-value (represented by the blue area on the graph) or the test statistic (the coordinate
along the horizontal axis) below to have the other value computed.
Student’s t-distribution
Other distributions: Normal • Chi-square • F
p-value: 0.195
t-value: -.904
d.f.: 9
p = 0.195
two tails
right tail
left tail
0 to t
-t to t
-4
-3
-2
0
1
2
3
4
t = -0.904
Link to calculation: http://www.statdistributions.com/t?t=-0.904&df=9&tail=3
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Hypothesis Testing for Population Mean
One-Sample t test
Example (2)
Abu Dhabi fire department aims to respond to fire calls in 4 minutes
or less, on average. A sample of 18 recent fire calls showed a mean
response time of 4 minutes 30 seconds with a standard deviation of 1
minute. Assume that response times are normally distributed. Would
these information provide sufficient evidence to show that the goal is
being met at 5% significance level?
1
Hypotheses:
H0 : µ ≤ 4 versus H1 : µ > 4
2
Test statistic: We have n = 18, df = 17, x = 4.5 and s = 1.
Also, we have a small sample with unknown σ but we know that
the distribution of the response time is normal so we can use the
t test.
x − µ0
4.5 − 4
√ = √ = 2.12
t=
s/ n
1/ 18
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Fall 2022
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Hypothesis Testing for Population Mean
One-Sample t test
Example (2)
3
P-value:
p − value = P (T > 2.12) = 0.025
4
Conclusion: Since p − value < 0.05, we reject H0 . There is no sufficient evidence to show that the goal is being met. Enter either the p-value (represented by the blue area on the graph) or the test statistic (the coordinate along the horizontal axis) below to have the other value computed. Student's t-distribution Other distributions: Normal • Chi-square • F p-value: 0.025 t-value: 2.12 d.f.: 17 two tails right tail left tail 0 to t -t to t p = 0.025 -4 -3 -2 -1 0 1 2 3 4 t = 2.12 Link to calculation: http://www.statdistributions.com/t?t=2.12&df=17&tail=2 Department of Mathematics and Statistics MTH213: Business Statistics Fall 2022 37 / 58 Hypothesis Testing for Population Mean One-Sample t test Confidence intervals and hypothesis tests Both confidence intervals and hypothesis tests are concerned with determining plausible values for a quantity such as µ. In a hypothesis test for a population mean µ, we specify a particular value of µ (say µ0 ) and determine whether that value is plausible. In contrast, a confidence interval for a population mean µ can be thought of as the collection of all values for µ that meet a certain criterion of plausibility, specified by the confidence level (1 − α) × 100%. The values contained within a (1 − α) × 100% confidence interval for a population mean µ are precisely those values for which the p-value of a two-tailed hypothesis test will be greater than α. Hence, the hypothesis H0 : µ = µ0 is not rejected at significance level α, whenever µ0 is contained in the (1−α)×100% confidence interval for µ. Equivalently, if µ0 is not contained in the (1 − α) × 100% confidence interval for µ, then H0 is rejected at level α. Department of Mathematics and Statistics MTH213: Business Statistics Fall 2022 38 / 58 Hypothesis Testing for Population Mean One-Sample t test Example In order to ensure efficient usage of a server, it is necessary to estimate the mean number of concurrent users. According to records, the average number of concurrent users at 81 randomly selected times is 38 with a standard deviation of 9.3). A 90% confidence interval for the average number of concurrent users is (36.3, 39.7). When testing whether the mean number of concurrent users is 35 at 10% significance level, then the hypotheses will be H0 : µ = 35 versus H1 : µ 6= 35 Based on the 90% confidence interval, we would reject H0 since µ0 = 35 does not fall inside the interval (p-value=0.004). Similarly, we should not reject H0 when testing whether the mean number of concurrent users is 39 since µ0 = 39 falls inside the interval. Department of Mathematics and Statistics MTH213: Business Statistics Fall 2022 39 / 58 Hypothesis Testing for Population Mean One-Sample t test Exercises 1 2 Knowledge of the amount of time a patient occupies a hospital bed -called length of stay (LOS)- is important for allocating resources. At one hospital, the mean LOS was determined to be 5 days. A hospital administrator believes the mean LOS may now be less than 5 days due to a newly adopted managed health care system. To check this, the LOSs (in days) for 100 randomly selected hospital patients were recorded and yielded a mean of 4.5 days and a standard deviation of 3.7 days. At 5% significance level, test the administrator’s belief. The quality assurance department at ADIB sampled 50 customers and found that the mean and the standard deviation of the waiting time were 5.75 and 1.2 minutes, respectively. Assume that the waiting time for customers follows a normal distribution. At the 1% significance level, can we conclude that the mean waiting time at ADIB is less than 6 minutes? Department of Mathematics and Statistics MTH213: Business Statistics Fall 2022 40 / 58 Hypothesis Testing for Population Mean One-Sample t test Exercises 3 4 A random sample of 400 response times taken during the peak period of the interactive inquiry system at Futile Finance yields a mean of 21 time units with a standard deviation of 12 time units. The service level agreement between the MIS Department and the users of the inquiry system is that the mean response time for the peak period should not exceed 20 time units. A test is to be made each day at the 1% level. Does the system pass today? A fast-food franchiser is considering building a restaurant at a certain location. Based on financial analyses, a site is acceptable only if the number of pedestrians passing the location averages at least 100 per hour. The number of pedestrians observed for each of 40 hours was recorded and yielded a mean of 94 and a standard deviation of 16. Can we conclude at the 1% significance level that the site is acceptable? Department of Mathematics and Statistics MTH213: Business Statistics Fall 2022 41 / 58 Hypothesis Testing for Population Mean One-Sample t test Exercises 5 6 In order to determine the number of workers required to meet demand, the productivity of newly hired trainees is studied. It is believed that trainees can process and distribute more than 450 packages per hour within one week of hiring. A sample of 50 trainees observed for one hour yielded a mean number of packages processed of 461 and a standard deviation of 39. At 5% significance level can we conclude that this belief is correct? Ultra Health Clinic claims that the average waiting time for a patient is 20 minutes or less. A random sample of 15 patients shows a mean wait time of 24.77 minutes with a standard deviation of 7.26 minutes. Assume that the waiting time is normally distributed. a b At the 5% level of significance, does the sample support or contradict the clinic’s claim? Is this conclusion sensitive to the choice of α? Department of Mathematics and Statistics MTH213: Business Statistics Fall 2022 42 / 58 Hypothesis Testing for Population Mean One-Sample t test Exercises 7 Spam e-mail has become a serious and costly nuisance. An office manager believes that the average amount of time spent by office workers reading and deleting spam exceeds 25 minutes per day. To test this belief, he takes a random sample of 18 workers and measures the amount of time each spends reading and deleting spam. The results yielded a mean time of 30 minutes and a standard deviation of 12 minutes. Assume that the population of times is normally distributed. a b c d Can the manager infer at the 1% significance level that he is correct? What type of error could be made in this case? Describe it in the context of the problem. Would it be possible to answer part (a) if the times were not normally distributed? Justify your answer. Repeat (a) and (b) using a 5% significance level. Department of Mathematics and Statistics MTH213: Business Statistics Fall 2022 43 / 58 Hypothesis Testing for Population Mean One-Sample t test Exercises 8 The quality-control manager at a light emitting diode (LED) factory needs to determine whether the mean life of a large shipment of LEDs is equal to 50,000 hours. A random sample of 64 LEDs indicates a sample mean life of 49,875 hours and a standard deviation of 1,500 hours. a b c d At the 0.05 level of significance, is there evidence that the mean life is different from 50,000 hours? Construct a 95% confidence interval estimate of the population mean life of the LEDs. Compare the results of (a) and (b). What conclusions do you reach? Repeat (a) through (c), assuming a sample standard deviation of 500 hours. Department of Mathematics and Statistics MTH213: Business Statistics Fall 2022 44 / 58 Hypothesis Testing for Population Mean One-Sample t test Exercises 9 The manager of a paint supply store wants to determine whether the mean amount of paint contained in 5-liter cans purchased from a known manufacturer is actually 5 liters. He selected a random sample of 50 cans and measured the exact amount of paint in each can. He found that the mean amount of paint per can is 4.975 liters and the standard deviation is 0.1 liter. a b c d e f g Is there evidence that the mean amount is different from 5 liters? Use α = 0.01. What will result if the true mean amount of paint is 5 liters? What will result if the true mean amount of paint is 4.95 liters? Suppose that the standard deviation is 0.06 liter. Repeat (a) and compare the results to those obtained in (a). What will result if the true mean amount of paint is 5 liters? What will result if the true mean amount of paint is 5.05 liters? What will result if the true mean amount of paint is 4.95 liters? Department of Mathematics and Statistics MTH213: Business Statistics Fall 2022 45 / 58 Hypothesis Testing for Population Mean One-Sample t test Exercises 10 The mean length of a small counterbalance bar is 43 millimeters. The production supervisor is concerned that the adjustments of the machine producing the bars have changed. He asks the engineer to select a random sample of 12 bars and measure each. The mean and the standard deviation of the length of the selected bars were 41.5 and 1.78, respectively. a b c d What is the required assumption, if any, of the test to be used in this case? Is it reasonable to conclude that there has been a change in the mean length of the bars? Use the 1% significance level. Another engineer suggested to use 5% significance level, what would be the conclusion? The production supervisor got puzzled by the results obtained by the two engineers. Explain to the supervisor why the conclusions were different and give him/her some insights and recommendations regarding the machine status. Department of Mathematics and Statistics MTH213: Business Statistics Fall 2022 46 / 58 Hypothesis Testing for Population Mean One-Sample t test Exercises 11 The “just-in-time” policy of inventory control (developed by the Japanese) is growing in popularity. Suppose that an automobile parts supplier claims to deliver parts to any manufacturer in an average time of less than 1 hour. In an effort to test the claim, a manufacturer recorded the times (in minutes) of 24 deliveries from this supplier as shown below. 55 45 a b c d 62 55 53 64 72 62 58 59 57 51 51 61 64 69 58 63 52 60 58 61 48 49 Construct a histogram or a boxplot (Box and Whiskers plot) for the delivery times and comment on the distribution. Can we conclude that the supplier’s assertion is correct at 5% significance level? What will result if the true mean delivery time is actually 55 minutes? What will result if the true mean delivery time is actually 62 minutes? Department of Mathematics and Statistics MTH213: Business Statistics Fall 2022 47 / 58 Hypothesis Testing for Population Proportion Table of Contents 1 Introduction to Hypothesis Testing 2 Hypothesis Testing for Population Mean One-Sample z test One-Sample t test 3 Hypothesis Testing for Population Proportion Department of Mathematics and Statistics MTH213: Business Statistics Fall 2022 48 / 58 Hypothesis Testing for Population Proportion One-Proportion z-Test Now, we consider testing whether a population proportion p equals some specified value p0 . A random sample of n subjects/objects is to be selected, and the number of successes in the sample, x, is determined. The logical statistic used to estimate and test the population proportion is the sample proportion defined as: x p̂ = n As mentioned in Chapter 6, for sufficiently large sample size, the sampling distribution ofrp̂ is approximately normal with mean p p(1 − p) and standard deviation . n To use the one-proportion z test, we must have np0 ≥ 5 and n(1 − p0 ) ≥ 5 Department of Mathematics and Statistics MTH213: Business Statistics Fall 2022 49 / 58 Hypothesis Testing for Population Proportion One-Proportion z-Test One-Proportion z-Test 1 Hypotheses: H0 : p ≤ p0 vs. H1 : p > p0 (Right-tailed test)
H0 : p ≥ p0 vs. H1 : p < p0 (Left-tailed test) H0 : p = p0 vs. H1 : p 6= p0 (Two-tailed test) 2 Test statistic: (np0 ≥ 5 and n(1 − p0 ) ≥ 5) zST AT = r 3 p̂ − p0 p0 (1 − p0 ) n , where p̂ = x n p-value: Similar to the one-sample z-test for the mean. Department of Mathematics and Statistics MTH213: Business Statistics Fall 2022 50 / 58 Hypothesis Testing for Population Proportion Example (1) In a sample of 150 households in a certain city, 99 had high-speed internet access. Can you conclude that at least 70% of the households in this city have high-speed internet access? Use α = 0.1. 1 Hypotheses: H0 : p ≥ 0.7 versus H1 : p < 0.7 2 99 Test statistic: We have n = 150 and x = 99 so p̂ = = 150 0.66. Notice that np0 = (150)(0.7) = 105 > 5 and n(1−p0 ) = (150)(0.3) = 45 > 5
so we can safely use the z-test.
p̂ − p0
0.66 − 0.7
zST AT = r
=r
= −1.07
p0 (1 − p0 )
(0.7)(0.3)
n
150
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Hypothesis Testing for Population Proportion
Example (1)
3
P-value:
p − value = P (Z < −1.07) = 0.142 4 Conclusion: Since p-value=0.142 > 0.1, we do not reject H0 and conclude
that at least 70% of the households in this city have high-speed
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internet access.
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2
Test statistic: We have n = 200 and p̂ = 0.825. Notice that
np0 = 150 > 5 and n(1 − p0 ) = 50 > 5
so we can safely use the z-test.
p̂ − p0
0.825 − 0.75
zST AT = r
=r
= 2.45
p0 (1 − p0 )
(0.75)(0.25)
n
200
Department of Mathematics and Statistics
MTH213: Business Statistics
Fall 2022
53 / 58
Hypothesis Testing for Population Proportion
Example (2)
3
P-value:
p − value = P (Z > 2.45) = 0.007
4
Conclusion:
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I Left-tailed (H1 : ρ < 0; i.e. Negative correlation) In this course, we are interested only in the two-tailed hypothesis test. The test is based on t distribution with n − 2 degrees of freedom. Chapter 12 MTH213: Business Statistics Fall 2022 20 / 41 Correlation Inference about the Correlation Coefficient Test of Significance of ρ 1 Hypotheses H0 : ρ = 0 H1 : ρ 6= 0 2 (No correlation) (Significant correlation) Test Statistic: r n−2 1 − r2 which has the t-distribution with n − 2 degrees of freedom. P-value: Computed using the Online t Calculator for two-tailed test with df = n − 2. t=r 3 Chapter 12 MTH213: Business Statistics Fall 2022 21 / 41 Correlation Inference about the Correlation Coefficient Note that the strength of evidence when testing a correlation depends on the sample size (n) as well as the magnitude of r. The same value of r might be quite insignificant for a small sample but strong evidence of an association for a much larger sample. The results of this test (t and p-value) are generated in the output produced by the Online Calculator. Chapter 12 MTH213: Business Statistics Fall 2022 22 / 41 Correlation Example: Copyline International Test whether there is a linear relationship between sales calls and copiers sold. Use 5% level. 1 Hypotheses: H0 : ρ = 0 versus H1 : ρ 6= 0 2 Test Statistic: r r n−2 15 − 2 t=r = 0.865 = 6.21 1 − r2 1 − 0.8652 3 P-value: p − value ≈ 0 4 Conclusion: We reject H0 , which means that there is a significant linear relationship between sales calls and copiers sold. Chapter 12 MTH213: Business Statistics Fall 2022 23 / 41 Correlation Exercise A large industrial plant has seven divisions that do the same type of work. A safety inspector visits each division of 20 workers quarterly. The number of work-hours devoted to safety training (x) and the number of work-hours lost due to industry-related accidents (y) are recorded for each separate division as shown. Division x y a b c 1 10 80 2 20 65 3 30 68 4 45 55 5 50 35 6 65 10 7 80 12 Find the correlation coefficient between the hours lost to accidents and hours in safety training. Interpret the correlation coefficient. Test whether there is a significant correlation between the hours lost to accidents and hours in safety training at 5% significance level. Chapter 12 MTH213: Business Statistics Fall 2022 24 / 41 Simple Linear Regression Table of Contents 1 Introduction 2 Correlation 3 Simple Linear Regression Chapter 12 MTH213: Business Statistics Fall 2022 25 / 41 Simple Linear Regression Regression One is interested in the association between a single variable (called response or dependent variable) and a single variable or a set of variables (called predictors, independent variables, regressors, etc.), described by an unknown mathematical function (called a regression function). The term “regression” was introduced by Francis Galton in 1855. The objectives of regression analysis: 1 2 3 Describe the association between the response and predictors. Predict the observations of the response given predictors. Assess the effects of the predictors on the response. Some Regression Applications: I Identifying factors that predict obesity among school children in UAE. I Investigating the impact of social networking usage on grades among college students. Chapter 12 MTH213: Business Statistics Fall 2022 26 / 41 Simple Linear Regression Simple Linear Regression Model The simple linear regression model is given by Y = β0 + β1 X + ε Y is the dependent variable and X is the independent variable. β0 and β1 are the intercept and the slope of the line. ε is a random error assumed to be normally distributed with a mean of 0 and constant variance σ 2 . I Without ε, any observed pair (x, y) would correspond to a point falling exactly on the line Y = β0 + β1 X, called the true (or population) regression line. The inclusion of the random error term allows (x, y) to fall either above the true regression line (when ε > 0) or below the line (when ε < 0). The least squares method is used to estimate the parameters of the model, which is based on minimizing the sum of squared vertical deviations from the observed data to the regression line. Chapter 12 MTH213: Business Statistics Fall 2022 27 / 41 Simple Linear Regression Estimating Regression Coefficients Estimation of Regression Coefficients The least squares estimated regression line is ŷ = b0 + b1 x where ŷ, read as “y hat”, is the predicted value of y and the coefficients b0 and b1 are the estimated regression coefficients given by P (x − x)(y − y) sy P =r b1 = 2 (x − x) sx b0 = y − b1 x where sy and sx are the standard deviations of y and x, respectively. Chapter 12 MTH213: Business Statistics Fall 2022 28 / 41 Simple Linear Regression Regression using Casio fx-991-ES Plus STEPS 1 CALCULATOR MODE 2 3: STAT 3 2: A+BX 4 5 Enter the X and Y values. Then Press AC Press Shift and then 1 6 5: REG (For Regression) 7 Chapter 12 1: A then = 2: B then = (b0) (b1) MTH213: Business Statistics Fall 2022 29 / 41 Simple Linear Regression Regression: Online Calculators Linear Correlation and Regression Online Calculator: http://vassarstats.net/corr stats.html 1 Enter the number of observations. 2 Enter the data (X and Y values) and click Calculate. 3 The regression coefficients are shown as Y Intercept (b0 ) and Slope (b1 ). b1 b0 ෝ = 𝟏𝟔. 𝟕𝟎𝟓𝟔 − 𝟏. 𝟓𝟔𝟓 𝒙 𝒚 Chapter 12 MTH213: Business Statistics Fall 2022 30 / 41 Simple Linear Regression Interpretation of Regression Coefficients The intercept, b0 is the predicted value of y when the value of x is zero. Note that when the scope of the model does not cover x = 0, b0 does not have any particular meaning as a separate term in the regression model. The slope, b1 is the predicted change in y as a result of a one-unit increase in x. (Slopes are always expressed in y-units per x-units.) Chapter 12 MTH213: Business Statistics Fall 2022 31 / 41 Simple Linear Regression Interpretation: Examples 1 The advertising expenditures (in millions of Dhs) predict revenue (in millions of Dhs). [ = 268 + 7.37 Ads Sales I b0 = 268: The firm would average 268 million Dhs of sales with zero advertising. I b1 = 7.37: Each extra 1 million Dhs of advertising will generate 7.37 million Dhs of sales on average. 2 Number of persons predicts dinner cost (in Dhs). d = 60 + 80 Persons Cost I b0 = 60: The intercept is not meaningful because Persons = 0 would not be observable. I b1 = 80: Each additional diner increases the mean dinner cost by 80 Dhs. Chapter 12 MTH213: Business Statistics Fall 2022 32 / 41 Simple Linear Regression Interpretation: Examples 3 Programming experience (in years) affects the completion time of a programming task (in hours). \ = 10 − 0.25 ProgExp CompTime I b0 = 10: Programmers with no experience are expected to complete the task in around 10 hours. I b1 = −0.25: For each additional year of experience, the completion time is expected to decrease by 0.25 hour (15 minutes). 4 The area of the apartment (in Square meter) predicts its monthly rent (in Dhs). d = 600 + 42 Area Rent I b0 = 600: The intercept is not meaningful because no apartment can have area=0 m2 . I b1 = 42: Each extra square meter adds 42 Dhs to monthly apartment rent. Chapter 12 MTH213: Business Statistics Fall 2022 33 / 41 Simple Linear Regression Example: Copyline International Chapter 12 MTH213: Business Statistics Fall 2022 34 / 41 Simple Linear Regression Example: Copyline International The estimated regression equation is: ŷ = 19.98 + 0.261x Interpretation of regression coefficients: I The intercept is b0 = 19.98 ≈ 20. This means that if x = 0 (i.e., no sales calls are made), around 20 copiers will be sold. However, note that x = 0 is outside the range of values included in the sample and, therefore, should not be used to estimate the number of copiers sold. The sales calls ranged from 36 to 180, so estimates should be limited to that range. Hence, in this case, the intercept is probably meaningless. I The slope coefficient b1 is 0.261. This means that for each additional sales call, the sales representative can expect to increase the number of copiers sold by about 0.261 (To put it another way, 20 additional sales calls in a month will result in about five more copiers being sold, found by 0.261(20) = 5.22). Chapter 12 MTH213: Business Statistics Fall 2022 35 / 41 Simple Linear Regression Using the Regression Equation If the regression relationship between y and x is significant, then ŷ = b0 + b1 x0 is the prediction of y when x = x0 . Example: How many copiers can Ali, a salesperson at Copyline International, expect to sell if he makes 100 calls? ŷ = 19.98 + 0.261x = 19.98 + 0.261(100) ≈ 46 Thus, Ali can expect to sell around 46 copiers. Attempting to use a regression equation to predict values outside of this range is often inappropriate, and may yield incredible answers. This practice is known as extrapolation. I Predicting the number of copiers sold when less than 36 or more than 180 calls are made is considered extrapolation. Chapter 12 MTH213: Business Statistics Fall 2022 36 / 41 Simple Linear Regression Exercises 1 The relationship between the sales per month (in thousands of Dhs) vs. number of employees for all the outlets of a large computer chain is linear, with only moderate scatter and no outliers. The correlation between Sales and Employees is 0.85, and the equation of the least squares model is: [ = 38 + 488 Employees Sales a b c d What does the y-intercept mean in this context? Is it meaningful? What does the slope mean in this context? The outlet in Dubai has 10 more employees than the outlet in Abu Dhabi. How much more sales do you expect it to have? Predict the monthly sales of an outlet store with 40 employees. Chapter 12 MTH213: Business Statistics Fall 2022 37 / 41 Simple Linear Regression Exercises 2 The owner of CarSwitch dealership, wants to study the relationship between the age of a Toyota Corolla (in years) and its selling price (in 1000 Dhs). Listed below is a random sample of 12 used Corollas sold at the dealership during the last year. Age Price a b c d e f g 9 32.4 7 24 11 14.4 12 16 8 20 7 40 8 30.4 11 32 10 32 12 24 6 34.4 6 32 Find and interpret the correlation coefficient. At 1% significance level, is there a significant linear relationship between the selling price and the age? Find the regression equation. What does the y-intercept mean in this context? Is it meaningful? What does the slope mean in this context? Predict the selling price of a five-year-old Corolla. Predict the selling price of a 20-year-old Corolla. Comment on how accurate you feel this prediction is. Chapter 12 MTH213: Business Statistics Fall 2022 38 / 41 Simple Linear Regression Exercises 3 A substance used in experimental research is shipped by airfreight to users in cartons of 1,000 ampules. The data, involving 10 shipments, were collected on the number of times the carton was transferred from one aircraft to another over the shipment route and the number of ampules found to be broken upon arrival. Transfers Broken Ampules 1 16 0 9 2 17 0 12 3 22 1 13 0 8 1 15 2 19 0 11 Find and interpret the correlation coefficient. Find the regression equation. c What does the y-intercept mean in this context? Is it meaningful? d What does the slope mean in this context? e At 5% significance level, test whether there is a linear association between number of times a carton is transferred and number of broken ampules. f The next shipment will entail two transfers. Predict the number of broken ampules for this shipment. gChapter 12 MTH213: Business Statistics Fall 2022 39 / 41 Predict the number of broken ampules for a shipment involving a b Simple Linear Regression Exercises 4 Disk drives have been getting larger. Their capacity is now often given in terabytes (TB). A survey of prices for external disk drives found the following data: Capacity (in TB) Price (in Dhs) a b c d e f g 0.15 105 0.2 897 0.25 120 0.32 150 1 225 2 330 3 420 4 975 Prepare and interpret a scatterplot of Price against Capacity. Find and interpret the correlation coefficient. It turned out that the 200 GB hard drive was a special “hardened” drive designed to resist physical shocks and work under water. Because it is completely different from the other drives, it was removed from the data. Repeat parts (a) and (b). Find the regression equation. Interpret the regression coefficients. What would you predict for the price of a 3.0 TB disk? You have found a 3.0 TB drive for 525 Dhs. Based on (f), is this a good buy? How much would you save? Chapter 12 MTH213: Business Statistics Fall 2022 40 / 41 Simple Linear Regression Exercises 5 Gym Outfitters sells and services exercise equipment to gyms and recreational centers. The company’s management would like to determine if there is a relationship between the number of minutes required to complete a routine service call (y) and the number of machines serviced (x). A random sample of 12 records revealed the following information. x y a b c d e f 11 115 8 60 9 80 10 90 7 55 6 65 8 70 4 33 10 95 5 50 5 40 12 110 Find and interpret the correlation coefficient. Find the regression equation. What does the y-intercept mean in this context? Is it meaningful? What does the slope mean in this context? Test whether there is a linear association between number of machines serviced and the service time. Use α = 0.05. If a gymnasium had six machines, how many minutes should Gym Outfitters expect a routine service call to require? Chapter 12 MTH213: Business Statistics Fall 2022 41 / 41 Zayed University MTH213 Hypothesis Testing - Basics - Solution Reject or Not Reject? In Exercises 1 - 6, determine whether you would reject or not reject the null hypothesis using the p-value approach. Question 1. H0 : µ ≥ 25 H1 : µ < 25 α = 0.05, x = 24, s = 9 and n = 100. Solution: • Check the assumptions: n ≥ 30 • Calculate the test statistic: zStat = x−µ 24 − 25 √ = √ = −1.11 s/ n 9/ 100 • Calculate the p value: p value = 0.133 • Compare α to the p value to decide whether to reject the null hypothesis: p value > 0.05, so we do not reject H0 .
Question 2.
H0 : µ ≥ 2.04
H1 : µ < 2.04 α = 0.1, x = 2.01, s = 0.1 and n = 400. Solution: • Check the assumptions: n ≥ 30 • Calculate the test statistic: zStat = x−µ 2.01 − 2.04 √ = √ = −6 s/ n 0.1/ 400 • Calculate the p value: p value ≈ 0 • Compare α to the p value to decide whether to reject the null hypothesis: p value < 0.1, so we reject H0 . Page 1 (of 3) Zayed University MTH213 Question 3. H1 : µ 6= 12.55 H0 : µ = 12.55 α = 0.1, x = 11.5, s = 2.3, n = 20 and the population is normal. Solution: • Check the assumptions: Population is normally distributed • Calculate the test statistic: tStat = x−µ 11.5 − 12.55 √ √ = = −2.04 s/ n 2.3/ 20 • Calculate the p value: p value = 0.055 • Compare α to the p value to decide whether to reject the null hypothesis: p value < 0.1, so we reject H0 . Question 4. H0 : µ ≤ 25 H1 : µ > 25
α = 0.05, x = 26.2, s = 9, n = 100 and the population is normal.
Solution:
• Check the assumptions:
Population is normally distributed
• Calculate the test statistic:
tStat =
x−µ
26.2 − 25
√
√ =
= 1.333
σ/ n
9/ 100
• Calculate the p value:
p value = 0.1
• Compare α to the p value to decide whether to reject the null hypothesis:
p value > 0.05, we do not reject H0 .
Page 2 (of 3)
Zayed University
MTH213
Question 5.
H1 : µ 6= 2.04
H0 : µ = 2.04
α = 0.05, x = 2.045, s = 0.1 and n = 400.
Solution:
• Check the assumptions:
n ≥ 30
• Calculate the test statistic:
zStat =
x−µ
2.045 − 2.04
√
√ =
=1
s/ n
0.1/ 400
• Calculate the p value:
p value = 0.317
• Compare α to the p value to decide whether to reject the null hypothesis:
p value > 0.05, so we do not reject H0 .
Question 6.
H0 : µ ≥ 12.55
H1 : µ < 12.55 α = 0.1, x = 11, s = 3, n = 25 and the population is normal. Solution: • Check the assumptions: Population is normally distributed • Calculate the test statistic: tStat = x−µ 11 − 12.55 √ = √ = −2.58 s/ n 3/ 25 • Calculate the p value: p value = 0.008 • Compare α to the p value to decide whether to reject the null hypothesis: p value < 0.1, so we reject H0 . Page 3 (of 3) Zayed University MTH213 Hypothesis Testing - Problem Set - Solution Question 1. A survey showed that the average cost of a hotel room in Dubai is 355 AED, but a researcher claims it is less. The researcher selects a sample of 30 hotel rooms and finds that the average cost is 350 AED and a standard deviation of 13.6 AED. At α = 0.05, is there enough evidence to reject the claim? Solution: • Set up the null and alternative hypotheses: H0 : µ ≥ 355 H1 : µ < 355 • Check the assumptions: n ≥ 30 • Calculate the test statistic: zStat = x−µ 350 − 355 √ = −2.01 √ = s/ n 13.6/ 30 • Calculate the p value: p value = 0.022 • Compare α to the p value to decide whether to reject the null hypothesis: p value < 0.05, so we reject H0 at significance level α = 0.05 • Interpret the statistical decision in terms of the stated problem: There is sufficient evidence to support the claim of the researcher. So, we believe that the average cost of a hotel room in Dubai is less than 355 AED. Page 1 (of 5) Zayed University MTH213 Question 2. The manager of a large U.A.E. factory states that the average hourly wage of the employees is $9.78 per hour. A sample of 38 employees has a mean hourly wage of $9.60 and a standard deviation of $1.42. At α = 0.05, is there enough evidence to support the manager’s claim? Solution: • Set up the null and alternative hypotheses: H0 : µ = 9.78 H1 : µ 6= 9.78 • Check the assumptions: n ≥ 30 • Calculate the test statistic: zStat = x−µ 9.60 − 9.78 √ = √ = −0.78 s/ n 1.42/ 38 • Calculate the p value: p value = 0.435 • Compare α to the p value to decide whether to reject the null hypothesis: p value > 0.05, so we do not reject H0 at significance level α = 0.05
• Interpret the statistical decision in terms of the stated problem:
There isn’t enough evidence to reject the manager’s claim that the average hourly wage of the
employees in his factory is $9.78 per hour.
Page 2 (of 5)
Zayed University
MTH213
Question 3. A maker of frozen meals claims that the average caloric content of its meals is 800, but a
researcher thinks it is higher. A researcher tested 20 meals and found that the average number of calories
was 812 with a standard deviation of 25. Assuming normality, is there enough evidence to reject the
claim at α = 0.05?
Solution:
• Set up the null and alternative hypotheses:
H0 : µ = 800
H1 : µ > 800
• Check the assumptions:
Population is normally distributed
• Calculate the test statistic:
tStat =
x−µ
812 − 800
√ =
√
= 2.146
s/ n
25/ 20
• Calculate the p value:
p value = 0.023
• Compare α to the p value to decide whether to reject the null hypothesis:
p value < 0.05, so we reject H0 at significance level α = 0.05 • Interpret the statistical decision in terms of the stated problem: There is enough evidence to reject the claim of the maker of frozen meals. Thus, the average caloric content of its meals is more than 800. Page 3 (of 5) Zayed University MTH213 Question 4. A report in Emarat Al Youm claimed that the average age of commercial jets in the United Arab Emirates is at most 14 years. An executive of a large airline company selects a sample of 26 planes and finds the average age of the planes is 15.6 years with a standard deviation of 2.7 years. Assume that the age of commercial jets in the United Arab Emirates is normally distributed. At α = 0.05, can it be concluded that the average age of the planes in his company is greater than the national average? Solution: • Set up the null and alternative hypotheses: H0 : µ ≤ 14 H1 : µ > 14
• Check the assumptions:
Population is normally distributed and σ is unknown
• Calculate the test statistic:
tStat =
x−µ
15.6 − 14
√ =
√
= 3.022
s/ n
2.7/ 26
• Calculate the p value:
p value = 0.003
• Compare α to the p value to decide whether to reject the null hypothesis:
p value < 0.05, so we reject H0 at significance level α = 0.05 • Interpret the statistical decision in terms of the stated problem: There is enough evidence to conclude that the average age of the planes in this company is greater than the national average. Page 4 (of 5) Zayed University MTH213 Question 5. A manufacturer states that the average lifetime of its light bulbs is at least 3 years, (36 months). Fifty bulbs are selected, and the average lifetime is found to be 34 months with a standard deviation is 8 months. Should the manufacturer’s statement be rejected at α = 0.05? Solution: • Set up the null and alternative hypotheses: H0 : µ ≥ 36 H1 : µ < 36 • Check the assumptions: n ≥ 30 • Calculate the test statistic: zStat = x−µ 34 − 36 √ = √ = −1.77 s/ n 8/ 50 • Calculate the p value: p value = 0.038 • Compare α to the p value to decide whether to reject the null hypothesis: p value > 0.05, so we reject H0 at significance level α = 0.05
• Interpret the statistical decision in terms of the stated problem:
There is enough evidence to reject the claim of the manufacturer’s statement. Hence, the average
lifetime of its light bulbs is less than 3 years.
Page 5 (of 5)
Chapter 12 Simple Linear Regression
Homework with Solutions
1) Calculate the correlation coefficient, r, for the data below.
x
y
-1 1
0
2
A) 0.819
8
19
5
11
3
8
2 4
4 9
B) 0.990
6
13
7
16
1)
0
2
C) 0.881
D) 0.792
2) Calculate the correlation coefficient, r, for the data below.
x
y
1
3
1 -4
A) -0.885
10
-16
7
-11
5
4
6
-9
-7 -6
B) -0.778
8
-14
9
-15
2)
2
-2
C) -0.671
D) -0.995
3) Calculate the correlation coefficient, r, for the data below.
x
y
5
7
7 -10
A) -0.132
14
4
11
-7
9
8
10
-6 -3
1
B) -0.549
12
-9
13
2
3)
6
3
C) -0.581
D) -0.104
4) A manager wishes to determine the relationship between the number of miles (in hundreds of
miles) the manager’s sales representatives travel per month and the amount of sales (in thousands
of dollars) per month. Calculate the correlation coefficient, r.
Miles traveled, x
Sales, y
A) 0.791
5
30
6
32
13 10
77 61
B) 0.632
11
64
18
60
6
4
14
47 54 119
C) 0.717
5) Which of the following values could not represent a correlation coefficient?
A) 1.032
B) -1
C) 0
6) Find the equation of the regression line for the given data.
x
y
-5 -3
-10 -8
4
9
1
1
-1 -2 0
-2 -6 -1
2
3
3
6
-4
-8
^
B) y = -0.552x + 2.097
^
^
D) y = 2.097x – 0.552
A) y = 2.097x + 0.552
^
C) y = 0.522x – 2.097
1
4)
D) 0.561
D) 0.927
5)
6)
7) Find the equation of the regression line for the given data.
x
y
-5 -3 4 1
11 6 -6 -1
-1 -2
3 4
0
1
7)
2 3 -4
-4 -5 8
^
B) y = -0.758x – 1.885
^
^
D) y = -1.885x + 0.758
A) y = 0.758x + 1.885
^
C) y = 1.885x – 0.758
8) The data below are the final exam scores of 10 randomly selected statistics students and the
number of hours they studied for the exam. Find the equation of the regression line for the given
data.
Hours, x
Scores, y
3
65
5
80
2
60
8
88
2
66
4
78
4
85
5
90
^^
6
90
8)
3
71
^
A) y = -56.113x – 5.044
B) y = 5.044x + 56.113
^
^
C) y = -5.044x + 56.113
D) y = 56.113x – 5.044
^
9)
^
10)
11) Given the equation of a regression line is y = 3.5x – 5.4, what is the best predicted value for y
given x = -1.2? Assume that the variables x and y have a significant correlation.
A) -12.3
B) -6.9
C) 12.3
D) -9.6
^
11)
12) Use the regression equation to predict the value of y for x = -2.8. Assume that the variables x and y
have a significant correlation.
12)
9) Given the equation of a regression line is y = 3x – 1, what is the best predicted value for y given
x = 4? Assume that the variables x and y have a significant correlation.
A) 13
B) 6
C) 1
D) 11
10) Given the equation of a regression line is y = -2.5x+ 5.3, what is the best predicted value for y
given x = 4.9? Assume that the variables x and y have a significant correlation.
A) -17.55
B) 6.95
C) -6.95
D) 17.55
x
y
-5 -3
-10 -8
4
9
1
1
-1 -2 0
-2 -6 -1
A) 0.551
2
3
3
6
-4
-8
B) -5.320
C) -6.424
D) 3.643
13) The data below are the final exam scores of 10 randomly selected statistics students and the
number of hours they studied for the exam. What is the best predicted value for y given x = 3?
Assume that the variables x and y have a significant correlation.
Hours, x
Scores, y
A) 72
3
65
5
80
2
60
8
88
2
66
4
78
4
85
5
90
B) 70
6
90
C) 69
2
3
71
D) 71
13)
Answer Key:
(1) B
(2) D
(3) D
(4) B
(5) A
(6) D
(7) D
(8) B
(9) D
(10) C
(11) D
(12) C
(13) D
3