Statistics Question

Assignment #10
The z-test
10.1
Assume that a treatment does have an effect and that the treatment effect is being
evaluated with a z hypothesis test. If all factors are held constant, how is the
outcome of the hypothesis test influenced by sample size? To answer this
question, do the following two tests and compare the results. For both tests, a
sample is selected from a normal population distribution with a mean of μ = 32 and
a standard deviation of σ = 9. After the treatment is administered to the individuals
in the sample, the sample mean is found to be M = 35. In each case, use a twotailed test with α = .05.
a.
For the first test, assume that the sample consists of n = 16 individuals.
b.
For the second test, assume that the sample consists of n = 64 individuals.
c.
Explain in your own words how the outcome of the hypothesis test is
influenced by the sample size.
(Note: Be sure and show a full diagram of the research design as shown in the Formula
Sheet and chart attached to this assignment packet. Without using Excel or Chegg, show all
steps and calculations you made for each test following the process outlined in the z-test
formula sheet handout. What statistical decision do you make in each case? Finally report
your results professionally in APA format as found in last step of the formula sheet attached
to this assignment packet).
10.2
Researchers at a National Weather Center in the northeastern United States
recorded the number of 90 degree days each year since records first started in
1875. The numbers form a normal shaped distribution with a mean of μ = 10 and a
standard deviation of σ = 2.1. To see if the data showed any evidence of global
warming, they also computed the mean number of 90 degree days for the most
recent n = 4 years and obtained M = 12.9 days. Do the data indicate that the past
four years have had significantly more 90 degree days than would be expected for a
random sample from this populaton? Use a one-tailed test with alpha = .05.
(Note: Be sure and show a full diagram of the research design as shown in the Formula
Sheet and chart attached to this assignment packet. Without using Excel or Chegg, show all
steps and calculations you made for each test following the process outlined in the z-test
formula sheet handout. What statistical decision do you make in each case? Finally report
your results professionally in APA format as found in last step of the formula sheet attached
to this assignment packet).
1
Single Sample z-test
I.
Assumptions for z-test
A.
one sample, randomly selected
B.
know population mean and population standard deviation ahead of time
C.
standard deviation is unchanged by treatment or experiment
D.
sample means are normally distributed; take all the possible sample means that
could happen by chance without treatment (usually normally distributed for
behavioral sciences if sample is greater than or equal to 30)
II.
Diagramming your research (shows the whole logic and process of hypothesis testing)
a.
Draw a picture of your research design (see diagramming your research
handout).
b.
There are always two explanations (i.e. hypotheses) of your research results, the
wording of which depends on whether the research question is directional (onetailed) or non-directional (two-tailed). State them as logical opposites.
c.
For statistical testing, ignore the alternative hypothesis and focus on the null
hypothesis, since the null hypothesis claims that the research results happened
by chance through sampling error.
d.
Assuming that the null is true (i.e. that the research results occurred by chance
through sampling error) allows one to do a probability calculation (i.e. all
statistical tests are nothing more than calculating the probability of getting your
research results by chance through sampling error).
e.
Observe that there are two outcomes which may occur from the results of the
probability calculation (high or low probability of getting your research results by
chance, depending on the alpha (α) level).
f.
Each outcome will lead to a decision about the null hypothesis, whether the null
is probably true (i.e. we then accept the null to be true) or probably not true (i.e.
we then reject the null as false).
III.
Hypotheses (i.e. the two explanations of your research results)
A.
Two-tailed (non-directional research question)
1.
Alternative hypothesis (H1): The independent variable (i.e. the treatment)
does make a difference in performance.
2.
Null hypothesis (H0): The independent variable (i.e. the treatment) does
not make a difference in performance.
B.
One-tailed (directional research question)
1.
Alternative hypothesis (H1): The treatment has an increased (right tail) or
a decreased (left tail) effect on performance.
2.
Null hypothesis (H0): The treatment has an opposite effect than expected
or no change in performance.
2
IV.
Determine critical regions (i.e. the z score boundary between the high or low probability
of getting your research results by chance) using table A-23
A.
Significance level (should be given or decided prior to the research; also called
the confidence, alpha, or p level)
1.
α or p = .05, .01, or .001
B.
One- or two-tailed test (using table A-23)
1.
One-tailed: use full alpha level amount for proportion in tail (Column C)
2.
Two-tailed: use half alpha level amount for proportion in tail (Column C)
C.
With one- or two-tailed p values, find the critical z value
1.
If two-tailed, then critical z value is ± z value
2.
If one-tailed, then determine if critical z value is +z (right tail) or -z (left
tail)
V.
Calculate the z-test statistic
A.
General Single Sample z-test statistical test formula
z=
the observed sample mean – the hypothesized population mean
standard error
B.
Calculations
1.
Compute standard error (average difference between sample &
population means)
Note: (standard error is simply an estimate of the average sampling error which may
occur by chance, since a sample can never give a totally accurate picture of a population)
σM =
2.
VI.
√𝑛
or √
𝜎2
𝑛
Compute z-test statistic (i.e. calculates the probability of getting your
research results by chance through sampling error)
z=
B.
𝜎
𝑀− µ
𝜎𝑀
Compare the calculated z-score to the critical z-score & make a decision about
the null hypothesis
1.
Reject the null (as false) and accept the alternative
or
2.
Accept null (as true)
Reporting the results of a single sample z test
“The treatment had a significant effect on scores (M = 25, SD = 4.22); z = +3.85, p < .05, two-tailed.” 3 non-treatment Step 3: Using Part IV of formula sheet, determine the critical value(s) Single Sample z test Diagram Sample Population µ=? σ=? Step 1: Diagram your study (noting the data you are given) Two-tailed test: both tails One-tailed test: left or right tail high high “and” half α “or” full α 2 explanations of your data (using directional or non-directional language depending on the research question) treatment H1: IV had an effect n=? M=? s=? single sample z test H0: IV had no effect (all results are due to chance primarily through sampling error) Prob. calcul ation 2 outcomes High Prob. (easy) 2 decisions Accept H0 (due to chance) α = .05 Low Prob. (hard) Reject H0 & Accept H1 (due to real effect) IV population before tx sample after tx Step 2: Create a table of calculations for each of the terms which you will need to solve the formulas (if needed) The independent variable is being manipulated with two treatments (hint: with an abstract problem like this, sometimes it helps to put in a real treatment, such as IV Pain Relievers with tx Ibuprofen and tx non-treatment) x x2 Step 4: Calculate the z statistic 1. σm = σ n Step 5: Compare critical z to the calculated z & make a decision about the H0 hypothesis (data for sample) __ ∑= __ ∑= ∑(x)2= 2. z = m−µ σm Step 6: Report results professionally (see last step of formula sheet)