solve 6 questions with step

STAT 1P98
Fall 2022
Written Assignment #2
due: Monday, November 14 at 4:30
This written assignment is to be submitted digitally to Crowdmark. Use .pdf, .jpg, or .png
formats for your assignment submissions.
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1. A sociologist intends to estimate the percentage of adults who believe in astrology. How
many adults does the sociologist need to survey if a confidence level of 99% and a margin
error of 4% are desired
a) if nothing is known about the percentage to be estimated.
b) if a recent survey shows 16% of respondents say they believe in astrology.
2. This question will have you explore the effect of the sample size on the sampling distribution of the mean.
a) Open Excel. Select Data Analysis, then select “Random Number Generation” from
the available options. Use the following values when prompted to enter them:
• Parameters: Between 1 and 10
• Number of Variables: 6
• Output: New Worksheet Ply
• Number of Random Numbers: 50
• Distribution: Uniform
There will be 50 rows and 6 columns filled with numbers in a new Sheet on Excel.
Consider each row as a random sample drawn uniformly from the numbers 1 to 10. In
this case, you have 50 samples of size 6.
Skip the next column, and in the first empty cell of the following empty column, enter
a formula for the mean of the sample in the first row, then drag this formula down to
the last row. In this case, you should enter the formula
=AVERAGE(A1:F1)
into cell H1. Dragging the formula down from cell H1 to H50 finds the remaining
sample means.
b) Create a histogram for the sampling distribution of the mean using the upper class
limits 1, 2, 3, . . . , 10. Edit the chart so that it reads “Created by hYour Namei” as the
title. Rename the “Bin” column so that it includes lower and upper class limits (make
the entries 0 to 1, 1.001 to 2, 2.001 to 3, 3.001 to 4, and so on). Set the gap width to
0%. Submit a screenshot of the histogram with its frequency distribution
table. You can refer to the document “Excel Commands” on Brightspace or refer to
your Written Assignment #1 if you need a reminder on creating histograms.
STAT 1P98
Fall 2022
c) Repeat a) and b) with Number of Variables as 20 instead of 6.
d) How does the sampling distribution of the mean change as the sample size increases?
3. A randomly selected group of 12 adults participated in a sleep study. The mean number of
hours slept was x = 6.8 hours. The number of hours slept by adults seems to be normally
distributed with mean 7 hours and standard deviation 2 hours.
a) Describe the sampling distribution of the sample mean x.
b) What is the probability of drawing a sample of 12 adults with a mean of 6.8 hours of
sleep or less? Give your answer as a percentage rounded to 3 decimal places.
4. In a random sample of 1480 adults, 455 of the respondents have sleepwalked.
a) Let p represent the proportion of all adults who have sleepwalked. Find a point estimate
for p rounded to 3 decimals places.
b) Find 90% and 99% confidence intervals for p, again rounded to 3 decimal places.
c) Using your 99% confidence interval from part b), make a conclusion about the results
of drawing the sample.
d) Describe what happens to the width of the confidence interval as the level of confidence
increases.
5. A group of 6 students are asked to estimate the length of one minute without reference
to any timers. The times they provided (in seconds) are given below:
53
67
63
55
75
58
Assuming the population is approximately normal and using a 0.10 significance level, test
the claim that these provided times are indeed from a population with mean 60 seconds.
a) State the null and alternative hypotheses.
b) Calculate the test statistic (round it to 3 decimal places).
c) Find the critical value (round it to 3 decimal places).
d) Make a conclusion based on your answers in parts b) and c). Do the results indicate
that students are reasonably good at estimating the length of one minute?
6. A researcher claims that the mean SAT score is greater than 515. A sample of 1800
students shows the sample mean SAT score is 519 with standard deviation 111. Test the
researcher’s claim using a 0.05 significance level.
a) State the null and alternative hypotheses.
b) Calculate the test statistic (round it to 3 decimal places).
c) Find the P -value (round it to 3 decimal places).
d) Make a conclusion based on your answers in parts b) and c).
e) Explain what would constitute a Type I error in this context.
f) Explain what would constitute a Type II error in this context.