Posted: July 15th, 2022

Statistics – Confidence Intervals and Statistic Tests

Part 1

1. Explain the difference between a 95% confidence interval and a 99% confidence interval in terms of probability.
   a) To construct a 95% confidence interval for a population mean µ, what is the correct critical value z*?
   b) To construct a 99% confidence interval for a population mean µ, what is the correct critical value z*?
2. Explain what the margin of error is and how to calculate it.
3. A survey of a group of students at a certain college, we call College ABC, asked: “About how many hours do you study in a week?” The mean response of the 400 students is 15.8 hours. Suppose that the study time distribution of the population is known to be normal with a standard deviation of 8.5 hours. Use the survey results to construct a 95% confidence interval for the mean study time at the College ABC.
4. Explain the difference between a null hypothesis and an alternative hypothesis.
5. Suppose that you are testing a null hypothesis H₀: µ = 10 against the alternate H₁: µ ≠ 10. A simple random sample of 35 observations from a normal population are used for a test. What values of the z statistic are statistically significant at the α = 0.05 level?

Part 2

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1. A study of a group of 40 male league bowlers chosen at random had an average score was 176. It is known that the standard deviation of the population is 9.
   a) Construct the 95% confidence interval for the mean score of all league bowlers.
   b) Construct the 95% confidence interval for the mean score of all league bowlers assuming that a sample of size 100 is used instead of 40, and the same mean and standard deviation occur.
   c) Give the margin of error for each interval.
   d) Explain why one confidence interval is larger than the other.

2. There are 100 apartments in a certain a San Francisco apartment building. The owner of the building wants to estimate the mean number of people living in an apartment. The owner draws a random sample of 40 apartments in the building. The number of people living in each apartment is as follows:

1      2      1      2      3      1      3      4      3      1

2      2      1      2      2      2      1      3      2      3

2      3      1      2      3      3      2      4      5      2

3      2      2      3      1      1      2      2      1      2

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a) Compute the sample mean and sample standard deviation.
b) Use the results from part (a) to construct a 95% confidence interval.

3. A doctor wishes to estimate the birth weights of infants. How large a sample must the doctor select if she desires to be 99% confident that the true population mean is at most 6 ounces away from the mean of the sample? Assume the standard deviation is 8 ounces. Hint: The margin of error should be at most 6.

4. In Problem 4 of Part 1 a class survey of 400 students was given in which students at College ABC claimed to study an average of 15.8 hours per week. Consider these students as a simple random sample from the particular population of College ABC students. We want to investigate the question: Does the survey provide good evidence that students study more than 15 hours per week on average? Assume the population of hours studied is normal with a standard deviation of 4.

Before working out this problem, it will help to look over the webpage, Hypothesis tests for means, http://stattrek.com/hypothesis-test/mean.aspx?Tutorial=AP
a) State the null and alternate hypothesis in terms of the mean study time in hours for the population.
b) Is this a one-tailed test or two-tailed test?
c) Determine the value of the test statistic.
d) Sketch a normal shape curve and identify the test statistic.
e) Indicate the p-value of the test. Use the standard normal table. Shade the area under the normal curve corresponding to the p-value. You can also use the website cited above to do this.
f) State your conclusion to the statistical problem in terms of the null hypothesis, and your conclusion to the practical problem.