Many graduate programs require students to pass a comprehensive exam (or qualifying exam, or something equivalent) before being admitted into candidacy to receive the degree. Suppose the population consists of all comprehensive exams (or equivalent) that are given to students in graduate programs, and of interest is the mean number of questions asked on the comprehensive exams. It is conjectured that the mean number of questions asked on all comprehensive exams is 8, and of interest is to test this conjecture versus the alternative that the mean number of questions asked on all comprehensive exams is different from 8

State the appropriate null and alternative hypotheses that should be tested.

H0: = 8 versus Ha: < 8

H0: = 8 versus Ha: 8

H0: = 8 versus Ha: > 8

H0: = 8 versus Ha: 8

Consider the information provided and the hypotheses specified in question 1.A simple random sample of 81 comprehensive exams was selected and the number of questions asked on each comprehensive exam in the sample was recorded. The mean number of questions asked for this sample of 81 comprehensive exams was 7.3 with a standard deviation of 3.7. The distribution of the data was bimodal and slightly skewed to the left. If appropriate, use this information to test the hypotheses stated in question 1 at the = .10 level of significance.

What are the assumptions and are they met in this situation?

A. We had a simple random sample, and the distribution is skewed.Therefore, the assumptions are not satisfied.

B. The distribution is skewed so the Central Limit Theorem does not apply.

C.We do not have a simple random sample.

D.We had a simple random sample, and the sample size is large enough for the Central Limit Theorem to apply (n = 81 > 15). Therefore, the assumptions are satisfied.

Since the population standard deviation is unknown, the test statistic is......

T = -1.703

Z = -1.703

T = 1.703

Z = 1.703

The p-value is .0925. What is the correct decision at = .10 level of significance.

Since p-value < .10 we reject the null hypothesis

Since p-value >.10 we reject the null hypothesis

Since p-value > .10 we fail to reject the null hypothesis

Since p-value < .10 we fail to reject the null hypothesis

Choose the appropriate conclusion.

There is insufficient evidence that mean number of questions asked on all comprehensive exams is not different from 8.

There is sufficient evidence that the mean number of questions asked on all comprehensive exams is different from 8.

There is insufficient evidence that the mean number of questions asked on all comprehensive exams is different from 8.

There is sufficient evidence that the mean number of questions asked on all comprehensive exams is equal to 8.

Of interest is to estimate the mean age of students at the time they take the comprehensive exam for all students enrolled in graduate programs that require students to take comprehensive exams. For this problem only, assume that the standard deviation of the ages of all such students is 3.3 years. What is the minimum number of graduate students that would need to be selected for the sample to allow the calculation of a 98% confidence interval with margin of error no larger than 1.5?

57

37

27

47

If the confidence level were decreased from 99% to 90%, what impact would this have on the margin of error and width of the confidence interval?

Both the margin of error and the width would increase.

The margin of error would increase and the width would decrease.

The margin of error would decrease and the width would increase.

Neither the margin of error nor the width would be affected.

Both the margin of error and the width would decrease.

Comprehensive Exams

Of interest is to determine the mean age of students at the time they take the comprehensive exam for all students enrolled in graduate programs that require students to take comprehensive exams. A simple random sample of 31 students enrolled in graduate programs was selected, and the age of each student when they took the comprehensive exam was recorded. The mean age at the time of taking the comprehensive exam for this sample of 31 students was 27.5 years with a standard deviation of 2.9 years; there were no outliers in the sample that would lead one to suspect heavy skewness in the distribution. If appropriate, use this information to calculate and interpret a 98% confidence interval for the mean age of students at the time they take the comprehensive exam for all students enrolled in graduate programs that require students to take comprehensive exams. Use this information to answer questions 3-5.

1. What are the two assumptions? Are the assumptions satisfied?

We do not have a large enough sample.

We have a simple random sample, and the sample size is large enough for the Central Limit Theorem to apply (n = 31 > 15), so the assumptions are satisfied

We do not have a simple random sample.

Both assumptions are not met.

4.Is the population standard deviation known or unknown?

Unknown

Known

I

5.Choose the correct interpretation.

We have 98% confidence that the mean age of students at the time they take the comprehensive exam for all students enrolled in graduate programs that require students to take comprehensive exams is between 26.22 and 28.78 years old.

We have 95% confidence that the mean age of students at the time they take the comprehensive exam for all students enrolled in graduate programs that require students to take comprehensive exams is between 36.22 and 38.78 years old.

What is the point estimate of the population mean?

Z

0

t

s

P

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Many graduate programs require students to pass a comprehensive exam (or qualifying exam, or something equivalent) before being admitted into candidacy to receive the degree.

Of interest is to determine the mean age of students at the time they take the comprehensive exam for all students enrolled in graduate programs that require students to take comprehensive exams.

Based on the information provided, what is thepopulationof interest?

mean age of all students enrolled in graduate programs that require students to take comprehensive exams.

all students enrolled in graduate programs that require students to take comprehensive exams.

all students at the university

all students enrolled in graduate programs

Consider the statement in question 2. Based on this, using both the appropriate symbol and in words, what is the parameter of interest?

= all students enrolled in graduate programs that require students to take comprehensive exams.

= the mean age of students at the time they take the comprehensive exam for all students enrolled in graduate programs that require students to take comprehensive exams.

= the mean age of students

= the mean age of students at the time they take the comprehensive exam for all students enrolled in graduate programs that require students to take comprehensive exams.

Students preparing for comprehensive exams usually spend many hours studying. Suppose it is known that the time students spend studying for comprehensive exams has a distribution that is skewed heavily to the right with a mean of 41.3 hours and a standard deviation of 6.8 hours. A simple random sample of 72 students is selected and the amount of time each spent studying for the comprehensive exam is determined.

1A State and check the two assumptions.

The sample size is large enough to apply the Central Limit Theorem (n = 72 > 40).

We have a simple random sample, and the sample size is large enough to apply the Central Limit Theorem (n = 72 > 40).

We do not have a simple random sample.

Population is skewed heavily to the right and therefore the Central Limit theorem does not apply.

1B Describe completely the sampling distribution of , the resulting mean amount of time spent studying for this sample of 72 students.

(41.3, 0.201).

(44.3, .801).

(41.3, 0.801).

(44.3, 0.201).