Math 227 Chapter 10 and 11 Review: Confidence Intervals and Hypothesis Testing

Math 227 Chapter 10 and 11 Review: Confidence Intervals and Hypothesis Testing

1. What is a type I error in hypothesis testing? What is a type II error?

2. A hypothesis test is conducted using two different significance levels. Computer technology gives us the following outputs.

z = 1.58 and z = 1.87

a) Which of the two results will give a smaller p-value for a right tail test? Draw a normal curve with the appropriate area shaded to support your answer.

b) Which of the two cases would be considered more unusual (less likely to happen by chance)?

3. When conducting a hypothesis test computer technology give a p-value of 0.063.

1. At a significance level of , is the result unusual?
2. At a significance level of , is the result unusual?
3. At a significance level of , is the result unusual?

4. A 2003 study of dreaming found that out of a random sample of 113 people, 92 reported dreaming in color. However, the rate reported in the 1940’s was 0.29.

a. = ? p = ?

b. Identify each of the symbols in part (a) as a statistic or a parameter.

c. Check to see whether the conditions for using a one proportion z-test are met assuming the rate in the 1940’s is still true. Do not conduct a hypothesis test.

5. The mean age of all 2550 students at a small college is 22.8 years with a standard deviation of 3.2 years and the distribution is right skewed. A random sample of 4 students’ ages is obtained at the mean is 23.2 years with a standard deviation of 2.4 years.

1. σ = ? = ? s = ? µ = ? n = ?
2. Identify each of the symbols in part (a) as a statistic or a parameter.
3. Are the conditions satisfied for this example?

6. A teacher giving a true/false test wants to make sure her students do better than they would if they were simply guessing, so she forms a hypothesis to test this. Her null hypothesis is that a student will get 50% of the questions on the exam correct. The alternative hypothesis is that the student is not guessing and should get more than 50% in the long run.

Set up:

Ho: p = 0.50

Ha: p > 0.50

A student gets 30 out of 50 questions, or 60% correct. Computer technology gives a p-value of 0.079.

a) Explain the meaning of the p-value in context.

b) Would you reject the null hypothesis at a significance level of 5%?

c) Would you reject the null hypothesis at a significance level of 10%?

7. Judging on the basis of experience, a politician claims that 48% in Pennsylvania have voted for an independent candidate in past elections. Suppose you surveyed 20 randomly selected people in Pennsylvania, and 11 of them reported having voted for an independent candidate. The null hypothesis is that the overall proportion of voters in Pennsylvania that have voted for an independent candidate is 48%.

a) Calculate the test statistic.

b) Based only on this test statistic, would you reject the null hypothesis that the overall proportion of voters is 48% based on a significance level of 5%?

8. A 2003 study of dreaming found that out of a random sample of 113 people, 92 reported dreaming in color. However, the rate reported in the 1940’s was 0.29. A researcher wanted to see if the proportion of dreaming in color has increased since the 1940’s. Let’s assume the necessary conditions are met. Perform a hypothesis test to test this claim. Use a 1% significance level.

A. Does this require a one-sample, two-sample, or paired t-test?

B. Conduct a hypothesis test

a. State the hypothesis.

b. Several outputs from computer technology are shown below. Circle the correct output for this hypothesis.

Write the p value:

Result A:

Result B:

Result C:

c. Do you reject or not reject the null hypothesis?

d. Write your conclusion.

e. Computer technology has provided the following confidence interval. Interpret the confidence interval. Then explain how the confidence interval supports the conclusion from the hypothesis test.

9. The population mean height for a 3 year old boy in the U.S. is 38 inches. Suppose a random sample of 15 non-U.S. boys are measured and the mean height for this group is 37.2 inches with a standard deviation of 3 inches. The non-U.S. boys were independently sampled. Assume that heights are normally distributed in the population.

A. Have the conditions been met to perform a hypothesis test? Explain.

B. Does this require a one-sample, two-sample, or paired t-test?

C. Perform a hypothesis test to determine if the mean height for non-U.S. boys is less than the U.S. population mean height. Use a significance level of 0.05.

a. State the hypothesis

b. Computer technology gives the following output.

c. Do you reject or not reject the null hypothesis?

d. Write your conclusion.

e. The 95% confidence interval is shown below. Interpret the confidence interval. Then explain how the confidence interval supports the conclusion from the hypothesis test.

10. A recent article (New England Journal of Medicine, 2010) reported the results of an experiment on reducing the likelihood that men develop prostate cancer. The investigators randomly assigned 3305 men to receive the drug Dutasteride and assigned 3424 men to receive a placebo. Of those receiving the drug, 659 developed prostate cancer. Of those men who received the placebo, 858 developed prostate cancer.

A. Does this require a one-sample, two-sample, or paired t-test?

B. Find the percentage of men that developed prostate cancer in each group. Let p1 be the proportion who received the drug and p2 the proportion who received the placebo.

C. Perform a hypothesis test to determine if the drug made a difference in lowering prostate cancer using a 5% significance level. Assume the conditions have been met.

a. State the hypothesis.

b. Several outputs from computer technology are shown below. Circle the correct output for this hypothesis.

Write the p value:

Result A

Result B

Result C

c. Do you reject or not reject the null hypothesis?

d. Write your conclusion.

e. Computer technology has provided the following confidence interval. Interpret the confidence interval. Then explain how the confidence interval supports the conclusion from the hypothesis test.

11. A researcher wants to know if a daughter’s height is about the same as the mother’s height. Twenty randomly selected mothers and daughters were chosen to participate. The results are shown below.

A. Does this require a one-sample, two-sample, or paired t-test?

B. Let the mother’s height be the first sample and the daughter’s height be the second sample. Computer technology determines that the mean height for the mother is 63.15 inches with a standard deviation of 2.22 inches. The daughter’s mean height is 63.55 inches with a standard deviation of 2.24 inches.

C. Perform a hypothesis test to determine a daughter’s height is about the same as her mother’s height using a 5% significance level. Assume the conditions have been met.

a. State the hypothesis.

b. The output from computer technology is given below.

Write the p value:

c. Do you reject or not reject the null hypothesis?

d. Write your conclusion.

e. Computer technology has provided the following confidence interval. Explain how the confidence interval supports the conclusion from the hypothesis test.

12. A veterinarian wants to compare the weight of two breeds of dogs. He believes that German Shepherds are larger than Doberman Pinchers. A random sample of 20 male German Shepherds found that their average weight was 112 pounds with a standard deviation of 28 pounds. A random sample of 14 male Dobermans found that their average weight was 107 pounds with a standard deviation of 20 pounds. Assume the weights of are normally distributed and that the conditions have been satisfied.

I. Does this require a one-sample, two-sample, or paired t-test?

II. Perform a hypothesis test to investigate the veterinarian’s claim that German Shepherds are larger than Doberman Pinchers. Use a 0.05 significance level. Assume the conditions have been met.

a. State the hypothesis (let weight of the German Shepherds be the first sample and the weight of the Dobermans be the second sample)

b. Computer technology gives the following output

Do you reject or not reject the null hypothesis?

c. Write your conclusion.

d. Computer technology has provided the following confidence interval. Interpret the confidence interval. Then explain how the confidence interval supports the conclusion from the hypothesis test.