# 10 Problems : Hypothesis Testing, Z-Tests, Standard Deviation, P-Values and Type I and Type II Errors

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1. A z-test is to be performed for a population mean. Express the decision criterion for the hypothesis in terms of _X , That is, determine for what values of _X

the null hypothesis would be rejected. A hypothesis test is to be performed to determine whether the mean waiting time during peak hours for customers in a supermarket has increased from the previous mean waiting time of 8.7 minutes. Preliminary data analyses indicate that it is reasonable to apply a z-test. The hypotheses are

H0: = 8.7 minutes

Ha: > 8.7 minutes

The population standard deviation is 3.5. The sample size is 48. The significance level is 0.05. Express the decision criterion for the hypothesis test in terms of _X

a. Reject H0 if _X > 9.69 minutes

b. Reject H0 if _X < 9.69 minutes

c. Reject HO if _X < 9.53 minutes

d. Reject H0 if _X > 9.53 minutes

2. A hypothesis test is to be performed for a population mean. Which of the following does the probability of a Type II error not depend on?

a. The sample size

b. The null-hypothesis mean

c. The significance level

d. The true mean

e. The sample mean

4. A one-sample z-test for a population mean is performed. Suppose that the P-value for the test is 0.04. For what significance levels (values of A) can the null hypothesis be rejected?

a. For A = 0.04

b. For A = 0.05, 0.10

c. For all values of A smaller than 0.04

d. For all values of A less than or equal to 0.04

e. For all values of A greater than or equal to 0.04

5. In 1990, the average math SAT score for students at one school was 475. Five years later, a teacher wants to perform a hypothesis test to determine whether the average math SAT score of students at the school has changed. He picks a random sample of 40 students and obtains their the mean math SAT score, which is 469. Do the data provide sufficient evidence to conclude that the mean math SAT score for all students at the school has changed from the previous mean of 475? Perform the appropriate hypothesis test using a significance level of 0.10. Assume that 0 = 73

a.Reject H0. At the 10% significance level, the data provide sufficient evidence to conclude that the mean math SAT score for students at the school has changed from the previous mean of 475.

b. Reject H0. At the 10% significance level, the data do not provide sufficient evidence to conclude that the mean math SAT score for students at the school has changed from the previous mean of 475.

c. Do not reject H0. At the 10% significance level, the data provide sufficient evidence to conclude that the mean math SAT score for students at the school has changed from the previous mean of 475.

d. Do not reject H0. At the 10% significance level, the data do not provide sufficient evidence to conclude that the mean math SAT score for students at the school has changed from the previous mean of 475.

6. Suppose that you wish to perform a hypothesis test for a population mean using the P-value method. Suppose that the population standard deviation is unknown. The correct procedure to use is the t-test. If you mistakenly use the standard normal table to obtain the P-value, will the value that you obtain be larger or smaller than the correct value?

a.The P-value obtained using the standard normal table will be larger than the correct P-value which will make it less likely that the null hypothesis will be rejected.

b. The P-value obtained using the standard normal table will be larger than the correct P-value which will make it more likely that the null hypothesis will be rejected.

c. The P-value obtained using the standard normal table will be smaller than the correct P-value which will make it less likely that the null hypothesis will be rejected.

d. The P-value obtained using the standard normal table will be smaller than the correct P-value which will make it more likely that the null hypothesis will be rejected

7. A hypothesis test is to be performed to determine whether the mean hematocrit (percentage by volume of the blood occupied by red blood cells) for women differs from the mean hematocrit for men which is known to be 47%. Preliminary data analyses indicate that it is reasonable to apply a z-test. The hypotheses are HO = 47%, HA not equal to 47% . Assume that the population standard deviation is 2.8%. The sample size is 10. The significance level is 0.01. Find the probability of a Type II error if in fact the mean hematocrit for women is 43%. Use 4 decimal places.

9. In 2003, the mean price of a summer suit at a certain clothing store was $98.60. A consumer information firm wants to conduct a hypothesis test to determine if the mean price in 2004 has changed from the 2003 mean. Assume that the sample size is 93 and the standard deviation is 0.63. At significance level 0.01, determine the probability of a Type I error.

10. In 2003, the mean price of a summer suit at a certain clothing store was $98.60. A consumer information firm wants to conduct a hypothesis test to determine if the mean price in 2004 has changed from the 2003 mean. Assume that the sample size is 93, the standard deviation is 0.63, and the significance level is 0.01. If the true mean 98.90, determine the probability of a Type II error. Use 4 decimal places.

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##### Solution Summary

Ten Problems involving Hypothesis Testing, Z-Tests, Standard Deviation, P-Values and Type I and Type II Errors are solved.

###### Education

- BSc , Wuhan Univ. China
- MA, Shandong Univ.

###### Recent Feedback

- "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
- "excellent work"
- "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
- "Thank you"
- "Thank you very much for your valuable time and assistance!"

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