To test H0: p = 0.75 versus H1: p 6 0.75, a simple random sample of n = 400 individuals

Explain what it means to make a Type II error.

Explain the term power of the test

To test H0: p = 0.30 versus H1: p 6 0.30, a simple random sample of n = 300 individuals is obtained and x = 86 successes are observed. (a) What does it mean to make a Type II error for this test? (b) If the researcher decides to test this hypothesis at the a = 0.05 level of significance, compute the probability of making a Type II error if the true population proportion is 0.28. What is the power of the test? (c) Redo part (b) if the true population proportion is 0.25.

To test H0: p = 0.40 versus H1: p 7 0.40, a simple random sample of n = 200 individuals is obtained and x = 84 successes are observed. (a) What does it mean to make a Type II error for this test? (b) If the researcher decides to test this hypothesis at the a = 0.05 level of significance, compute the probability of making a Type II error if the true population proportion is 0.44. What is the power of the test? (c) Redo part (b) if the true population proportion is 0.47.

To test H0: p = 0.65 versus H1: p 7 0.65, a simple random sample of n = 100 individuals is obtained and x = 69 successes are observed. (a) What does it mean to make a Type II error for this test? (b) If the researcher decides to test this hypothesis at the a = 0.01 level of significance, compute the probability of making a Type II error if the true population proportion is 0.70. What is the power of the test? (c) Redo part (b) if the true population proportion is 0.72

To test H0: p = 0.75 versus H1: p 6 0.75, a simple random sample of n = 400 individuals is obtained and x = 280 successes are observed. (a) What does it mean to make a Type II error for this test? (b) If the researcher decides to test this hypothesis at the a = 0.01 level of significance, compute the probability of making a Type II error if the true population proportion is 0.71. What is the power of the test? (c) Redo part (b) if the true population proportion is 0.68.

To test H0: p = 0.45 versus H1: p 0.45, a simple random sample of n = 500 individuals is obtained and x = 245 successes are observed. (a) What does it mean to make a Type II error for this test? (b) If the researcher decides to test this hypothesis at the a = 0.1 level of significance, compute the probability of making a Type II error if the true population proportion is 0.49. What is the power of the test? (c) Redo part (b) if the true population proportion is 0.47.

To test H0: p = 0.25 versus H1: p 0.25, a simple random sample of n = 350 individuals is obtained and x = 74 successes are observed. (a) What does it mean to make a Type II error for this test? (b) If the researcher decides to test this hypothesis at the a = 0.05 level of significance, compute the probability of making a Type II error if the true population proportion is 0.23. What is the power of the test? (c) Redo part (b) if the true population proportion is 0.28.

Worried about Retirement? In April 2009, the Gallup organization surveyed 676 adults aged 18 and older and found that 352 believed they would not have enough money to live comfortably in retirement. The folks at Gallup want to know if this represents sufficient evidence to conclude a majority (more than 50%) of adults in the United States believe they will not have enough money in retirement. (a) What does it mean to make a Type II error for this test? (b) If the researcher decides to test this hypothesis at the a = 0.05 level of significance, determine the probability of making a Type II error if the true population proportion is 0.53. What is the power of the test? (c) Redo part (b) if the true proportion is 0.55.

Ready for College? There are two major college entrance exams that a majority of colleges accept for admission, the SAT and ACT. ACT looked at historical records and established 21 as the minimum score on the ACT reading portion of the exam for a student to be considered prepared for social science in college. Note: Being prepared means there is a 75% probability of successfully completing a social science course in college. An official with the Illinois State Department of Education wonders whether a majority of the students in her state are prepared to take social science. She obtains a simple random sample of 500 records of students who have taken the ACT and finds that 269 are prepared for social science in college. Does this represent significant evidence that a majority (more than 50%) of the students in the state of Illinois are prepared for social science in college upon graduation? (a) What does it mean to make a Type II error for this test? (b) If the researcher decides to test this hypothesis at the a = 0.05 level of significance, determine the probability of making a Type II error if the true population proportion is 0.52. What is the power of the test? (c) Redo part (b) if the true proportion is 0.55.

Quality of Education In August 2002, 47% of parents who had children in grades K12 were satisfied with the quality of education the students receive. In September 2010, the Gallup organization conducted a poll of 1013 parents who had children in grades K12 and asked if they were satisfied with the quality of education the students receive. Of the 1013 surveyed, 437 indicated they were satisfied. Does this suggest the proportion of parents satisfied with the quality of education has decreased? (a) What does it mean to make a Type II error for this test? (b) If the researcher decides to test this hypothesis at the a = 0.10 level of significance, determine the probability of making a Type II error if the true population proportion is 0.42. What is the power of the test? (c) Redo part (b) if the true proportion is 0.46.

Eating Together In December 2001, 38% of adults with children under the age of 18 reported that their family ate dinner together 7 nights a week. In a recent poll, 403 of 1122 adults with children under the age of 18 reported that their family ate dinner together 7 nights a week. Has the proportion of families with children under the age of 18 who eat dinner together 7 nights a week decreased? (a) What does it mean to make a Type II error for this test? (b) If the researcher decides to test this hypothesis at the a = 0.10 level of significance, determine the probability of making a Type II error if the true population proportion is 0.35. What is the power of the test? (c) Redo part (b) if the true proportion is 0.32.

Taught Enough Math In 1994, 52% of parents of children in high school felt it was a serious problem that high school students were not being taught enough math and science. A recent survey found that 256 of 800 parents of children in high school felt it was a serious problem that high school students were not being taught enough math and science. Do parents feel differently today than they did in 1994? (a) What does it mean to make a Type II error for this test? (b) If the researcher decides to test this hypothesis at the a = 0.05 level of significance, determine the probability of making a Type II error if the true population proportion is 0.50. What is the power of the test? (c) Redo part (b) if the true proportion is 0.48.

Taught Enough Math In 1994, 52% of parents of children in high school felt it was a serious problem that high school students were not being taught enough math and science. A recent survey found that 256 of 800 parents of children in high school felt it was a serious problem that high school students were not being taught enough math and science. Do parents feel differently today than they did in 1994? (a) What does it mean to make a Type II error for this test? (b) If the researcher decides to test this hypothesis at the a = 0.05 level of significance, determine the probability of making a Type II error if the true population proportion is 0.50. What is the power of the test? (c) Redo part (b) if the true proportion is 0.48. (a) What does it mean to make a Type II error for this test? (b) If the researcher decides to test this hypothesis at the a = 0.05 level of significance, determine the probability of making a Type II error if the true population proportion is 0.25. What is the power of the test? (c) Redo part (b) if the true proportion is 0.20.

Effect of A Redo Problem 3(b) with a = 0.01. What effect does lowering the level of significance have on the power of the test? Why does this make sense?

Effect of A Redo Problem 4(b) with a = 0.01. What effect does lowering the level of significance have on the power of the test? Why does this make sense?

Power Curve Draw a power curve for the scenario in Problem 3 by finding the power of the test for p = 0.2, 0.22, 0.24, 0.26, 0.28 [done in part (b)], and 0.29.

. Power Curve Draw a power curve for the scenario in Problem 4 by finding the power of the test for p = 0.41, 0.42, 0.44 [done in part (b)], 0.46, 0.48, and 0.50.

. Power in Tests on Means To test H0: m = 50 versus H1: m 6 50 a simple random sample of size n = 24 is obtained from a population that is known to be normally distributed, and the sample standard deviation is found to be 6. (a) A researcher decides to test the hypothesis at the a = 0.05 level of significance. Determine the sample mean that separates the rejection region from the nonrejection region. [Hint: Follow the same approach as that laid out in the summary on page 518, but use Students t-distribution to find the critical value.] (b) Suppose the true population mean is m = 48.9. Use technology to find the area under the t-distribution to the right of the sample mean found in part (a) assuming m = 48.9. [Hint: This can be accomplished by performing a onesample t-test.] This represents the probability of making a Type II error, b. What is the power of the test?

What happens to the power of the test as the true value of the parameter gets closer to the value of the parameter stated in the null hypothesis? Why is this result reasonable?

What effect does increasing the sample size have on the power of the test, assuming all else remains unchanged?

How do the probability of making a Type II error and effect size play a role in determining an appropriate sample size when performing a hypothesis test?

Explain the difference between accepting and not rejecting a null hypothesis.

According to the American Time Use Survey, the mean number of hours each day Americans age 15 and older spend eating and drinking is 1.22. A researcher wanted to know if Americans age 15 to 19 spent less time eating and drinking. After surveying 50 Americans age 15 to 19 and running the appropriate hypothesis test, she obtained a P-value of 0.0329. State the null and alternative hypothesis this researcher used and interpret the P-value.

Explain the procedure for testing a hypothesis using the Classical Approach. What is the criterion for judging whether to reject the null hypothesis?

Explain the procedure for testing a hypothesis using the P-value Approach. What is the criterion for judging whether to reject the null hypothesis?