Answers to the odd-numbered exercises appear in Appendix B.

Review Your Knowledge

6.01 A hypothesis is a proposed ____ for observed ____.

6.02 The procedure by which the observation of a ____ is used to evaluate a hypothesis about a ____ is called ____.

6.03 If what is observed in a sample is close to what is expected if the hypothesis is true, there is little reason to question the ____.

6.04 A hypothesis is a statement about a ____ not a ____.

6.05 The null hypothesis is abbreviated as ____ and the alternative hypothesis as ____.

6.06 The null and alternative hypotheses must be ____ and ____.

6.07 The null hypothesis is a ____ prediction and a ____ statement.

6.08 The null hypothesis says that the explanatory variable does/does not have an impact on the outcome variable.

6.09 The hypothesis a researcher believes is really true is the ____ hypothesis.

6.10 One ____ prove that a negative statement is true.

6.11 It takes just one example to ____ a negative statement.

6.12 When the null hypothesis is rejected, the researcher is forced to accept the ____.

6.13 Because the null and alternative hypotheses are mutually exclusive, if one is not true, then the other is ____.

6.14 If one fails to disprove the null hypothesis, one can’t say it has been ____.

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6.15 One shouldn’t expect the sample mean to be exactly the same as the population mean because of ____.

6.16 The mnemonic to remember the six steps of hypothesis testing is ____.

6.17 The six steps of hypothesis testing, in order, are ____.

6.18 In the first step of hypothesis testing, one picks a ____.

6.19 If the ____ of a hypothesis test aren’t met, one can’t be sure what the results mean.

6.20 If a robust assumption is violated, one ____ proceed with the test.

6.21 A two-tailed test has ____ hypotheses.

6.22 A two-tailed test allows one to test for a positive or a negative effect of the ____ on the ____.

6.23 It is easier to reject the null hypothesis with a ____-tailed test than a ____-tailed test.

6.24 Once the data are collected, it is OK/not OK to change from a two-tailed test to a one-tailed test.

6.25 The decision rule involves finding the ____ of the test statistic.

6.26 When the value of the test statistic meets or exceeds the critical value of the test statistic, one ____ the null hypothesis.

6.27 Explaining, in plain language, what the results of a statistical test mean is called ____.

6.28 A single-sample z test is used to compare a ____ mean to a population ____.

6.29 In order to use a single-sample z test, one must know the ____ standard deviation.

6.30 The random sample assumption for a single-sample z test says that the ____ is a random sample from the ____.

6.31 Independence of observations within a group means that the cases don’t ____ each other.

6.32 The normality assumption states that the ____ is normally distributed in the ____.

6.33 The null hypothesis for a single-sample z test says that the ____ is a specific value.

6.34 The alternative hypothesis for a single-sample z test says that the population mean is not what the ____ indicated it was.

6.35 The common zone of the sampling distribution of the mean is centered around a z score of ____.

6.36 Sample means will commonly fall in the ____ of the sampling distribution of the mean.

6.37 It is rare that a sample mean will fall in the ____ of the sampling distribution of the mean.

6.38 If the observed mean falls in the common zone, then what was expected to happen if the ____ is true did happen.

6.39 If the sample mean falls in the rare zone, then this is a ____ event if the null hypothesis is true.

6.40 Statisticians say that something that happens more than ____% of the time is common and less than or equal to ____% of the time is rare.

6.41 The z scores that are the critical values for a two-tailed, single-sample z test with alpha set at .05 are ____ and ____.

6.42 If ____ ≤ ____, reject the null hypothesis.

6.43 If ____ ≥ ____, reject the null hypothesis.

6.44 The alpha level is the probability that an outcome that is ____ to occur if the null hypothesis is true does occur.

6.45 ____ is the abbreviation for alpha.

6.46 If alpha equals ____, then a rare event is something that happens at most only 5% of the time.

6.47 The numerator in calculating a single-sample z test is the difference between the ____ and the ____.

6.48 The denominator in calculating a single-sample z test is ____.

6.49 The first question to be addressed in an interpretation is whether one ____ the null hypothesis.

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6.50 If one rejects the null hypothesis, one can decide the direction of the difference by comparing the ____ to the ____.

6.51 If the result of a single-sample z test is statistically significant, that means the sample mean is ____ from the population mean.

6.52 APA format indicates what statistical test was done, how many ____ there were, what the value of the ____ was, what ____ was selected, and whether the null hypothesis was ____.

6.53 ____, in APA format, means the null hypothesis was rejected.

6.54 ____, in APA format, means the null hypothesis was not rejected.

6.55 If one fails to reject the null hypothesis for a single-sample z test, one does/does not need to be concerned about the direction of the difference between the sample mean and the population mean.

6.56 If one fails to reject the null hypothesis, one says there is ____ evidence to conclude that the independent variable affects the dependent variable.

6.57 It is a correct conclusion in hypothesis testing if one rejects the null hypothesis and the null hypothesis should ____.

6.58 It is an incorrect conclusion in hypothesis testing if one ____ the null hypothesis and it should have been rejected.

6.59 With hypothesis testing, one can / can’t be sure that the conclusion about the null hypothesis is correct.

6.60 Type ____ error occurs when one rejects the null hypothesis but shouldn’t have.

6.61 If α = .05, then the probability of making a Type I error is ____.

6.62 If the cost of making a Type I error is high, one might set alpha at ____.

6.63 Compared to α = .01, α = .10 has a ____ rare zone, making it ____ to reject the null hypothesis.

6.64 When alpha is set low, say, at .01, the chance of being able to reject the null hypothesis is larger/smaller.

6.65 In hypothesis testing, one wants to keep the probability of Type I error ____ and still have a reasonable chance to ____ the null hypothesis.

6.66 The term for the error that occurs when the null hypothesis should be rejected but isn’t is ____.

6.67 The probability of Type I error is usually set at ____.

6.68 The probability of Type II error is commonly set at ____.

6.69 If one fails to reject the null hypothesis, one needs to worry about ____ error but not ____ error.

6.70 As the size of the effect increases, the probability of Type II error ____.

6.71 Power is the probability of rejecting the null hypothesis when ____.

6.72 1.00 = ____ + power.

6.73 If one rejects the null hypothesis, one needs to worry about ____ error but not ____ error.

6.74 If one makes a Type I error or a Type II error, then the conclusion about the ____ is wrong.

Apply Your Knowledge

Select the right statistical test.

6.75 A scientific supply company has developed a new breed of lab rat, which it claims weighs the same as the classic white rat. The population mean (and standard deviation) for the classic white rat is 485 grams (50 grams). A researcher obtained a sample of 76 of the new breed of rats, weighed them, and found M = 515 grams. What test should he do to see if the company’s claim is true?

6.76 The mean vacancy rate for apartment rentals in the United States is 10%, with a standard deviation of 4.6. An urban studies major obtained a sample of 15 rustbelt cities and found that the mean vacancy rate was 13.3%. What statistical test should she use to see if the mean vacancy rate for these cities differs from the U.S. average?

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Check the assumptions.

6.77 A researcher has a first-grade readiness test that is administered to kindergarten students and scored at the interval level. The population mean is 60, with a standard deviation of 10. He has administered it, individually, to a random sample of 58 kindergarten students in a city, M = 66, and wants to use a single-sample z test, two-tailed with α = .05, in order to see whether first-grade readiness in this city differs from the national level. Check the assumptions and decide if it is OK for him to proceed with the single-sample z test.

6.78 A Veterans Administration researcher has developed a test that is meant to predict combat soldiers’ vulnerability to developing post-traumatic stress disorder (PTSD). She has developed the test so that μ = 45 and σ = 15. She is curious if victims of violent crime are as much at risk for PTSD as combat veterans. So, she obtains a sample of 122 recent victims of violent crime and administers the test to each one of them. She’s planning to use a single-sample z test, two-tailed and with alpha set at .05, to compare the sample mean (42.8) to the population mean. Check the assumptions and decide if it is OK to proceed with the single-sample z test.

List the hypotheses.

6.79 List the null and alternative hypotheses for Exercise 6.77.

6.80 List the null and alternative hypotheses for Exercise 6.78.

State the decision rule.

6.81 State the decision rule for Exercise 6.77.

6.82 State the decision rule for Exercise 6.78.

Calculate σ_M.

6.83 Calculate σ_M using the data from Exercise 6.77.

6.84 Calculate σ_M using the data from Exercise 6.78.

Calculate z.

6.85 Calculate z for M = 100, μ = 120, and σ_M = 17.5.

6.86 Calculate z for M = 97, μ = 85, and σ_M = 4.5.

Calculate σ_M. Use it to calculate z.

6.87 Use the following information to calculate (a) σ_M and (b) z. M = 12, μ = 10, σ = 5, and N = 28.

6.88 Use the following information to calculate (a) σ_M and (b) z. M = 15, μ = 21, σ = 1.5, and N = 63.

Determine if the null hypothesis was rejected and use APA format.

6.89 Given N = 23 and z = 2.37, (a) decide if the null hypothesis was rejected, and (b) report the results in APA format. Use α = .05, two-tailed.

6.90 Given N = 87 and z = –1.96, (a) decide if the null hypothesis was rejected, and (b) report the results in APA format. Use α = .05, two-tailed.

Determine if the difference was statistically significant and the direction of the difference.

6.91 M = 16, μ = 11, and the results were reported in APA format as p < .05. (a) Was there a statistically significant difference between the sample mean and the population mean? (b) What was the direction of the difference?

6.92 M = 20, μ = 24, and the results were reported in APA format as p > .05. (a) Was there a statistically significant difference between the sample mean and the population mean? (b) What was the direction of the difference?

Given the results, interpret them. Be sure to tell what was done in the study, give some facts, and indicate what the results mean.

6.93 A researcher obtained a sample of 123 American women who said they wanted to lose weight. She weighed each of them and found M = 178 pounds. The mean weight for women in the United States is 164 pounds and the population standard deviation is known. The researcher used a single-sample z test and found z (N = 123) = 3.68, p < .05. Interpret her results to see if women who want to lose weight differ from the general population in terms of weight.

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6.94 In the population of children in a school district, the mean number of days tardy per year is 2.8. A sociologist obtained a sample of children from single-parent families and found the mean number of days tardy was 3.2. He used a single-sample z test to analyze the data, finding z (N = 28) = 1.68, p > .05. Interpret the results to see if coming from a single-parent family is related to tardiness.

Type I vs. Type II error

6.95 A journalist was comparing the horsepower of a sample of contemporary American cars to the population value of horsepower for American cars of the 1970s. He concluded that there was a statistically significant difference, such that contemporary cars had more horsepower. Unfortunately, his conclusion was in error. (a) What type of error did the journalist make? (b) What conclusion should he have reached?

6.96 A Verizon researcher compared the number of text messages sent by a sample of teenage boys to the population mean for all Verizon users. She found no evidence to conclude that there was a difference. Unfortunately, her conclusion was in error. (a) What type of error did she make? (b) What conclusion should she have reached?

Given β or power, calculate the other.

6.97 If β = .75, power = ____.

6.98 If power = .90, β = ____.

Doing a complete statistical test

6.99 A dietitian wondered if being on a diet was related to sodium intake. She knew that the mean daily sodium intake in the United States was 3,400 mg, with a standard deviation of 270. She obtained a random sample of 172 dieting Americans and found, for sodium consumption, M = 2,900 mg. Complete all six steps of hypothesis testing.

6.100 At a large state university, the population data show that the average number of times that students meet with their academic advisors is 4.2 with σ = 1.8. The dean of student activities at this university wondered what the relation was between being involved in student clubs and organizations (such as the band, student government, or the ski club) and being involved academically in the university. She assumed that meeting with an academic advisor indicated students took their academics seriously. From the students who had been involved in student clubs and organizations for all four years of college, she obtained a random sample of 76 students, interviewed them individually, and found M = 4.8 for the number of times they met with their academic advisors. Complete all six steps of hypothesis testing.

Expand Your Knowledge

For Exercises 6.101 to 6.104, decide which option has a higher likelihood of being able to reject the null hypothesis.

6.101 (a) β = .60; (b) power = .60

6.102 (a) β = .30; (b) power = .50

6.103 (a) μ₀ = 10, μ₁ = 15; (b) μ₀ = 10, μ₁ = 20

6.104 (a) μ₀ = –17, μ₁ = –23; (b) μ₀ = –17; μ₁ = –18

For Exercises 6.105 to 6.108, label each conclusion as “correct” or “incorrect.” If incorrect, label it as a “Type I error” or a “Type II error.” Use the following scenario. A biochemist has developed a test to determine if food is contaminated or not. The test operates with the null hypothesis that a food is not contaminated.

6.105 Noncontaminated food is tested and it is called “noncontaminated.”

6.106 Noncontaminated food is tested and it is called “contaminated.”

6.107 Contaminated food is tested and it is called “noncontaminated.”

6.108 Contaminated food is tested and it is called “contaminated.”

6.109 A researcher is testing the null hypothesis that μ = 35. She is doing a two-tailed test, has set alpha at .05, and has 121 cases in her sample. What is beta?

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6.110 A researcher is testing the null hypothesis that μ = 120, with σ_M = 20. In reality, the population mean is 140. (a) Draw a figure like Figure 6.12. Designate in the figure the areas representing Type II error and power. (b) Does one need to worry about Type I error?

For 6.111 and 6.112, determine how many tails there are.

6.111 A health science researcher believes that athletes have a lower resting heart rate than the general population. He knows, for the general population, that μ = 78 and σ = 9. He is planning to do a study in which he obtains a random sample of athletes, measures their resting heart rates, and uses a single-sample z test to see if the mean heart rate for the sample of athletes is lower than the general population mean. (a) Should he do a one-tailed test or a two-tailed test? (b) Write the null and alternative hypotheses.

6.112 A lightbulb manufacturer believes that her compact fluorescent bulbs last longer than incandescent bulbs. She knows that the mean number of hours is 2,350 hours for a 60-watt incandescent bulb, with a standard deviation of 130 hours (these are population values). She gets a random sample of her compact fluorescent bulbs and measures the mean number of hours they last. She is going to use a single-sample z test to compare the sample mean to the population mean. (a) Should she do a one-tailed test or a two-tailed test? (b) Write the null and alternative hypotheses.

6.113 The population of random, two-digit numbers ranges from 00 to 99, has μ = 49.50 and σ = 28.87. A statistician takes a random sample from this population and wants to see if his sample is representative. He figures that if it is representative, the sample mean will be close to 49.50. He plans to use a single-sample z test to test the sample against the population. Can this test be used?