SECTION 6.3 Exercises

For Exercises 6.51 and 6.52, see pages 320–321; for 6.53 to 6.55, see pages 323–324; for 6.56 to 6.59, see pages 325–326; for 6.60 to 6.62, see pages 328–329; for 6.63 and 6.64, see page 331; and for 6.65 and 6.66, see page 332.

Question 6.67

6.67 What’s wrong?

Here are several situations in which there is an incorrect application of the ideas presented in this section. Write a short explanation of what is wrong in each situation and why it is wrong.

  1. A manager wants to test the null hypothesis that average weekly demand is not equal to 100 units.
  2. A random sample of size 25 is taken from a population that is assumed to have a standard deviation of 9. The standard deviation of the sample mean is 9/25.
  3. A researcher tests the following null hypothesis: .

6.67

(a) The null hypothesis should contain the = sign, so the hypothesis should be “… the average weekly demand is equal to 100 units.” (b) The square root is missing; the standard deviation should be . (c) The hypotheses are always about the parameter, not the statistic, so it should be : .

Question 6.68

6.68 What’s wrong?

Here are several situations in which there is an incorrect application of the ideas presented in this section. Write a short explanation of what is wrong in each situation and why it is wrong.

  1. A report says that the alternative hypothesis is rejected because the -value is 0.002.
  2. A significance test rejected the null hypothesis that the sample mean is 120.
  3. A report on a study says that the results are statistically signifcant and the -value is 0.87.
  4. The statistic had a value of 0.014, and the null hypothesis was rejected at the 5% level because .

Question 6.69

6.69 What’s wrong?

Here are several situations in which there is an incorrect application of the ideas presented in this section. Write a short explanation of what is wrong in each situation and why it is wrong.

  1. The statistic had a value of for a two-sided test. The null hypothesis is not rejected for because .
  2. A two-sided test is conducted to test , and the observed sample mean is . The null hypothesis is rejected because .
  3. The statistic had a value of 1.2 for a two-sided test. The -value was calculated as .
  4. The observed sample mean is 5 for a sample size . The population standard deviation is 2. For testing the null hypothesis mean of , a statistic of is calculated.

6.69

(a) For , we reject when is either bigger than 1.96 or smaller than . (b) You cannot just compare to ; we need to do a hypothesis test. (c) The -value is always the tail probability, so it should be . (d) is missing from the formula.

Question 6.70

6.70 Interpreting -value.

The reporting of -values is standard practice in statistics. Unfortunately, misinterpretations of -values by producers and readers of statistical reports are common. The previous two exercises dealt with a few incorrect applications of the -value. This exercise explores the -value a bit further.

  1. Suppose that the -value is 0.03. Explain what is wrong with stating, “The probability that the null hypothesis is true is 0.03.”
  2. Suppose that the -value is 0.03. Explain what is wrong with stating, “The probability that the alternative hypothesis is true is 0.97.”
  3. Generally, the -value can be viewed as a measure of discrepancy of the null hypothesis to the data. In terms of a probability language, a -value is a conditional probability. Define the event as “observing a test statistic as extreme or more extreme than actually observed.” Consider two conditional probabilities: versus . Refer to page 197 for the introduction to conditional probability. Explain which of these two conditional probabilities represents a -value.

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Question 6.71

6.71 Hypotheses.

Each of the following situations requires a significance test about a population mean . State the appropriate null hypothesis and alternative hypothesis in each case.

  1. David’s car averages 28 miles per gallon on the highway. He now switches to a new motor oil that is advertised as increasing gas mileage. After driving 2500 highway miles with the new oil, he wants to determine if his gas mileage actually has increased.
  2. The diameter of a spindle in a small motor is supposed to be 4 millimeters. If the spindle is either too small or too large, the motor will not perform properly. The manufacturer measures the diameter in a sample of motors to determine whether the mean diameter has moved away from the target.
  3. Many studies have shown that the content of many herbal supplement pills are not filled with what is advertised but rather have signifcant amounts of filler material such as powdered rice and weeds.17 The percentages of real produce versus filler vary by company. A consumer advocacy group randomly selects bottles and tests each pill for its percentage of ginseng. The group is testing the pills to see if there is evidence that the percent of ginseng is less than 90%.

6.71

(a) . (b) . (c) .

Question 6.72

6.72 Hypotheses.

In each of the following situations, a significance test for a population mean is called for. State the null hypothesis and the alternative hypothesis in each case.

  1. A university gives credit in French language courses to students who pass a placement test. The language department wants to know if students who get credit in this way differ in their understanding of spoken French from students who actually take the French courses. Experience has shown that the mean score of students in the courses on a standard listening test is 26. The language department gives the same listening test to a sample of 35 students who passed the credit examination to see if their performance is different.
  2. Experiments on learning in animals sometimes measure how long it takes a mouse to find its way through a maze. The mean time is 22 seconds for one particular maze. A researcher thinks that a loud noise will cause the mice to complete the maze faster. She measures how long each of 12 mice takes with a noise as stimulus.
  3. The examinations in a large accounting class are scaled after grading so that the mean score is 75. A self-confident teaching assistant thinks that his students have a higher mean score than the class as a whole. His students this semester can be considered a sample from the population of all students he might teach, so he compares their mean score with 75.

Question 6.73

6.73 Hypotheses.

In each of the following situations, state an appropriate null hypothesis and alternative hypothesis . Be sure to identify the parameters that you use to state the hypotheses. (We have not yet learned how to test these hypotheses.)

  1. A sociologist asks a large sample of high school students which academic subject they like best. She suspects that a higher percent of males than of females will name economics as their favorite subject.
  2. An education researcher randomly divides sixth-grade students into two groups for physical education class. He teaches both groups basketball skills, using the same methods of instruction in both classes. He encourages Group A with compliments and other positive behavior but acts cool and neutral toward Group B. He hopes to show that positive teacher attitudes result in a higher mean score on a test of basketball skills than do neutral attitudes.
  3. An economist believes that among employed young adults, there is a positive correlation between income and the percent of disposable income that is saved. To test this, she gathers income and savings data from a sample of employed persons in her city aged 25 to 34.

6.73

(a) = percent of males, = percent of females. . (b) = mean score for Group A, = mean score for group B. . (c) = correlation between income and the percent of disposable income that is saved. .

Question 6.74

6.74 Hypotheses.

Translate each of the following research questions into appropriate and .

  1. Census Bureau data show that the mean household income in the area served by a shopping mall is $62,500 per year. A market research firm questions shoppers at the mall to find out whether the mean household income of mall shoppers is higher than that of the general population.
  2. Last year, your company’s service technicians took an average of 2.6 hours to respond to trouble calls from business customers who had purchased service contracts. Do this year’s data show a different average response time?

Question 6.75

6.75 Exercise and statistics exams.

A study examined whether exercise affects how students perform on their final exam in statistics. The -value was given as 0.68.

  1. State null and alternative hypotheses that could be used for this study. (Note that there is more than one correct answer.)
  2. Do you reject the null hypothesis? State your conclusion in plain language.
  3. What other facts about the study would you like to know for a proper interpretation of the results?

6.75

(a) Answers will vary. (b) Do not reject the null hypothesis. There is no evidence that exercise affects how students perform on their final exam in statistics. (c) How did they measure exercise—was it observational or experimental, how did they get the sample, etc.

Question 6.76

6.76 Financial aid.

The fnancial aid office of a university asks a sample of students about their employment and earnings. The report says that “for academic year earnings, a signifcant difference was found between the sexes, with men earning more on the average. No difference was found between the earnings of black and white students.”18 Explain both of these conclusions, for the effects of sex and of race on mean earnings, in language understandable to someone who knows no statistics.

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Question 6.77

6.77 Who is the author?

Statistics can help decide the authorship of literary works. Sonnets by a certain Elizabethan poet are known to contain an average of new words (words not used in the poet’s other works). The standard deviation of the number of new words is . Now a manuscript with five new sonnets has come to light, and scholars are debating whether it is the poet’s work. The new sonnets contain an average of words not used in the poet’s known works. We expect poems by another author to contain more new words, so to see if we have evidence that the new sonnets are not by our poet, we test

Give the test statistic and its -value. What do you conclude about the authorship of the new poems?

6.77

. -value (< 0.0002). The new poems are by another author.

Question 6.78

6.78 Study habits.

The Survey of Study Habits and Attitudes (SSHA) is a psychological test that measures the motivation, attitude toward school, and study habits of students. Scores range from 0 to 200. The mean score for U.S. college students is about 115, and the standard deviation is about 30. A teacher who suspects that older students have better attitudes toward school gives the SSHA to 25 students who are at least 30 years of age. Their mean score is .
(a) Assuming that for the population of older students, carry out a test of

Report the -value of your test, draw a sketch illustrating the -value, and state your conclusion clearly. (b) Your test in part (a) required two important assumptions in addition to the assumption that the value of σis known. What are they? Which of these assumptions is most important to the validity of your conclusion in part (a)?

Question 6.79

6.79 Corn yield.

The 10-year historical average yield of corn in the United States is about 160 bushels per acre. A survey of 50 farmers this year gives a sample mean yield of bushels per acre. We want to know whether this is good evidence that the national mean this year is not 160 bushels per acre. Assume that the farmers surveyed are an SRS from the population of all commercial corn growers and that the standard deviation of the yield in this population is bushels per acre. Report the value of the test statistic , give a sketch illustrating the -value and report the -value for the test of

Are you convinced that the population mean is not 160 bushels per acre? Is your conclusion correct if the distribution of corn yields is somewhat non-Normal? Why?

6.79

. -value = 0.0238. There is evidence that the population mean is not 160 bushels per acre. Yes, the conclusion is still correct for non-Normality because of the central limit theorem.

Question 6.80

6.80 E-cigarette use among the youth.

E-cigarettes are battery operated devices that aim to mimic standard cigarettes. They don’t contain tobacco but operate by heating nicotine into a vapor that is inhaled. Here is an excerpt from a 2014 UK public health report in which the use of e-cigarettes among children (ages 11 to 18) is summarized:

In terms of prevalence, among all children “ever use” of e-cigarettes was low but did increase between the two surveys. In 2011 it was 3.3%, rising to in 2012. Current use (>1 day in the past 30 days) signifcantly increased from 1.1 to , and current “dual use” (e-cigarettes and tobacco) increased from 0.8 to from 2011 to 2012.19

  1. The report doesn’t state the null and alternative hypotheses for each of the reported estimates with -values. What are the implicit competing hypotheses?
  2. Can you say that the changes in usage are signifcant at the 1% level? Explain.

Question 6.81

6.81 Academic probation and TV watching.

There are other statistics that we have not yet met. We can use Table D to assess the significance of any statistic. A study compares the habits of students who are on academic probation with students whose grades are satisfactory. One variable measured is the hours spent watching television last week. The null hypothesis is “no difference” between the means for the two populations. The alternative hypothesis is two-sided. The value of the test statistic is .

  1. Is the result signifcant at the 5% level?
  2. Is the result signifcant at the 1% level?

6.81

(a) No. (b) No.

Question 6.82

6.82 Impact of on significance.

The Statistical Significance applet illustrates statistical tests with a fixed level of significance for Normally distributed data with known standard deviation. Open the applet and keep the default settings for the null and the alternative hypotheses, the sample size , the standard deviation , and the significance level . In the “I have data and the observed is ” box, enter the value 1. Is the difference between and signifcant at the 5% level? Repeat for equal to 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9. Make a table giving and the results of the significance tests. What do you conclude?

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Question 6.83

6.83 Effect of changing on significance.

Repeat the previous exercise with significance level . How does the choice of affect which values of are far enough away from to be statistically signifcant?

6.83

At , it is significant at . If is smaller, has to be farther away from to be significant.

Question 6.84

6.84 Changing to a two-sided alternative.

Repeat the previous exercise but with the two-sided alternative hypothesis. How does this change affect which values of are far enough away from to be statistically signifcant at the 0.01 level?

Question 6.85

6.85 Changing the sample size.

Refer to Exercise 6.82. Suppose that you increase the sample size from 10 to 40. Again make a table giving and the results of the significance tests at the 0.05 significance level. What do you conclude?

6.85

The test is significant when . Larger samples are able to detect smaller differences between and .

Question 6.86

6.86 Impact of on the -value.

We can also study the -value using the Statistical Significance applet. Reset the applet to the default settings for the null and the alternative hypotheses, the sample size , the standard deviation , and the significance level . In the “I have data, and the observed is ” box, enter the value 1. What is the -value? It is shown at the top of the blue vertical line. Repeat for equal to 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9. Make a table giving and -values. How does the -value change as moves farther away from ?

Question 6.87

6.87 Changing to a two-sided alternative, continued.

Repeat the previous exercise but with the two-sided alternative hypothesis. How does this change affect the -values associated with each ? Explain why the -values change in this way.

6.87

The -value doubles for each value because our -value now represents two tail areas.

Question 6.88

6.88 Other changes and the -value.

Refer to the previous exercise.

  1. What happens to the -values when you change the significance level to 0.01? Explain the result.
  2. What happens to the -values when you change the sample size from 10 to 40? Explain the result.

Question 6.89

6.89 Why is it signifcant at the 5% level?

Explain in plain language why a significance test that is signifcant at the 1% level must always be signifcant at the 5% level.

6.89

If the -value is less than 0.01, it must also be less than 0.05.

Question 6.90

6.90 Finding a -value.

You have performed a two-sided test of significance and obtained a value of .

  1. Use Table A to find the -value for this test.
  2. Use software to find the -value even more accurately.

Question 6.91

6.91 Test statistic and levels of significance.

Consider a significance test for a null hypothesis versus a two-sided alternative. Give a value of that will give a result signifcant at the 1% level but not at the 0.5% level.

6.91

Any number between 2.576 and 2.807 (or −2.576 and −2.807).

Question 6.92

6.92 Finding a -value.

You have performed a one-sided test of significance for greater-than alternative and obtained a value of .

  1. Use Table A to find the approximate -value for this test.
  2. Use software to find the -value even more accurately.