EXAMPLE 15 test for using the critical-value method

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In April 2014, the Pew Research Center published a report called U.S. Views of Technology and the Future,15 in which the results of a survey of Americans' views on the future of technology were examined. Among other questions, respondents were asked whether they agreed that “technological changes will lead toward a future where people's lives are mostly better.” The results are shown in Table 13. Assume the samples are independent.

Table 10.49: Table 13 Proportions of males and females who agree that technological change will lead to a better future
Males Females
Number agreeing
Sample size
Sample proportion
Population proportion
  1. Find the point estimate of the difference in the population proportions of males and females, .
  2. Compute the pooled estimate of the common proportion, .
  3. Calculate the value of the test statistic .
  4. Check whether the conditions for performing the test for are met.
  5. Test whether the population proportion of males who agree that technology will lead to a better future is greater than the population proportion of females who agree. Use the critical-value method at level of significance .

610

Solution

  1. The point estimate is
  2. image
    Figure 10.16: FIGURE 16 TI-83/84 results.
  4. We check the conditions for performing the test for . We have: , , , and . We may thus proceed with the hypothesis test.
  5. The test for follows the steps below.

  • Step 1 State the hypotheses.

    The key words “greater than,” together with the fact that sample 1 represents the males, indicate that we have a right-tailed test:

    where and represent the population proportion of males and females, respectively, who agree that technology will lead to a better future.

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    Figure 10.17: FIGURE 17 is extreme, leading to rejection of .

  • Step 2 Find and state the rejection rule.

    For a right-tailed test with level of significance , Table 12 gives us and our rejection rule: Reject if .

  • Step 3 Calculate .

    From (c), we have (also see Figure 16).

  • Step 4 State the conclusion and the interpretation.

    ; therefore, reject (see Figure 17). There is evidence at level of significance that the population proportion of males who agree that technology will lead to a better future is greater than the population proportion of females who agree.

NOW YOU CAN DO

Exercises 5–8.