Apr 4, 2017

Hypothesis Testing — practice test 8A

20 cards
, 38 answers
    Problems with solutions in Elementary Statistics — Basics of Hypothesis Testing; Testing a Claim about a Proportion; Testing a Claim about a Mean: 1) σ known, 2) σ not known; Testing Claim about Variation.

    Directions:
    Give your answers to the short-answer items. Choose the correct answer for multiple-choice items.

  • Provide an appropriate response.
    • 1. Define Type I and Type II errors. Give an example of a Type I error which would have serious consequences. Give an example of a Type II error which would have serious consequences. What should be done to minimize the consequences of a serious Type I error?

      Possible answer
      Type I: The mistake of rejecting the null hypothesis when it is true. Type II: The mistake of failing to reject the null hypothesis when it is false. Answers for examples will vary. To minimize a Type I error with serious consequences, make α smaller. Also, make the sample size larger to minimize both α and β.

      Possible example for Type I: Pharmaceutical Company A produces a new drug believed to be superior to the one it will replace. If a hypothesis test suggests this improved drug is better, the company will spend millions of dollars preparing and marketing its new drug, when in the reality the former one was better.
      Possible example for Type II: Pharmaceutical Company A produces a new drug. This time a hypothesis test supports the null hypothesis with the result that Company A abandons production of this improved drug. In the reality, the improved drug is better; and A loses millions of dollars, not only because of the cost of production of the improved drug but also because its existing drug in not competitive.

    • Solve the problem.
      • 2. Write the claim that is suggested by the given statement, then write a conclusion about the claim. Do not use symbolic expressions or formal procedures; use common sense.

        A math teacher tries a new method for teaching her introductory statistics class. Last year the mean score on the final test was 73. This year the mean on the same final was 76.

        Possible answer
        The claim is that the new teaching method is more effective than the old method and that on average students will score higher when she uses the new teaching method than when she uses the old teaching method. The small difference in the two means in not strong evidence that the new method is more effective. Even if both methods were equally effective, such a difference could easily occur by chance.

      • Express the null hypothesis H0 and the alternative hypothesis H1 in symbolic form. Use the correct symbol (μ, p, σ) for the indicated parameter.
        • 3. An entomologist writes an article in a scientific journal which claims that fewer than 12 in ten thousand male fireflies are unable to produce light due to a genetic mutation. Use the parameter p, the true proportion of fireflies unable to produce light.
          1. H0: p = 0.0012, H1: p > 0.0012
          2. H0: p = 0.0012, H1: p < 0.0012
          3. H0: p > 0.0012, H1: p ≤ 0.0012
          4. H0: p < 0.0012, H1: p ≥ 0.0012

      • Assume that the data has a normal distribution and the number of observations is greater than fifty. Find the critical z value used to test a null hypothesis.
        • 4. α = 0.08; H1 is μ ≠ 3.24
          1. 1.75
          2. ± 1.41
          3. 1.41
          4. ± 1.75

      • Find the value of the test statistic z using z = (p̂ – p) / √(p*q/n).
        • 5. A claim is made that the proportion of children who play sports is less than 0.5, and the sample statistics include n = 1933 subjects with 30% saying that they play a sport.
          1. –35.90
          2. –17.59
          3. 17.59
          4. 35.90

      • Use the given information to find the P-value.
        • 6. The test statistic in a left-tailed test is z = –2.05.
          1. 0.5000
          2. 0.0453
          3. 0.0202
          4. 0.4798

      • Formulate the indicated conclusion in nontechnical terms. Be sure to address the original claim.
        • 7. Carter Motor Company claims that its new sedan, the Libra, will average better than 21 miles per gallon in the city. Assuming that a hypothesis test of the claim has been conducted and that the conclusion is to reject the null hypothesis, state the conclusion in nontechnical terms.
          1. There is not sufficient evidence to support the claim that the mean is greater than 21 miles per gallon.
          2. There is sufficient evidence to support the claim that the mean is greater than 21 miles per gallon.
          3. There is sufficient evidence to support the claim that the mean is less than 21 miles per gallon.
          4. There in not sufficient evidence to support the claim that the mean is less than 21 miles per gallon.

      • Assume that a hypothesis test of the given claim will be conducted. Identify the type I or type II error for the test.
        • 8. The principal of a middle school claims that test scores of the seventh-graders at his school vary less than the test scores of seventh-graders at a neighboring school, which have variation described by σ = 14.7. Identify the type I error for the test.
          1. The error of rejecting the claim that the standard deviation is less than 14.7 when it actually is 14.7.
          2. The error of failing to reject the claim that the standard deviation is less than 14.7 when it actually is 14.7.
          3. The error of rejecting the claim that the standard deviation is less than 14.7 when it actually is less than 14.7.

      • Solve the problem.
        • 9. True or False: In a hypothesis test, an increase in α will cause a decrease in the power of the test provided the sample size is kept fixed.
          1. True
          2. False

      • Identify the null hypothesis, alternative hypothesis, test statistic, P-value, conclusion about the null hypothesis, and final conclusion that addresses the original claim.
        • 10. A supplier of 3.5˝ disks claims that no more than 1% of the disks are defective. In a random sample of 600 disks, it is found that 3% are defective, but the supplier claims that this is only a sample fluctuation. At the 0.01 level of significance, test the supplier’s claim that no more than 1% are defective.

          Possible answer
          H0: p = 0.01.
          H1: p > 0.01.
          Test statistic: z = 4.92.
          P-value: p = 0.0001.
          Reject null hypothesis. There is sufficient evidence to warrant rejection of the claim that no more than 1% are defective.
          Note: Since the term “no more than” is used, the translation is p ≤ 0.01. Therefore, the competing hypothesis is p > 0.01.

        • Find the P-value for the indicated hypothesis test.
          • 11. A random sample of 139 forty-year-old men contains 26% smokers. Find the P-value for a test of the claim that the percentage of forty-year-old men that smoke is 22%.
            1. 0.2802
            2. 0.1271
            3. 0.2542
            4. 0.1401

        • Determine whether the given conditions justify testing a claim about a population mean μ.
          • 12. The sample size is n = 22, σ = 6.20, and the original population is normally distributed.
            1. No
            2. Yes

        • Identify the null hypothesis, alternative hypothesis, test statistic, P-value, conclusion about the null hypothesis, and final conclusion that addresses the original claim.
          • 13. A random sample of 100 pumpkins is obtained and the mean circumference is found to be 40.5 cm. Assuming that the population standard deviation is known to be 1.6 cm, use a 0.05 significance level to test the claim that the mean circumference of all pumpkins is equal to 39.9 cm.

            Possible answer
            H0: µ = 39.9.
            H1: µ ≠ 39.9.
            Test statistic: z = 3.75.
            P-value: p = 0.0002.
            Reject H0. There is sufficient evidence to warrant rejection of the claim that the mean equals 39.9 cm.

          • Determine whether the hypothesis test involves a sampling distribution of means that is a normal distribution, Student t distribution, or neither.
            • 14. Claim: µ = 105. Sample data: n = 18, x̅ = 101, s = 15.1.The sample data appear to come from a normally distributed population with unknown µ and σ.
              1. Neither
              2. Student t
              3. Normal

          • Assume that a simple random sample has been selected from a normally distributed population. Find the test statistic, P-value, critical value(s), and state the final conclusion.
            • 15. Test the claim that the mean lifetime of car engines of a particular type is greater than 220,000 miles. Sample data are summarized as n = 23, x̅ = 226,450 miles, and s = 11,500 miles. Use a significance level of α = 0.01.

              Possible answer
              α = 0.01.
              Test statistic: t = 2.6898.
              P-value: 0.005 < P-value < 0.01.
              Critical value: t = 2.508.
              Because TS t > 2.508, we reject the null hypothesis. There is sufficient evidence to accept the claim that the mean lifetime of car engines is greater than 220,000 miles.

            • Test the given claim using the traditional method of hypothesis testing. Assume that the sample has been randomly selected from a population with a normal distribution.
              • 16. A large software company gives job applicants a test of programming ability and the mean for that test has been 160 in the past. Twenty-five job applicants are randomly selected from one large university and they produce a mean score and standard deviation of 183 and 12, respectively. Use a 0.05 level of significance to test the claim that this sample comes from a population with a mean score greater than 160.

                Possible answer
                Test statistic: t = 9.583.
                Critical value: t = 1.711.
                Reject the null hypothesis. There is sufficient evidence to support the claim that the mean is greater than 160.

                1. 17. In tests of a computer component, it is found that the mean time between failures is 520 hours. A modification is made which is supposed to increase the time between failures. Tests on a random sample of 10 modified components resulted in the following times (in hours) between failures.
                  518 548 561 523 536
                  499 538 557 528 563
                  At the 0.05 significance level, test the claim that for the modified components, the mean time between failures is greater than 520 hours.

                  Possible answer
                  Test statistic: t = 2.612.
                  Critical value: t = 1.833.
                  Reject H0. There is sufficient evidence to support the claim that the mean is greater than 520 hours.

                2. Find the critical value or values of x2 based on the given information.
                  • 18. H1: σ ≠ 9.3, n = 28, α = 0.05.
                    1. –40.113, 40.113
                    2. 14.573, 43.194
                    3. –14.573, 14.573
                    4. 16.151, 40.113

                3. Use the traditional method to test the given hypothesis. Assume that the population is normally distributed and that the sample has been randomly selected.
                  • 19. The standard deviation of math test scores at one high school is 16.1. A teacher claims that the standard deviation of the girls’ test scores is smaller than 16.1. A random sample of 22 girls results in scores with a standard deviation of 13.3. Use a significance level of 0.01 to test the teacher’s claim.

                    Possible answer
                    Test statistic: X2 = 14.331.
                    Critical value: X2 = 8.897.
                    Fail to reject H0. There is not sufficient evidence to support the claim that the standard deviation of the girls’ test scores is smaller than 16.1.

                    1. 20. Heights of men aged 25 to 34 have a standard deviation of 2.9. Use a 0.05 significance level to test the claim that the heights of women aged 25 to 34 have a different standard deviation. The heights (in inches) of 16 randomly selected women aged are listed below.
                      62.13 65.09 64.18 66.72 63.09 61.15 67.50 64.65
                      63.80 64.21 60.17 68.28 66.49 62.10 65.73 64.72

                      Possible answer
                      Test statistic: X2 = 9.2597.
                      Critical value: X2 = 6.262, 27.488.
                      Fail to reject H0. There is not sufficient evidence to support the claim that heights of women aged 25 to 34 have a standard deviation different from 2.9 in.

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