Section 9.4 Hypothesis Tests for Proportions 457 a. What are the null and alternate hypotheses? H0 : p = 0.4, H1: p ≠ 0.4 b. What is the value of the sample proportion ̂p? 0.36 c. Can H0 be rejected at the 0.05 level? Explain. No d. Someone wants to use these data to test H0 : p = 0.5 versus H1: p < 0.5. Find the test statistic z and use the method of this section to find the P-value. Do you reject H0 at the �� = 0.05 level? −2.42; Yes 27. Interpret computer output: The following MINITAB output presents the results of a hypothesis test for a population proportion p. �������� ���� �� = �� �� ���� �� > �� �� ��-���������� �� ���� ��-���������� �� �� ������������ �� ����% ���������� ���������� ������ ������ �������������� �������������� �������� a. What are the null and alternate hypotheses? H0 : p = 0.6, H1: p > 0.6 b. What is the value of the sample proportion ̂p? 0.618829 c. Can H0 be rejected at the 0.05 level? Explain. No d. Someone wants to use these data to test H0 : p = 0.65 versus H1: p < 0.65. Find the test statistic z and use the method of this section to find the P-value. Do you reject H0 at the �� = 0.05 level? −1.93; yes 28. Interpret computer output: The following MINITAB output presents the results of a hypothesis test for a population proportion p. �������� ���� �� = �� �� ���� �� ������ ���������� �� �� ��-���������� *�� ���� ��-���������� �� ������ �� ���� �� ���� ������������ �� �� ������������ ����~ ���� .�� ������������, �� ������������/ a. What are the null and alternate hypotheses? H0 : p = 0.7, H1: p ≠ 0.7 b. What is the value of the sample proportion ̂p? 0.519231 c. Can H0 be rejected at the 0.05 level? Explain. Yes d. Someone wants to use these data to test H0 : p = 0.6 versus H1: p ≠ 0.6. Find the test statistic z and use the method of this section to find the P-value. Do you reject H0 at the �� = 0.05 level? −1.19; no 29. Satisfied with college? A simple random sample of 500 students at a certain college were surveyed and asked whether they were satisfied with college life. Two hundred eighty of them replied that they were satisfied. The Dean of Students claims that more than half of the students at the college are satisfied. To test this claim, a test of the hypotheses H0 : p = 0.5 versus H1: p > 0.5 is performed. a. Show that the P-value is 0.004. b. The P-value is very small, so H0 is rejected. Someone claims that because P is very small, the population proportion p must be much greater than 0.5. Is this a correct interpretation of the P-value? No c. Someone else claims that because the P-value is very small, we can be fairly certain that the population proportion p is greater than 0.5, but we cannot be certain that it is a lot greater. Is this a correct interpretation of the P-value? Yes 30. Who will you vote for? A simple random sample of 1500 voters were surveyed and asked whether they were planning to vote for the incumbent mayor for re-election. Seven hundred ninety-eight of them replied that they were planning to vote for the mayor. The mayor claims that more than half of all voters are planning to vote for her. To test this claim, a test of the hypotheses H0 : p = 0.5 versus H1: p > 0.5 is performed. a. Show that the P-value is 0.007. b. The P-value is very small, so H0 is rejected. A pollster claims that because the P-value is very small, we can be fairly certain that the population proportion p is greater than 0.5, but we cannot be certain that it is a lot greater. Is this a correct interpretation of the P-value? Yes c. The mayor’s campaign manager claims that because the P-value is very small, the population proportion of voters who plan to vote for the mayor must be much greater than 0.5. Is this a correct interpretation of the P-value? No 31. Don’t perform a test: A few weeks before election day, a TV station broadcast a debate between the two leading candidates for governor. Viewers were invited to send a tweet to indicate which candidate they plan to vote for. A total of 3125 people tweeted, and 1800 of them said that they planned to vote for candidate A. Explain why these data should not be used to test the claim that more than half of the voters plan to vote for candidate A. 32. Don’t perform a test: Over the past 100 days, the price of a certain stock went up on 60 days and went down on 40 days. Explain why these data should not be used to test the claim that this stock price goes down on less than half of the days. Extending the Concepts 33. Exact test: When np0 < 10 or n(1 − p0) < 10, we cannot use the normal approximation, but we can use the binomial distribution to perform what is known as an exact test. Let p be the probability that a given coin lands heads. The coin is tossed 10 times and comes up heads 9 times. Test H0 : p = 0.5 versus H1: p > 0.5, as follows. a. Let n be the number of tosses and let X denote the number of heads. Find the values of n and X in this example. n = 10, X = 9 b. The distribution of X is binomial. Assuming H0 is true, find n and p. n = 10, p = 0.5 c. Because the alternate hypothesis is p > 0.5, large values of X support H1. Find the probability of observing a value of X as extreme as or more extreme than the value actually observed, assuming H0 to be true. This is the P-value. 0.0107 d. Do you reject H0 at the �� = 0.05 level? Yes
navidi_monk_elementary_statistics_2e_ch7-9
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