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Probability and Statistics Chapter 8: Hypothesis Testing Answers to Even Problems 8-2 Basics of Hypothesis Testing 2. Estimates and hypothesis tests are both methods of inferential statistics, but they have different objectives. We could use the sample weights to construct a confidence interval estimate of the mean weights of M&Ms, but hypothesis testing is used to test some claim made about the mean weight of M&Ms. 4. The P-value of 0.001 is preferred because it corresponds to the sample evidence that most strongly supports the alternative hypothesis that the XSORT method is effective. 6. a. π > 0.5 b. π»0 : π = 0.5 and π»1 : π > 0.5 8. a. π β₯ 50 b. π»0 : π = 50 and π»1 : π < 50 10. There is sufficient evidence to support the claim that when parents use the XSORT method of gender selection, the proportion of baby girls is greater than 0.5. 12. There is sufficient evidence to reject the claim that pulse rates of adult females have a standard deviation of at least 50. 14. π§ = β1.27, 16. π‘ = β2.358 18. P-value = 0.0228, critical value: π§ = β1.645 20. P-value = 0.1336, critical values: π§ = β1.96, π§ = 1.96 22. P-value = 0.0124, critical values: π§ = β1.96, π§ = 1.96 24. P-value = 0.0020, critical value: π§ = 1.645 26. a. Fail to reject π»0 b. There is not sufficient evidence to support the claim that fewer than 20% of M&M candies are green 28. a. Reject π»0 b. There is sufficient evidence to warrant rejection of the claim that women have heights with a standard deviation equal to 5.00 cm. 8-3 Testing a Claim About a Proportion 411 2. πΜ = 1003 or 0.410. The symbol πΜ is used to represent a sample proportion. 4. a. The symbol π represents the population proportion, but the P-value is a probability of getting sample results that are at least as extreme as those obtained (assuming the null hypothesis is true). b. If the P-value is very low (such as less than or equal to 0.05), βthe null must goβ means that we should reject the null hypothesis. c. The statement that βif P is high, the null will flyβ suggests that with a high P-value, the null hypothesis has been provide or is supported, but we should never make such a conclusion. 8-4 Testing a Claim About a Mean 2. df denotes the number of degrees of freedom. For the sample of 12 times, df = 11. 4. Use a 90% confidence level. The given confidence interval does contain the value of 90 seconds, so it is possible that the value of π is equal to 90 seconds or some lower value, so there is not sufficient evidence to support the claim that the mean is greater than 90 seconds. 8-5 Testing a Claim About a Standard Deviation or Variance 2. a. The normality requirement for a hypothesis test of a claim about a standard deviation is much more strict, meaning that the distribution of the population must be much closer to a normal distribution. b. With only 10 sample values, a histogram doesnβt really give us a good picture of the distribution, so a normal quantile plot would be better. Also, we should determine that there are no outliers. 4. a. π»0 : π = 1.8 min and π»1 : π < 1.8 min b. π 2 = 0.694 c. Reject the null hypothesis. d. There is sufficient evidence to support the claim that the standard deviation of waiting times of all customers is less than 1.8 minutes. e. The change to a single waiting line is effective because the variation among waiting times is less than it was with multiple lines.