8-4 Testing a Claim About a Standard Deviation or Variance 423 4. Minting Dollar Coins: Confidence Interval If we use the data given in Exercise 1, we get this 90% confidence interval estimate of the standard deviation of weights with the new minting process: 0.01686 g 6 s 6 0.03128 g. What does this confidence interval tell us about the new filling process? Testing Claims About Variation. In Exercises 5–16, test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim. Assume that a simple random sample is selected from a normally distributed population. 5. Pulse Rates of Men A simple random sample of 153 men results in a standard deviation of 11.3 beats per minute (based on Data Set 1 “Body Data” in Appendix B). The normal range of pulse rates of adults is typically given as 60 to 100 beats per minute. If the range rule of thumb is applied to that normal range, the result is a standard deviation of 10 beats per minute. Use the sample results with a 0.05 significance level to test the claim that pulse rates of men have a standard deviation equal to 10 beats per minute; see the accompanying StatCrunch display for this test. What do the results indicate about the effectiveness of using the range rule of thumb with the “normal range” from 60 to 100 beats per minute for estimating s in this case? 6. Pulse Rates of Women Repeat the preceding exercise using the pulse rates of women listed in Data Set 1 “Body Data” in Appendix B. For the sample of pulse rates of women, n = 147 and s = 12.5. See the accompanying JMP display that results from using the original list of pulse rates instead of the summary statistics. (Hint: The bottom three rows of the display provide P-values for a two-tailed test, a left-tailed test, and a right-tailed test, respectively.) What do the results indicate about the effectiveness of using the range rule of thumb with the “normal range” from 60 to 100 beats per minute for estimating s in this case? 7. Body Temperature Example 5 in Section 8-3 involved a test of the claim that humans have body temperatures with a mean equal to 98.6°F. The sample of 106 body temperatures has a standard deviation of 0.62°F. The conclusion in that example would change if the sample standard deviation s were 2.08°F or greater. Use a 0.01 significance level to test the claim that the sample of 106 body temperatures is from a population with a standard deviation less than 2.08°F. What does the result tell us about the validity of the hypothesis test in Example 5 in Section 8-3? 8. Birth Weights A simple random sample of birth weights of 30 girls has a standard deviation of 829.5 g. Use a 0.01 significance level to test the claim that birth weights of girls have the same standard deviation as birth weights of boys, which is 660.2 g (based on Data Set 6 “Births” in Appendix B). 9. M&Ms Plain M&M candies are fairly consistent in their sizes, but Data Set 38 “Candies” shows that their weights do vary. Here are the statistics for the weights of a sample of the M&M candies listed in that data set: n = 20, x = 0.8724 g, s = 0.0337 g. If it is a goal to manufacture M&Ms so that they have weights with a standard deviation less than 0.0400 g, does the sample of M&Ms appear to satisfy that goal? Use a significance level of 0.01. 10. Minting of Pennies Data Set 40 “Coin Weights” lists weights (grams) of pennies minted after 1983. Here are the statistics for those weights: n = 37, x = 2.49910 g, s = 0.01648 g. Use a 0.05 significance level to test the claim that the sample is from a population of pennies with weights having a standard deviation greater than 0.01000 g.
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