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9.52 At Ajax Spring Water, a half-liter bottle of soft drink is supposed to contain a mean of 520 ml. The filling process follows a normal distribution with a known process standard deviation of 4 ml. (a) Which sampling distribution would you use if random samples of 10 bottles are to be weighed? Why? (b) Set up hypotheses and a two-tailed decision rule for the correct mean using the 5 percent level of significance. (c) If a sample of 16 bottles shows a mean fill of 515 ml, does this contradict the hypothesis that the true mean is 520 ml? (a) t- distribution, since the sample size of 10 is less than 30. (b) H0: The true mean is 520, that is, m = 520 Ha: The true mean is different from 520, that is, m ≠ 520 Two-tailed t- test at a = 0.05 Degrees of freedom = 9; Critical t- score = 2.2621 Decision rule: Reject H0 if the sample t- value < -2.2621 or > 2.2621 (c) SE = s/n = 4/16 = 1 t = (x-bar – m)/SE = (515 – 520)/1 = -5 Since -5 < -2.2621, we reject H0 and accept Ha It appears that the true mean is different from 520 mL. 9.60 Ages for the 2005 Boston Red Sox pitchers are shown below. (a) Assuming this is a random sample of major league pitchers, at the 5 percent level of significance does this sample show that the true mean age of all American League pitchers is over 30 years? State your hypotheses and decision rule and show all work. (b) If there is a difference, is it important? (c) Find the p-value and interpret it. (Data are from http://www.boston.redsox.mlb.com.) RedSox Ages of Boston Red Sox Pitchers, October 2005 Arroyo 28 Foulke 33 Mantei 32 Timlin 39 Clement 31 Gonzalez 30 Miller 29 Wakefield 39 Embree 35 Halama 33 Myers 36 Wells 42 (a) H0: The true mean age of the players is not greater than 30 years Ha: The true mean age of the players is > 30 years Upper-tailed t- test at a = 0.05 Degrees of freedom = 11; Critical t- score = 1.796 Decision rule: Reject H0 if the sample t- value > 1.796 (c) For the given sample of 12 players, mean = x-bar = 33.92 years and s = 4.38 years SE = s/n = 4.38/12 = 1.2644 t = (x-bar – m)/SE = (33.92 – 30)/1.2644 = 3.1 Since 3.1 > 1.796, we reject H0 and accept Ha It appears that the true mean age of the players is greater than 30 years. (b) The difference is significant since 3.1 is not close to 1.796 but much greater. (c) p- value = 0.005. This is the probability that t can be as extreme as the test statistic value.