Section 8.2

#9. "More than 25% of Internet users pay bills online." This is a statement about a percentage (not an average), so the parameter of interest is p. The statement itself would translate to "p > 0.25." That means the other hypothesis must be "p ≤ 0.25." Since the second one has an equal sign in it, it must be the null hypothesis. So we have: H_{0}: p ≤ 0.25 and H_{1}: p > 0.25.

#11. "The mean weight of women who won Miss America titles is equal to 121 lb." This is a statement about an average (not a proportion), so the parameter of interest is µ. The statement itself would translate to "µ = 121." That means the other hypothesis must be "µ ≠ 121." Since the first one has an equal sign in it, it must be the null hypothesis. So we have: H_{0}: µ = 121 and H_{1}: µ ≠ 121.

#13. "IQ scores of college professors have a standard deviation less than 15." This is a statement about the standard deviation, so the parameter of interest is sigma. The statement itself would translate to "sigma < 15 ." That means the other hypothesis must be "sigma ≥ 15." Since the second one has an equal sign in it, it must be the null hypothesis. So we have: H_{0}: sigma ≥ 15 and H_{1}: sigma < 15.

#15. "Plain M&M candies have a mean weight that is at least 0.8535 g." This is a statement about an average (not a proportion), so the parameter of interest is µ. The statement itself would translate to "µ ≥ 0.8535." That means the other hypothesis must be "µ < 0.8535." Since the first one has an equal sign in it, it must be the null hypothesis. So we have: H_{0}: µ ≥ 0.8535 and H_{1}: µ < 0.8535.

#17. "Two-tailed test, alpha = 0.05." We need to find the z-scores that trap an area of 0.05 in the two tails, so 0.025 in both the upper and lower tails. That is, we need to chop off the top 0.025 area, which means the lower 0.975 area. We look up the area of 0.975 in the z-chart, which gives z=1.96. So the critical values are ±1.96.

#19. "Right-tailed test, alpha = 0.01." We need to find the z-score that traps an area of 0.01 in the upper tail, which means the lower area must be 0.9900 area. We look up the area of 0.9900 in the z-chart, which gives z=2.33. So the critical value is 2.33.

#25. The stated value of p is 0.25, which means q is 0.75. The value of p-hat is 224/1018 = 0.2200, and the value of n is 1018. So z = (0.2200-0.2500)/sqrt(0.25*0.75/1018) = -0.03/0.0136 = -2.21

#27. The stated value of p is 0.75, which means q is 0.25. The value of p-hat is 516/580 = 0.8900, and the value of n is 580. So z = (0.8900-0.7500)/sqrt(0.75*0.25/580) = 0.14/0.0180 = 7.78

#37. The original claim was "The proportion of male golfers is less than 0.5." Since this would not have an equal sign in it, it must be H_{1}. Our first conclusion is to reject H_{0}, which means we vote in favor of H_{1}. Our second conclusion, then, is that there is sufficient evidence to conclude that the proportion of male golfers is less than 0.5.

#39. The original claim was "The proportion of red M&Ms is different from 0.13." Since this would not have an equal sign in it, it must be H_{1}. Our first conclusion is to fail to reject H_{0}, which means we vote in favor of H_{0}. Our second conclusion, then, is that we accept that the proportion of red M&Ms is 0.13. (Or we might say that there is not sufficient evidence to conclude that the proportion is not 0.13.)