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# Managerial Statistics

1) Suppose that an independent laboratory has tested trash bags and has found that no 30-gallon bags that are currently on the market have a mean breaking strength of 50 pounds or more. On the basis of these results, the producer of the new, improved trash bag feels sure that its 30-gallon bag will be the strongest such bag on the market if the new trash bag’s mean breaking strength can be shown to be at least 50 pounds. The mean of the sample of 40 trash bag breaking strengths in Table 1. 9 is x=50. 575. If we let u denote the mean of the breaking strengths of all trash bags of the new type and assume that o equals 1. 5: a. Calculate 95 percent and 99 percent confidence intervals for u. b. Using the 95 percent confidence interval, can we be 95 percent confident that u is at least 50 pounds? Explain c. Using the 99% confidence interval, can we be 99% confident that u is at least 50 pounds? explain d. Based on your answers to parts b and c, how convinced are you that the new 30-gallon trash bag is the strongest such bag on the market? (a) (i) 95% confidence interval for ? :n = 40x-bar = 50. 575s = 1. 65% = 95Standard Error, SE = ? /On = 0. 2609z- score = 1. 9600Width of the confidence interval = z * SE = 0. 113Lower Limit of the confidence interval = x-bar – width = 50. 0637Upper Limit of the confidence interval = x-bar + width = 51. 0863The confidence interval is [50. 0637 pounds, 51. 0863 pounds](ii) 99% confidence interval for ? :n = 40x-bar = 50. 575s = 1. 65% = 99Standard Error, SE = ? /On = 0. 2609z- score = 2. 5758Width of the confidence interval = z * SE = 0. 6720Lower Limit of the confidence interval = x-bar – width = 49. 9030Upper Limit of the confidence interval = x-bar + width = 51. 2470The confidence interval is [49. 9030 pounds, 51. 2470 pounds](b) Yes, we can be 95% confident that ? s at least 50 pounds, since the entire 95% confidence interval lies above 50 pounds (c) No, we can’t be 99% confident that ? is at least 50 pounds, since a part of the 99% confidence interval lies below 50 pounds (d) At 95% confidence level, we can say that the new 30-gallon trash bag is the strongest such bag on the market. But we cannot conclude the same at 99% confidence level. 2) Quality Progress, February 2005, reports on the results achieved by Bank of America in improving customer satisfaction and customer loyalty by listening to the “voice of the customer. A key measure of customer satisfaction is the response on a scale from 1 to 10 to the question: “Considering all the business you do with Bank of America, what is your overall satisfaction with Bank of America? ” Suppose that a random sample of 350 current customers results in 195 customers with a response of 9 or 10 representing “customer delight. ” Find a 95% confidence interval for the true proportion of all current Bank of America customers who would respond with a 9 or 10. Are we 95% confident that this proportion exceeds . 8, the historical proportion of customer delight for Bank of America? (a) 95% confidence interval for p:n = 350p = 0. 5571% = 95Standard Error, SE = O{p(1 – p)/n} = 0. 0266z- score = 1. 9600Width of the confidence interval = z * SE = 0. 0520Lower Limit of the confidence interval = P – width = 0. 5051Upper Limit of the confidence interval = P + width = 0. 6092The confidence interval is [0. 5051, 0. 6092](b) Yes, we can be 95% confident that p exceeds 0. 48, since the entire 95% confidence interval lies above 0. 48.

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