8.6 THE TRASH BAG CASE TrashBag Consider the trash bag problem. Suppose that an

Question

8.6
THE TRASH BAG CASE TrashBag
Consider the trash bag problem. 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 m denote the mean of the breaking strengths of all possible trash bags of the new type and assume that s equals 1.65:
a Calculate 95 percent and 99 percent confidence intervals for m.
b Using the 95 percent confidence interval, can we be 95 percent confident that m is at least
50 pounds? Explain.
c Using the 99 percent confidence interval, can we be 99 percent confident that m 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?

Explanation / Answer

(a)The intervals go from: mean - z*sigma/sqrt(N) to mean + z*sigma/sqrt(N) 95% interval. The z value, from a table, is 1.96: 50.575 - 1.96*1.65/sqrt(40) to 50.575 + 1.96*1.65/sqrt(40) 50.064 to 51.086 99% interval. The z value, from a table, is 2.5758: 50.575 - 2.5758*1.65/sqrt(40) to 50.575 + 2.5758*1.65/sqrt(40) 49.903 to 51.247 -------------------------------------------------------------------------------------- (b)Yes, we can be 95% confident that the mean is at least 50, since the entire interval lies above 50 pounds. ------------------------------------------------------------------------------------- (c)No, we cannot be 99% confident that the mean is at least 50, since part of the interval lies below 50 pounds. -------------------------------------------------------------------------------------- (d)Yes, I am convinced that this bag is the strongest, since the 95% interval lies above 50 pounds, so we can be quite sure that it is the strongest.

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