Consider the trash bag problem. Suppose that an independent laboratory has teste

Question

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 36 trash bag breaking strength in table 1.9 is T = 50.563. If we let mu denote the mean of the breaking strength of all possible trash bags of the new type and assume that sigma equals 1.61: (a) Calculate 95 percent and 99 percent confidence intervals for mu. (Round your answers to 3 decimal places.) (b) Using the 95 percent confidence interval, can we be 95 percent confident that at least 50 pounds? Explain. ______, 95 percent interval is ______ 50. (c) Using the 99 percent confidence interval, can we be 99 percent confident that mu is at least 50 pounds? Explain. ______, 99 percent interval extends ______ 50. (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? ______ confident, since the 95 percent Cl ______ 50 while the 99 percent Cl contains 50.

Explanation / Answer

mean = 50.563
n = 36
sigma = 1.61

Here, the null and alternate hypothesis are
H0: mu <= 50
H1: mu > 50

As this is one tailed test, we are concerned with the upper limit. Null hypothesis will be rejected only if the value of mean of sample is greater than this upper limit.

This actually means null hypothesis cannot be rejected for the value between 0 and upper limit.

(a)
For 95% CI, z-value = 1.64
lower limit = 0
upper limit = 50 + 1.64*1.61/sqrt(36) = 50.4401

For 99% CI, z-value = 2.33
lower limit = 0
upper limit = 50 + 2.33*1.61/sqrt(36) = 50.6252

(b)
Using 95%, we reject null hypothesis. This means we are confident enough to confirm the mean is atleast 50 pounds.

(c) Using 99% CI, we fail to reject the null hypothesis. This means we are not confident enough to confirm the mean is at least 50 pounds.

(d) 95% confident that the bag is atleast 50 pounds strong.

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