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 and the standard deviation of the sample of 40 trash bag breaking
strengths in Table 1.10 are: the mean of the sample = 50.585 and standard deviation
sample = 1.6438. If we let the mean population denote the mean of the breaking strengths
of all possible trash bags of the new type,
a. Calculate 95 percent and 99 percent confidence intervals for the mean of the
population.
s
The 1   %C.I . is given as: x  t / 2, n 1
n
For a 95% CI,   1  0.95  0.05   / 2  0.025
n = 40, x  50.585, s  1.6438
t0.025,39  2.023
CI  50.585  2.023 *1.6438 / 40  50.585  0.525793
For a 99% CI,   1  0.99  0.01   / 2  0.005
t0.005,39  2.708
CI  50.585  2.708 *1.6438 / 40  50.585  0.70383
b. Using the 95 percent confidence interval, can we be 95 percent confident that the mean
of the population is at least 50 pounds? Explain.
For a 95% CI, LCL = 50.585-0.525793 = 50.05921
UCL = 50.585+0.525793 = 51.11709
Since both the LCL and the UCL lie above the population mean of 50 pounds, we can be
95% confident that the mean of the population is at least 50 pounds.
c. Using the 99 percent confidence interval, can we be 99 percent confident that the mean
of the population is at least 50 pounds? Explain.
For a 99% CI, LCL = 50.585-0.70383 = 49.88117
UCL = 50.585+0.70383 = 51.28883
Since the LCL is below the population mean of 50 pounds, we cannot be 99% confident
that the mean of the population is at least 50 pounds.
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?
Based on the results of parts b and c and the sample test, we can be 95% confident that
the new 30-gallon bag is the strongest such bag on the market. However, we cannot claim
the same on a 99% level.