Consider the trash bag problem. Suppose that an independent laboratory has teste
ID: 3181792 • Letter: C
Question
Consider the trash bag problem. Suppose that an independent laboratory has tested trash bags and has found that no 30-galion 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-gailon 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 p denote the mean of the breaking strengths of all possible trash bags of the new type and assume that a equals 1.65: (a) Calculate 95 percent and 99 percent confidence intervals for p. (b) Using the 95 percent confidence interval, can we be 95 percent confident that mu is at least 50 pounds? because if we are 95 percent confident that the interval contains mu, and the entire interval is [above] 50, then we are 95 percent confident that mu is 50 pounds. (c) Using the 99 percent confidence interval, can we be 99 percent confident that if is at least 50 pounds? because the interval extends (below) 50. That is, there is that possibility that, if the 99 percent interval does contains mu, mu could be in the portion of the interval that extends (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? since the 95 percent confidence interval 50 and the 99 percent confidence interval contains 50.Explanation / Answer
Solution:
a) The (1-)%C.I is given as: x±t/2, n-1 s/n
For a 95% CI, = 1-0.95 = 0.05=>/2 = 0.025
n= 40, x = 50.575, s= 1.65
t0.025, 39 = 2.023
therefore CI = 50.575±2.023*1.65/40 = 50.575±0.52777
for a 99% CI, = 1-0.99 = 0.01=>/2 = 0.005
t0.005, 39 = 2.708
therefore CI = 50.575±2.708*1.65/40 = 50.575±0.70648
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 = 50.575-0.52777 = 50.04723
UCL = 50.575+0.52777 = 51.10277
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 = 50.575-0.70648 = 49.86852
UCL = 50.575+0.70648 = 51.28148
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.