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 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 bags breaking strengths in Table 1.9 is x = 50.575. If we let u denote the mean of the breaking strengths of all possible trash bags of the new type and assume that o equals 1.65
a. Calculate 95 percent and 99 percent confidence intervals for u.
100(1 - .5) = 95, and the confidence of coefficient is (1 – x) = .95, which implies that x = .01 and x/2 = .005. Therefore we need to find the normal point z.005
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 percent confidence interval, can we be 99 percent 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?
show me how
a. To calculate the confidence intervals for u, we need to find the standard error of the sample mean and then use the z-values for the desired confidence level.
The standard error of the sample mean, denoted as SE(x̄), is calculated as:
SE(x̄) = o / sqrt(n)
Where o is the standard deviation of the population (given as 1.65) and n is the sample size (given as 40).
SE(x̄) = 1.65 / sqrt(40) = 0.26
For a 95% confidence interval, we need to find the z-value when the cumulative probability from the left tail is 0.025. This is because we want a two-tailed confidence interval, so we divide the significance level (0.05) by 2.
From a standard normal distribution table or z-table, we can find that the z-value for a cumulative probability of 0.025 is approximately -1.96.
The 95% confidence interval for u can be calculated as:
CI_95 = x̄ ± (z * SE(x̄))
= 50.575 ± (1.96 * 0.26)
≈ (50.060, 51.090)
For a 99% confidence interval, we need to find the z-value when the cumulative probability from the left tail is 0.005. This is because we want a two-tailed confidence interval, so we divide the significance level (0.01) by 2.
From a standard normal distribution table or z-table, we can find that the z-value for a cumulative probability of 0.005 is approximately -2.58.
The 99% confidence interval for u can be calculated as:
CI_99 = x̄ ± (z * SE(x̄))
= 50.575 ± (2.58 * 0.26)
≈ (49.923, 51.228)
b. With the 95% confidence interval, we can say that we are 95% confident that the true mean breaking strength of the new trash bag (u) lies between 50.060 and 51.090 pounds. Since the lower limit of the confidence interval (50.060) is greater than 50 pounds, we can be 95% confident that u is at least 50 pounds.
c. With the 99% confidence interval, we can say that we are 99% confident that the true mean breaking strength of the new trash bag (u) lies between 49.923 and 51.228 pounds. Since the lower limit of the confidence interval (49.923) is also greater than 50 pounds, we can be 99% confident that u is at least 50 pounds.
d. Based on the answers to parts b and c, we can be quite confident that the new 30-gallon trash bag is one of the strongest bags on the market. With a 95% confidence level, we can be 95% confident that it has a breaking strength of at least 50 pounds, and with a 99% confidence level, we can be 99% confident. However, it's important to note that confidence intervals estimate the range of possible values for the population mean, and there is still a small chance that the true mean breaking strength falls below 50 pounds.