Question

A random sample of 50 bottles is selected, and the sample mean amount of water per 1-gallon bottle is 0.994 gallon.

(a) Construct a 99% confidence interval estimate for the population mean amount of water included in a 1-gallon bottle. Express the confidence interval as ≤μ≤ (rounded to five decimal places as needed).

(b) On the basis of these results, do you think that the distributor has a right to complain to the water bottling company? Explain your reasoning, considering the 99% confidence interval.

(c) Must you assume that the population amount of water per bottle is normally distributed here? Explain your choice among the options provided.

(d) Construct a 90% confidence interval estimate for the population mean amount of water in a 1-gallon bottle. Express the confidence interval as ≤μ≤ (rounded to five decimal places as needed). Then, consider how this new confidence interval affects your answer to part (b).

Answer 1

Without the standard deviation, we cannot calculate the 99% and 90% confidence intervals for the water volume in the bottles. Typically, a t-distribution is used with the sample size and mean for such calculations, and the Central Limit Theorem allows using the sample mean if the sample size is sufficient.

Step-by-step explanation:

To construct a 99% confidence interval for the population mean amount of water included in a 1-gallon bottle, we need the sample mean, the standard deviation, and the sample size. The question does not provide the standard deviation, so we cannot complete the calculation. However, typically we'd use the t-distribution for a sample size of 50 and the sample mean of 0.994 gallons to compute the interval, if standard deviation were known.

For part (b), without the confidence interval, we cannot definitively say if the distributor has a right to complain. If the confidence interval does not include 1 gallon, it may indicate that the bottles contain less than advertised, supporting the distributor's complaint.

For part (c), the Central Limit Theorem advises that with a large sample size (typically n > 30), the distribution of the sample mean will be approximately normal, regardless of the population distribution. However, this holds true only if the standard deviation is known or the sample size is large enough.

Similarly, to construct a 90% confidence interval, we would use the sample mean and the appropriate t-score for 90% confidence with our sample size. The narrower interval might lead to a different conclusion about the distributor's complaint.