Question

A local bottler in Hawaii wishes to ensure that an average of 23 ounces of passion fruit juice is used to fill each bottle. In order to analyze the accuracy of the bottling process, she takes a random sample of 42 bottles. The mean weight of the passion fruit juice in the sample is 22.41 ounces. Assume that the population standard deviation is 1.44 ounces.

a) What is the population mean for the passion fruit juice weight?

b) What is the margin of error for the mean weight of the passion fruit juice?

c) Construct a 95% confidence interval for the mean weight of the passion fruit juice.

d) If the bottler wants to decrease the margin of error, should she increase or decrease the sample size?

Answer 1

a) The population mean for the passion fruit juice weight is 23 ounces. b) The margin of error is calculated using the formula: Margin of Error = (Z-score) * (Standard Deviation / Square Root of Sample Size). c) The 95% confidence interval is constructed using the formula: Confidence Interval = (Sample Mean) ± (Margin of Error). d) Increasing the sample size decreases the margin of error.

Step-by-step explanation:

a) The population mean for the passion fruit juice weight is the desired average weight of the passion fruit juice in each bottle, which is 23 ounces.

b) The margin of error for the mean weight of the passion fruit juice can be calculated using the formula: Margin of Error = (Z-score) * (Standard Deviation / Square Root of Sample Size). Since the sample size is 42 and the population standard deviation is 1.44 ounces, we can use a Z-score of 1.96 (for 95% confidence) to calculate the margin of error.

c) To construct a 95% confidence interval for the mean weight of the passion fruit juice, we can use the formula: Confidence Interval = (Sample Mean) ± (Margin of Error). Plugging in the given values, the confidence interval is 22.41 ± 0.389 ounces.

d) To decrease the margin of error, the bottler should increase the sample size. This is because a larger sample size provides more accurate estimates of the population parameters.