Questions: The United States Postal Service wants to estimate the average number of days that it takes for a package to be shipped from the United States to an overseas military base. In a random sample of 60 such packages, the average shipping time was 12.2 days, with a standard deviation of 1.42 days. A histogram and boxplot of the data are below. Construct a 90% confidence interval for the average number of days that it takes for a package to be shipped from the United States to an overseas military base. Make sure to include all steps. Write your answers in the space provided on your paper. Step 1: Check conditions. Be specific as to why each condition is or isn't met. Even if one of the conditions is not met, continue with constructing the confidence interval. Step 2: Calculate the confidence interval. Make sure to show all work and to write down any RStudio commands that you used to find values. Step 3: Write a sentence or two interpreting the confidence interval in the context of the problem.

Solution Steps

1. Random Sample: The problem states a random sample of 60 packages was taken. This satisfies the random condition.

2. Independence: We can assume that the shipping time of one package does not influence the shipping time of another. Additionally, the sample size of 60 is less than 10% of all packages shipped overseas. Hence, independence is reasonably met.

3. Normality: The sample size is 60, which is greater than 30. By the Central Limit Theorem, we can assume the sampling distribution of the sample mean is approximately normal, even if the underlying population distribution is not perfectly normal. The histogram and boxplot provided show some right skewness and an outlier. The skewness is not so extreme, so together with the sample size of 60, it appears reasonable for the normality assumption to be met.

We are constructing a 90% confidence interval. The formula for a confidence interval for a population mean is:

• x̄ ± t*(s/√n)

Where:

• x̄ is the sample mean (12.2 days)
• t* is the critical t-value for the desired confidence level and degrees of freedom (n-1)
• s is the sample standard deviation (1.42 days)
• n is the sample size (60)

Degrees of freedom = 60 - 1 = 59. For a 90% confidence level and 59 degrees of freedom, the critical t-value is approximately 1.671 (can be found using a t-table or software like RStudio with the command qt(0.95, 59)).

Plugging in the values:

• 12.2 ± 1.671 * (1.42/√60)
• 12.2 ± 0.306
• (11.894, 12.506)

We are 90% confident that the true average number of days it takes to ship a package from the United States to an overseas military base is between 11.894 days and 12.506 days.

Final Answer: