Review the data file and answer the questions. Your answers should be submitted in the form of a Word document. The Word document should include any charts, tables, or graphs you create. You are also required to submit an Excel file, R code or copy of hand calculations to support your answers.

Question 1

Suppose the average height of American adult females is 5 feet 4 inches (64 inches) and the standard deviation is 4 inches. If we randomly sample 144 females:

1. What will the expected value of the average height of that sample be? (i.e. the mean of the sampling distribution)
2. What will the standard deviation of the average height in that sample be? (i.e. the standard deviation of the sampling distribution)
3. How big of a sample would we need to have a standard deviation of 0.2 inches?
4. Find the probability that the mean of the sample is less than 65 inches.
5. Find the probability that the mean of the sample is between 62 and 65 inches.
6. Why can we state that the sample distribution of the sample mean is normal? Explain.

Question 2

Auditors of Old Kent Bank are interested in comparing the reported value of customer savings account balances with their own findings regarding the actual value of such assets. Rather than reviewing the records of each savings account at the bank, the auditors decide to examine a representative sample of savings account balances. The frame from which they will sample is shown on Balances.xlsx Download Balances.xlsx.

1. What sample size would be required for the auditor to be 90% confident that their estimate of the average savings account balance at this bank is within $150 of the true mean? Assume that their best estimate of the population standard deviation is $450.
2. Using the R function sample(), choose a simple random sample of the size found in a). Use the command set.seed(321) to fix the seed before generating the sample. Calculate the sample statistics for the same, use the command Summary, and provide the Summary output.
3. Compute the observed sampling error based on the sample you have drawn from the population. That is, calculate the sample standard deviation for your sample and divide it by the square root of the sample size.

Question 3

From prior experience, a freight company knows that the shipping containers that it receives have an average weight of 26 tons and a standard deviation of 5 tons. Consider a small cargo ship with a cargo capacity of 1600 tons.

1. What is the average weight and standard deviation of 60 randomly selected containers?
2. Why can we state that the distribution of the total weight of 60 randomly selected containers is normal?
3. What is the probability that the ship's cargo capacity of 1600 tons is exceeded if 60 randomly selected containers are loaded onto it? Show how work for probability calculations (either R code or Excel formula)