Question

Question 1 Suppose the average height of American adult males is 5 feet 10 inches (70 inches) and the standard deviation is 5 inches. If we randomly sample 100 men: What will the expected value of the average height of that sample be? (i.e. the mean of the sampling distribution) What will the standard deviation of the average height in that sample be? (i.e. the standard deviation of the sampling distribution) How big of a sample would we need to have a standard deviation of 0.1 inches? Find probability that the mean of the sample is less than 71 inches. Find probability that the mean of the sample is between 69 and 71 inches. Why can we state that the sample distribution of the sample mean is normal? 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 below $75.30 $614.11 $696.34 $572.08 $748.23 $21.20 $99.79 $1,233.38 $530.40 $378.37 $596.14 $239.65 $2,995.38 $1,069.06 $929.80 $259.98 $123.65 $68.92 $192.35 $754.45 $309.00 $163.31 $71.75 $904.92 $40.70 $161.12 $459.38 $171.48 $402.81 $157.44 $41.81 $87.08 $489.97 $468.12 $400.57 $319.40 $533.82 $1,801.35 $1,666.50 $37.16 $85.92 $91.43 $193.14 $106.95 $214.62 $10.62 $582.18 $39.65 $123.66 $76.33 $291.73 $398.48 $659.18 $101.24 $1,740.47 $322.26 $1,509.34 $1,599.04 $358.62 $492.05 $1,052.68 $596.33 $100.54 $1,288.70 $421.46 $1,799.51 $581.21 $571.63 $180.58 $98.82 $358.68 $38.93 $874.78 $2,761.93 $750.44 $376.60 $269.48 $456.79 $216.81 $305.49 What sample size would be required for the auditor to be 95% sure 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 $300. 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. Present the values. Compute the observed sampling error based on the sample you have drawn from the population. Question 3 From prior experience, a freight company knows that the shipping containers that it receives have an average weight of 1.50 tons and a standard deviation of 0.25 tons. Consider a cargo ship with a cargo capacity of 93 tons. What is the average weight and standard deviation of 60 randomly selected containers? Why can we state that the distribution of the total weight of 60 randomly selected containers is normal? What is the probability that the ship’s cargo capacity of 93 tons is exceeded if 60 randomly selected containers are loaded onto it? Specify the R command and output generated to calculate the probability.

Answer 1

Que.1

a.

The expected value of the average height of that sample be 5 feet 10 inches = 70 inches.

b.

The standard deviation of the average height in that sample be

c.

We require sample size = 2500, in order to get standard error 0.1

d.

Probability that the mean of the sample is less than 71 inches :

e.

Probability that the mean of the sample is between 69 and 71 inches :

f.

According to central limit theorem, as sample size increases, mean of the sample follows normal distribution regardless of the distribution of sample.