# statistic for business and economics

1. A soft drink filling machine is set up to fill bottles with 12 ounces of soft drink. The standard deviation s is known to be 0.4 ounces. The quality control department periodically selects samples of 16 bottles and measures their contents. Assume the distribution of filling volumes is normal.

Determine the upper and lower control limits and explain what they indicate. The means of six samples were 11.8, 12.2, 11.9, 11.9, 12.1, and 11.8 ounces. Construct an x-bar

chart and indicate whether or not the process is in control.

2. An automobile manufacturer must make an immediate decision on the car size that should account for the majority of the firm’s production two years from now. The firm perceives three possible states of nature at that time: S1, gasoline will be rationed; S2, gasoline will be readily available at close to current prices; and S3, gasoline will be readily available, but at much higher prices. The firm has determined the following profit payoff table (in $1,000s).

Decision

Alternative

make mostly large cars make mostly medium cars make mostly small cars

States of Nature

S 1 S 2 S 3 -200 1,900 200 400 1,400 700

900 800

1,400 An economist at the auto company has advised the firm that the probabilities of the states of nature are P(S 1 ) = .2, P(S 2 ) = .5, and P(S 3 ) = .3. Find the expected monetary value for the three decisions.

Which decision should be chosen under the expected monetary value criterion?

Determine the expected value of perfect information.

3. A sample of 4 clusters is to be taken from a population with N = 50 clusters and M = 800 elements in the population. The values of M i and X i for each cluster are shown below.

Cluster M i X i

1 8 70

2 14 120

3 16 80

4 12 130

Total 50 400

Determine the point estimate of the population mean.

Determine the standard error of the mean.

Develop a 95% confidence interval for the population mean.

Determine the point estimator of the population total.

Approximate a 95% confidence interval estimate of the population total.