 # General Statistics Problems

## Question

1) A local Board of Elections wants to estimate the average number of voters per district for the last election. A random sample of 81 districts shows a sample mean of 168 voters with a standard deviation of 45 voters. Find a 99% confidence interval (to the nearest tenth) for the population mean number of voters per district.
2) Some professional football players seem to earn tremendous amounts of money. However, their careers professional players are short. One sports magazine reported that historically the average career length is 4.7 years with a standard deviation of 2.1 years. A random sample of 50 retired players showed a sample mean career length of 5.3 years. Construct a hypothesis test to determine whether the average career in professional football is longer than 4.7 years. Use a 1% level of significance.
3) The Rocky Mountain Company claims that their cigarettes contain an average of only 10mg. of tar. A random sample of 52 Rocky Mountain cigarettes shows the average tar content to be 11.5 mg with a standard deviation of 6.5 mg. Construct a hypothesis test to determine whether the average tar content of Rocky Mountain cigarettes exceeds 10 mg. Use a 1% level of significance.
4) The following is the Correlation and Regression output from a Multiple Regression run in Excel. The dependent variable is “Median Home Value” (MEDHOMEVAL). The other variables are the Independent variables: “Robbery rate” (ROBBERY), “%Working outside county of residence”
(WORKOUTSIDE), ROWHOUSE (%), “% with 5 or more units” (5+_UNITS), “% of Housing very Crowded” (1.5_PER_RM), & “% Houses that use LP Gas” (LP_GAS). Please answer the following questions:
a) Which variables pass the “P-Value test”?
b) Which variable pairs are “multicollinear”?
c) Compute the Coefficient of determination for each independent variable.
d) Write out the complete regression equation.
e) How many additional variables would you be allowed to add to this equation based on the heuristic rule? ## Solution Preview

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4)

A) In order to run the “P-value test”, we would need a confidence level for comparisons; in most cases, we use 95%. Based on this, all variables would pass except ROW_HOUSE.

B) Variables that are multicollinear have strong, positive correlations. These quantities are expressed in the first table and the variable pairs are: (ROW_HOUSE, ROBBERY), (5+_UNITS,ROBBERY), (5+_UNITS,MEDHOMEVAL), and (1.5+PER+RM,5+_UNITS).

C) The coefficient of determination for this multiple regression analysis is essentially calculated by squaring the each value of the first column. The reason we do this is because each independent variable is correlated to the dependent variable, independently, in the first column.
r^2 (ROBBERY)=.254²=.064516
r^2 (WORKOUTSIDE)=.235²=.055225
r^2 (ROW_HOUSE)=.377²=.142129
r^2 (5+_UNITS)=.648²=.419904
r^2 (1.5_PER_RM)=.652²=.425104
r^2 (LP_GAS)=-.617²=.380689...
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