Question

A popular U.S. automobile manufacturer has 10,000 dealerships located throughout the country. The automobile manufacturer has multiple brands within its portfolio: a value brand that caters to younger clientele, a moderate brand that caters to middle class customers and finally, a premium brand which is marketed to wealthy clientele. The company's leadership, located at corporate headquarters, is very interested in the relationship between the median salary of potential customers and the company's revenue. Specifically, the company is concerned that if potential customers' salaries continue to not increase in the future, the company's revenue will remain stagnant, which will in turn steer away potential investors and shareholders. The company's research department recently collected data for analysis in order to support leadership's upcoming discussion with shareholders and investors about the company's future revenue forecast. Sales figures from a random sample of 1000 dealerships were collected. The research division also conducted statistical analysis, using data provided by the Bureau of Labor and Statistics, to calculate the median salary of people living in the vicinity of these 1,000 dealerships. The Dealership Number, State, Median Salary, Annual Sales, Number of Vehicles Sold, Square Footage and Quality Award Winner data were collected for these 1000 dealerships.

StatCrunch Data Set

Corporate headquarters decides to use the salary data of potential customers from its random sample of 1000 dealerships to estimate the mean salary of potential customers living near all of its 10,000 dealerships. Construct a 95% One-Sample T confidence interval for the mean salary of potential customers.

Assume that all necessary Central Limit Theorem conditions for a One-Sample T confidence interval have been met.

What is the 95% lower limit?

Answer 1

Construct a 95% One-Sample T confidence interval for the mean salary of potential customers.

The general formula for computing confidence interval of the mean is as follows:

with df = n-1

where

is the sample mean salary

s is the standard deviation of the mean salary

n is the sample size = 1000

From the table, we find the t statistic

From the 1000 samples from the StatCrunch Car dealership data set

Average Median salary = 38007.14 calculated using excel function =average(C2:C1001)

the standard deviation of the median salary = 12341.9253  calculated using excel function =STDEV.S(C2:C1001)

Inserting the values in the formula for CI

Thus 95% One-Sample T confidence interval for the mean salary of potential customers is 765.874. The 95% upper limit is 38773.014 and the 95% lower limit is 37241.266.

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