Question

A study of the career paths of hotel general managers sent questionnaires to an SRS of 300 hotels belonging to major U.S. hotel chains. There were 187 responses. The average time these 187 general managers had spent with their current company was 9.93 years. (Take it as known that the standard deviation of time with the company for all general managers is 3.5 years.) (a) Find the margin of error for an 85% confidence interval to estimate the mean time a general manager had spent with their current company: 9.3109 years (b) Find the margin of error for a 99% confidence interval to estimate the mean time a general manager had spent with their current company: years (c) In general, increasing the confidence level the margin of error (width) of the confidence interval. (Enter: ''DECREASES'', ''DOES NOT CHANGE'' or ''INCREASES'', without the quotes.)

Answer 1

Out of N = 187 sample size readings, the average time these 187 general managers had spent with their current company was M = 9.93 years. (Take it as known that the standard deviation of time with the company for all general managers is = 3.5 years.

a) Since the sample size is greater than 30 and the population standard deviation is known hence Z-statistic is applicable for the Margin of error calculation.

The Error is calculated as:

where Zc is calculated at given confidence level here the confidence level id 0.85 hence using excel formula for normal distribution the Zc is calculated using formula =NORM.S.INV(0.925) which results in Zc = 1.44

b) Again at 99% or 0.99, the margin of error is calculated as:

Zc at 0.99 confidence level is computed using excel formula =NORM.S.INV(0.995) which results in Zc = 2.58

c) So, as we can see that increasing the confidence level INCREASES the width of the margin of error because we can see that Zc is directly proportional to the margin of error and Zc increases as the confidence level increases.