A study of the career paths of hotel general managers sent questionnaires to an SRS of 160 hotels belonging to major U.S. hotel chains. There were 114 responses. The average time these 114 general managers had spent with their current company was 11.78 years. Give a 99%; confidence interval for the mean number of years general managers of major-chain hotels have spent with their current company. (Take it as known that the standard deviation of time with the company for all general managers is 3.2 years.)

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25-01-2021
Mathematics

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A study of the career paths of hotel general managers sent questionnaires to an SRS of 160 hotels belonging to major U.S. hotel chains. There were 114 responses. The average time these 114 general managers had spent with their current company was 11.78 years. Give a 99%; confidence interval for the mean number of years general managers of major-chain hotels have spent with their current company. (Take it as known that the standard deviation of time with the company for all general managers is 3.2 years.)

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joaobezerra joaobezerra · Answer 1 · 2021-01-26T01:44:15+01:00

Answer:

The 99% confidence interval for the mean number of years general managers of major-chain hotels have spent with their current company is between 11.01 years and 12.55 years.

Step-by-step explanation:

We have that to find our [tex]\alpha[/tex] level, that is the subtraction of 1 by the confidence interval divided by 2. So:

[tex]\alpha = \frac{1-0.99}{2} = 0.005[/tex]

Now, we have to find z in the Ztable as such z has a pvalue of [tex]1-\alpha[/tex].

So it is z with a pvalue of [tex]1-0.005 = 0.995[/tex], so [tex]z = 2.575[/tex]

Now, find the margin of error M as such

[tex]M = z*\frac{\sigma}{\sqrt{n}}[/tex]

In which [tex]\sigma[/tex] is the standard deviation of the population and n is the size of the sample.

[tex]M = 2.575*\frac{3.2}{\sqrt{114}} = 0.77[/tex]

The lower end of the interval is the sample mean subtracted by M. So it is 11.78 - 0.77 = 11.01 years

The upper end of the interval is the sample mean added to M. So it is 11.78 + 0.77 = 12.55 years

The 99% confidence interval for the mean number of years general managers of major-chain hotels have spent with their current company is between 11.01 years and 12.55 years.