Answered

The deadline passes, and students will have the work automatically graded and the scores transferred to the ebook in Canvas.

Assignment Scoring
Your last submission is used for your score.
[9 / 20 Points]

---

BBUNDERSTAT12 7.1.023

Previous Answers
Practice Another

---

One way to examine how banks rate is to look at annual profits per employee. The following data represents annual profits per employee (in units of 1 thousand dollars per employee) for various companies in financial services. Assume [tex]$\sigma=9.8$[/tex] thousand dollars.

\begin{tabular}{lllllllllll}
46.0 & 50.9 & 33.4 & 45.3 & 30.6 & 33.4 & 32.0 & 34.8 & 42.5 & 33.0 & 33.6 \\
36.9 & 27.0 & 47.1 & 33.8 & 28.1 & 28.5 & 29.1 & 36.5 & 36.1 & 26.9 & 27.8 \\
28.8 & 29.3 & 31.5 & 31.7 & 31.1 & 38.0 & 32.0 & 31.7 & 32.9 & 23.1 & 54.9 \\
43.8 & 36.9 & 31.9 & 25.5 & 23.2 & 29.8 & 22.3 & 26.5 & 26.7 & &
\end{tabular}

(a) Use a calculator or appropriate computer software to find [tex]$\bar{x}$[/tex] for the preceding data. (Round your answer to two decimal places.)
[tex]$\square$[/tex] thousand dollars

(b) Let us say that the preceding data are representative of the entire sector of (successful) financial services corporations. Find a [tex]$75 \%$[/tex] confidence interval for [tex]$\mu$[/tex], the average annual profit per employee for all successful banks. (Round your answers to two decimal places.)
Lower limit: [tex]$\square$[/tex] thousand dollars
Upper limit: [tex]$\square$[/tex] thousand dollars

(c) Let us say that you are the manager of a local bank with a large number of employees. Suppose the annual profits per employee are less...



Answer :

GinnyAnswer
Sure, let's break down the solution to this problem step-by-step.

### Part (a): Finding [tex]\(\bar{x}\)[/tex]

First, let's calculate the mean of the data set. The data points in units of thousands of dollars per employee are:

[tex]\[ 46.0, 50.9, 33.4, 45.3, 30.6, 33.4, 32.0, 34.8, 42.5, 33.0, 33.6, 36.9, 27.0, 47.1, 33.8, 28.1, 28.5, 29.1, 36.5, 36.1, 26.9, 27.8, 28.8, 29.3, 31.5, 31.7, 31.1, 38.0, 32.0, 31.7, 32.9, 23.1, 54.9, 43.8, 36.9, 31.9, 25.5, 23.2, 29.8, 22.3, 26.5, 26.7 \][/tex]

When you find the average of these values, it results in:
[tex]\[ \bar{x} = 33.45 \, \text{thousand dollars} \][/tex]
This is the mean annual profit per employee.

### Part (b): 75% Confidence Interval for [tex]\(\mu\)[/tex]

To find the 75% confidence interval for the mean, we follow these steps:

1. Mean ([tex]\(\bar{x}\)[/tex]): From part (a), we know [tex]\(\bar{x} = 33.45\)[/tex] thousand dollars.
2. Standard deviation ([tex]\(\sigma\)[/tex]): Given as 9.8 thousand dollars.
3. Sample size (n): The total number of data points is 41.

To find the standard error ([tex]\(SE\)[/tex]), we use the formula:
[tex]\[ SE = \frac{\sigma}{\sqrt{n}} \][/tex]

Now, we need the z-value corresponding to a 75% confidence level. For a 75% confidence interval, we find:
[tex]\[ z = 1.15 \][/tex] (approximately, since we're using the standard normal distribution)

Next, calculate the margin of error:
[tex]\[ \text{Margin of Error} = z \times SE \][/tex]

Finally, calculate the lower and upper bounds of the confidence interval:
[tex]\[ \text{Lower Limit} = \bar{x} - \text{Margin of Error} \][/tex]
[tex]\[ \text{Upper Limit} = \bar{x} + \text{Margin of Error} \][/tex]

Plugging in the values, the confidence interval is:
[tex]\[ \text{Lower Limit} = 31.71 \, \text{thousand dollars} \][/tex]
[tex]\[ \text{Upper Limit} = 35.19 \, \text{thousand dollars} \][/tex]

### Summary

1. Mean [tex]\(\bar{x}\)[/tex]:
[tex]\[ \bar{x} = 33.45 \, \text{thousand dollars} \][/tex]

2. 75% Confidence Interval:
- Lower Limit:
[tex]\[ 31.71 \, \text{thousand dollars} \][/tex]
- Upper Limit:
[tex]\[ 35.19 \, \text{thousand dollars} \][/tex]

I hope this step-by-step explanation clarifies how we arrived at the mean and the 75% confidence interval for the data!