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Sagot :
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!
### 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!
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