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Answer :
To construct a 90% confidence interval for the population mean number of minutes played per game, follow these steps:
1. Define the information given:
- The data set consists of 10 observations.
- The known population standard deviation is
[tex]$$\sigma = 5.25 \text{ minutes}.$$[/tex]
- The sample mean is calculated from the data:
[tex]$$\bar{x} = 33.11 \text{ minutes}.$$[/tex]
- The number of observations is
[tex]$$n = 10.$$[/tex]
- The confidence level is [tex]$90\%$[/tex], which means the significance level is
[tex]$$\alpha = 0.10.$$[/tex]
For a two-tailed test, the critical value is found using
[tex]$$\frac{\alpha}{2} = 0.05.$$[/tex]
2. Determine the critical value:
- The [tex]$z$[/tex]-value corresponding to a cumulative probability of [tex]$1 - 0.05 = 0.95$[/tex] is
[tex]$$z = 1.6449.$$[/tex]
3. Compute the standard error of the mean:
- The standard error is given by
[tex]$$SE = \frac{\sigma}{\sqrt{n}} = \frac{5.25}{\sqrt{10}} \approx 1.6602.$$[/tex]
4. Find the margin of error:
- The margin of error is calculated by multiplying the critical value by the standard error:
[tex]$$\text{Margin of Error} = z \times SE \approx 1.6449 \times 1.6602 \approx 2.73.$$[/tex]
5. Construct the confidence interval:
- The lower limit of the interval is:
[tex]$$\bar{x} - \text{Margin of Error} \approx 33.11 - 2.73 = 30.38.$$[/tex]
- The upper limit of the interval is:
[tex]$$\bar{x} + \text{Margin of Error} \approx 33.11 + 2.73 = 35.84.$$[/tex]
- Therefore, the [tex]$90\%$[/tex] confidence interval is:
[tex]$$\left(30.38, 35.84\right).$$[/tex]
6. Interpretation:
- We are [tex]$90\%$[/tex] confident that the population mean number of minutes played per game for professional basketball players lies between [tex]$30.38$[/tex] minutes and [tex]$35.84$[/tex] minutes.
Thus, the final answer is:
[tex]$$\text{90\% Confidence Interval: } (30.38, 35.84).$$[/tex]
And the interpretation is:
"We are 90% confident that the population mean number of minutes played per game for professional basketball players is between 30.38 and 35.84 minutes."
1. Define the information given:
- The data set consists of 10 observations.
- The known population standard deviation is
[tex]$$\sigma = 5.25 \text{ minutes}.$$[/tex]
- The sample mean is calculated from the data:
[tex]$$\bar{x} = 33.11 \text{ minutes}.$$[/tex]
- The number of observations is
[tex]$$n = 10.$$[/tex]
- The confidence level is [tex]$90\%$[/tex], which means the significance level is
[tex]$$\alpha = 0.10.$$[/tex]
For a two-tailed test, the critical value is found using
[tex]$$\frac{\alpha}{2} = 0.05.$$[/tex]
2. Determine the critical value:
- The [tex]$z$[/tex]-value corresponding to a cumulative probability of [tex]$1 - 0.05 = 0.95$[/tex] is
[tex]$$z = 1.6449.$$[/tex]
3. Compute the standard error of the mean:
- The standard error is given by
[tex]$$SE = \frac{\sigma}{\sqrt{n}} = \frac{5.25}{\sqrt{10}} \approx 1.6602.$$[/tex]
4. Find the margin of error:
- The margin of error is calculated by multiplying the critical value by the standard error:
[tex]$$\text{Margin of Error} = z \times SE \approx 1.6449 \times 1.6602 \approx 2.73.$$[/tex]
5. Construct the confidence interval:
- The lower limit of the interval is:
[tex]$$\bar{x} - \text{Margin of Error} \approx 33.11 - 2.73 = 30.38.$$[/tex]
- The upper limit of the interval is:
[tex]$$\bar{x} + \text{Margin of Error} \approx 33.11 + 2.73 = 35.84.$$[/tex]
- Therefore, the [tex]$90\%$[/tex] confidence interval is:
[tex]$$\left(30.38, 35.84\right).$$[/tex]
6. Interpretation:
- We are [tex]$90\%$[/tex] confident that the population mean number of minutes played per game for professional basketball players lies between [tex]$30.38$[/tex] minutes and [tex]$35.84$[/tex] minutes.
Thus, the final answer is:
[tex]$$\text{90\% Confidence Interval: } (30.38, 35.84).$$[/tex]
And the interpretation is:
"We are 90% confident that the population mean number of minutes played per game for professional basketball players is between 30.38 and 35.84 minutes."
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