Thank you for visiting The following data set represents the average number of minutes played for a random sample of professional basketball players in a recent season 35 9. This page is designed to guide you through key points and clear explanations related to the topic at hand. We aim to make your learning experience smooth, insightful, and informative. Dive in and discover the answers you're looking for!
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."
Thank you for reading the article The following data set represents the average number of minutes played for a random sample of professional basketball players in a recent season 35 9. We hope the information provided is useful and helps you understand this topic better. Feel free to explore more helpful content on our website!
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Rewritten by : Jeany
