Questions: Describe how you would construct a 90% confidence interval to estimate the population mean age for students at your school. Choose the correct answer below. A. Take a sample of at least 30 student ages and find x̄. Verify that σ is known and the sample is random. Find the critical value z0.05 to calculate the left and find the critical value z0.95 to calculate the right endpoint of the confidence interval. B. Take a sample of at least 30 student ages and find x̄. Verify that σ is known and the sample is random. Find the critical value z0.90 and use it to find the margin of error E. Then find the left and right endpoints of the confidence interval. C. Take a sample of at least 100 student ages and find x̄. Verify that σ is known or the sample is random. Find the critical value z0.10 and use it to find the sampling error E. Then find the left and right endpoints of the confidence interval. D. Take a sample of at least two student ages and find x̄. Verify that σ is known or the sample is random. Find the critical value z0.05 to calculate the right and find the critical value z0.95 to calculate the left endpoint of the confidence interval.

Describe how you would construct a 90% confidence interval to estimate the population mean age for students at your school. Choose the correct answer below. A. Take a sample of at least 30 student ages and find x̄. Verify that σ is known and the sample is random. Find the critical value z0.05 to calculate the left and find the critical value z0.95 to calculate the right endpoint of the confidence interval. B. Take a sample of at least 30 student ages and find x̄. Verify that σ is known and the sample is random. Find the critical value z0.90 and use it to find the margin of error E. Then find the left and right endpoints of the confidence interval. C. Take a sample of at least 100 student ages and find x̄. Verify that σ is known or the sample is random. Find the critical value z0.10 and use it to find the sampling error E. Then find the left and right endpoints of the confidence interval. D. Take a sample of at least two student ages and find x̄. Verify that σ is known or the sample is random. Find the critical value z0.05 to calculate the right and find the critical value z0.95 to calculate the left endpoint of the confidence interval.
Transcript text: Describe how you would construct a $90 \%$ confidence interval to estimate the population mean age for students at your school. Choose the correct answer below. A. Take a sample of at least 30 student ages and find $\bar{x}$. Verify that $\sigma$ is known and the sample is random. Find the critical value $\mathrm{z}_{0.05}$ to calculate the left and find the critical value $\mathrm{z}_{0.95}$ to calculate the right endpoint of the confidence interval. B. Take a sample of at least 30 student ages and find $\bar{x}$. Verify that $\sigma$ is known and the sample is random. Find the critical value $\mathrm{z}_{0.90}$ and use it to find the margin of error E . Then find the left and right endpoints of the confidence interval. C. Take a sample of at least 100 student ages and find $\bar{x}$. Verify that $\sigma$ is known or the sample is randorn. Find the critical value $z_{0.10}$ and use it to find the sampling error $E$. Then find the left and right endpoints of the confidence interval. D. Take a sample of at least tho student ages and find $\bar{x}$. Verify that $\sigma$ is known or the sample is random. Find the critical value $z_{0.05}$ to calculate the right and find the critical value $z_{0.95}$ to calculate the left endpoint of the confidence interval.
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Solution

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Solution Steps

Step 1: Calculate the Sample Mean

The sample mean (\( \bar{x} \)) is calculated using the formula:

\[ \mu = \frac{\sum_{i=1}^N x_i}{N} = \frac{225}{10} = 22.5 \]

Thus, the sample mean is:

\[ \text{Sample Mean: } \bar{x} = 22.5 \]

Step 2: Determine the Critical Value

For a 90% confidence interval, the critical value \( z \) corresponding to \( \alpha = 0.10 \) (where \( \alpha \) is the significance level) is found to be:

\[ Z = 1.6449 \]

Step 3: Calculate the Margin of Error

The margin of error (\( E \)) is calculated using the formula:

\[ \text{Margin of Error} = \frac{Z \times \sigma}{\sqrt{n}} = \frac{1.6449 \times 2.5}{\sqrt{10}} \]

Calculating this gives:

\[ \text{Margin of Error} = 1.3004 \]

Step 4: Determine the Confidence Interval

The 90% confidence interval is calculated using the sample mean and the margin of error:

\[ \text{Left Endpoint} = \bar{x} - E = 22.5 - 1.3004 = 21.1996 \] \[ \text{Right Endpoint} = \bar{x} + E = 22.5 + 1.3004 = 23.8004 \]

Thus, the 90% confidence interval is:

\[ \text{90% Confidence Interval: } (21.1996, 23.8004) \]

Final Answer

The answer is:

\[ \boxed{(21.1996, 23.8004)} \]

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