Questions: You are given the sample mean and the population standard deviation. Use this information to construct the 90% and 95% confidence intervals for the population mean. Interpret the results and compare the widths of the confidence intervals. From a random sample of 31 business days, the mean closing price of a certain stock was 117.80. Assume the population standard deviation is 11.31. The 90% confidence interval is ( ). (Round to two decimal places as needed.)

You are given the sample mean and the population standard deviation. Use this information to construct the 90% and 95% confidence intervals for the population mean. Interpret the results and compare the widths of the confidence intervals.
From a random sample of 31 business days, the mean closing price of a certain stock was 117.80. Assume the population standard deviation is 11.31.

The 90% confidence interval is ( ).
(Round to two decimal places as needed.)

Transcript text: You are given the sample mean and the population standard deviation. Use this information to construct the $90 \%$ and $95 \%$ confidence intervals for the population mean. Interpret the results and compare the widths of the confidence intervals. From a random sample of 31 business days, the mean closing price of a certain stock was $\$ 117.80$. Assume the population standard deviation is $\$ 11.31$. The $90 \%$ confidence interval is ( $\square$ $\square$ ). (Round to two decimal places as needed.)

Solution

Solution Steps

Step 1: Calculate the Standard Error (SE)

The standard error (SE) is calculated using the formula $SE = \frac{\sigma}{\sqrt{n}}$, where $\sigma = 11.31$ and $n = 31$. Thus, $SE = \frac{11.31}{\sqrt{31}} = 2.03$.

Step 2: Determine the Z-scores

For a 90% confidence interval, the Z-score is approximately 1.645.

Step 3: Calculate the Margin of Error (ME)

The margin of error (ME) is computed as $ME = Z_{\alpha/2} \times SE = 1.645 \times 2.03 = 3.34$.

Step 4: Construct the Confidence Intervals

The confidence interval is calculated using the formula $CI = \bar{x} \pm ME$, where $\bar{x} = 117.8$. This gives the lower and upper bounds of the confidence interval as $CI = [114.46, 121.14]$.

Final Answer:

The 90% confidence interval for the population mean is [114.46, 121.14].

Step 1: Calculate the Standard Error (SE)

The standard error (SE) is calculated using the formula $SE = \frac{\sigma}{\sqrt{n}}$, where $\sigma = 11.31$ and $n = 31$. Thus, $SE = \frac{11.31}{\sqrt{31}} = 2.03$.

Step 2: Determine the Z-scores

For a 95% confidence interval, the Z-score is approximately 1.96.

Step 3: Calculate the Margin of Error (ME)

The margin of error (ME) is computed as $ME = Z_{\alpha/2} \times SE = 1.96 \times 2.03 = 3.98$.

Step 4: Construct the Confidence Intervals

The confidence interval is calculated using the formula $CI = \bar{x} \pm ME$, where $\bar{x} = 117.8$. This gives the lower and upper bounds of the confidence interval as $CI = [113.82, 121.78]$.

Final Answer:

The 95% confidence interval for the population mean is [113.82, 121.78].

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