A survey of 66 randomly selected students finds that they spend a mean of per semester. Assume that
the population standard deviation is per month.
Confidence Interval: What is the confidence interval to estimate the population mean? Round your
answers to two decimal places.

Ask by Peters Mccarthy. in the United States

Mar 14,2025

Question

A survey of 66 randomly selected students finds that they spend a mean of per semester. Assume that
the population standard deviation is per month.
Confidence Interval: What is the confidence interval to estimate the population mean? Round your
answers to two decimal places.

Ask by Peters Mccarthy. in the United States

Mar 14,2025

UpStudy AI · Accepted Answer

**Step 1. Identify the given values**

We are given:
- Sample size: $$ n = 66 $$
- Sample mean: $$ \bar{x} = 395 $$
- Population standard deviation: $$ \sigma = 54 $$
- Confidence level: $$ 90\% $$ which implies a critical value $$ z_{\alpha/2} $$ corresponding to $$ \alpha = 0.10 $$ (i.e. $$ \alpha/2 = 0.05 $$). The critical value is $$ z_{0.05} \approx 1.645 $$.

**Step 2. Compute the standard error of the mean**

The standard error (SE) is given by:
$$
\text{SE} = \frac{\sigma}{\sqrt{n}} = \frac{54}{\sqrt{66}}
$$
Calculating:
$$
\sqrt{66} \approx 8.124
$$
$$
\text{SE} \approx \frac{54}{8.124} \approx 6.646
$$

**Step 3. Compute the margin of error**

The margin of error (ME) is calculated as:
$$
\text{ME} = z_{\alpha/2} \times \text{SE} = 1.645 \times 6.646 \approx 10.935
$$

**Step 4. Construct the confidence interval**

The $$ 90\% $$ confidence interval for the population mean is:
$$
\bar{x} \pm \text{ME}
$$
Thus, the lower limit is:
$$
395 - 10.935 \approx 384.07
$$
and the upper limit is:
$$
395 + 10.935 \approx 405.94
$$

**Step 5. Write the final answer**

The $$ 90\% $$ confidence interval to estimate the population mean is:
$$
(384.07, \, 405.94)
$$
The 90% confidence interval for the population mean is (384.07, 405.94).

A survey of 66 randomly selected students finds that they spend a mean of per semester. Assume that
the population standard deviation is per month.
Confidence Interval: What is the confidence interval to estimate the population mean? Round your
answers to two decimal places.

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A survey of 66 randomly selected students finds that they spend a mean of per semester. Assume that the population standard deviation is per month. Confidence Interval: What is the confidence interval to estimate the population mean? Round your answers to two decimal places.

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A survey of 66 randomly selected students finds that they spend a mean of per semester. Assume that
the population standard deviation is per month.
Confidence Interval: What is the confidence interval to estimate the population mean? Round your
answers to two decimal places.