Question

sample of size n=38 has sample mean x=58 and sample standard deviation s=7.6. (a) Construct a 99% confidence interval for the population mean u. Enter the values for the lower and upper limits and the mean to graph. Round the answers to one decimal place.

Answer 1

To construct a 99% confidence interval for the population mean \( u \) based on the sample data provided, we can use the following formula for the confidence interval:

\[
\text{Confidence Interval} = \bar{x} \pm z^* \left( \frac{s}{\sqrt{n}} \right)
\]

Where:
- \( \bar{x} \) = sample mean
- \( z^* \) = z-value corresponding to the desired confidence level
- \( s \) = sample standard deviation
- \( n \) = sample size

### Step 1: Determine the z-value for a 99% confidence level
For a 99% confidence level, the z-value (critical value) can be found using z-tables or calculators. The z-value for a 99% confidence interval is approximately:

\[
z^* \approx 2.576
\]

### Step 2: Calculate the margin of error
Now, plug in the values into the formula for the margin of error:

\[
\text{Margin of Error} = z^* \left( \frac{s}{\sqrt{n}} \right)
\]

Substituting the known values:

\[
\text{Margin of Error} = 2.576 \left( \frac{7.6}{\sqrt{38}} \right)
\]

Calculating \( \frac{s}{\sqrt{n}} \):

\[
\frac{7.6}{\sqrt{38}} \approx \frac{7.6}{6.1644} \approx 1.230
\]

Now calculate the Margin of Error:

\[
\text{Margin of Error} \approx 2.576 \times 1.230 \approx 3.17
\]

### Step 3: Construct the confidence interval
The lower and upper limits of the confidence interval can be calculated as follows:

\[
\text{Lower limit} = \bar{x} - \text{Margin of Error} = 58 - 3.17 \approx 54.8
\]
\[
\text{Upper limit} = \bar{x} + \text{Margin of Error} = 58 + 3.17 \approx 61.2
\]

### Final Results
The 99% confidence interval for the population mean \( u \) is:

\[
(54.8, 61.2)
\]

Thus, the values you should enter for the lower limit, upper limit, and mean are:

- Lower limit: **54.8**
- Upper limit: **61.2**
- Mean: **58**

All values rounded to one decimal place.