Questions: A sample of size n=37 has sample mean x̄=58 and sample standard deviation s=9.2. Part 1 of 2 (a) Construct a 98% confidence interval for the population mean μ. Enter the values for the lower and upper limits and the mean to graph. Round the answers to one decimal place.

A sample of size n=37 has sample mean x̄=58 and sample standard deviation s=9.2.

Part 1 of 2
(a) Construct a 98% confidence interval for the population mean μ. Enter the values for the lower and upper limits and the mean to graph. Round the answers to one decimal place.

Transcript text: A sample of size $n=37$ has sample mean $\bar{x}=58$ and sample standard deviation $s=9.2$. Part 1 of 2 (a) Construct a $98 \%$ confidence interval for the population mean $\mu$. Enter the values for the lower and upper limits and the mean to graph. Round the answers to one decimal place.

Solution

Solution Steps

Step 1: Calculate the Z Critical Value

To construct a $ 98\% $ confidence interval for the population mean $ \mu $, we first need to determine the Z critical value. The formula for the Z critical value is given by:

\[ Z = \Phi^{-1}\left(1 - \frac{\alpha}{2}\right) \]

where $ \alpha = 1 - 0.98 = 0.02 $. Thus, we find:

\[ Z \approx 2.3263 \]

Step 2: Calculate the Margin of Error

Next, we calculate the margin of error $ E $ using the formula:

\[ E = Z \cdot \frac{s}{\sqrt{n}} \]

where $ s = 9.2 $ (sample standard deviation) and $ n = 37 $ (sample size). Substituting the values, we get:

\[ E \approx 2.3263 \cdot \frac{9.2}{\sqrt{37}} \approx 3.5185 \]

Step 3: Construct the Confidence Interval

Now, we can construct the $ 98\% $ confidence interval for the population mean $ \mu $ using the sample mean $ \bar{x} = 58 $:

\[ \text{Lower Limit} = \bar{x} - E \approx 58 - 3.5185 \approx 54.5 \] \[ \text{Upper Limit} = \bar{x} + E \approx 58 + 3.5185 \approx 61.5 \]

Thus, the $ 98\% $ confidence interval is: