Question

If n=25,
¯
x
=40, and s=15, construct a confidence interval at a 95% confidence level. Assume the data came from a normally distributed population.

Give your answers to one decimal place.

Answer 1

To construct a confidence interval for the mean when the sample size is known, we can use the formula for the confidence interval:

\[
\text{CI} = \bar{x} \pm z \left(\frac{s}{\sqrt{n}}\right)
\]

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

### Given Values:
- \(n = 25\)
- \(\bar{x} = 40\)
- \(s = 15\)

### Step 1: Determine the z-value for a 95% confidence level
For a 95% confidence level, the z-value (cutoff for the standard normal distribution) is typically 1.96.

### Step 2: Calculate the standard error (SE)
The standard error (SE) is calculated as follows:

\[
SE = \frac{s}{\sqrt{n}} = \frac{15}{\sqrt{25}} = \frac{15}{5} = 3
\]

### Step 3: Calculate the margin of error (ME)
Now, we calculate the margin of error (ME):

\[
ME = z \times SE = 1.96 \times 3 = 5.88
\]

### Step 4: Construct the confidence interval
Now we can construct the confidence interval:

\[
\text{Lower limit} = \bar{x} - ME = 40 - 5.88 = 34.12
\]
\[
\text{Upper limit} = \bar{x} + ME = 40 + 5.88 = 45.88
\]

### Final Confidence Interval
The 95% confidence interval for the mean is:

\[
(34.1, 45.9)
\]

Thus, the answer to one decimal place is:

\[
34.1 \text{ to } 45.9
\]