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:
\[ 34.1 \text{ to } 45.9 \]
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