To construct a 95% confidence interval for a sample mean (\(\bar{x}\)), we will use the formula for the confidence interval when the population standard deviation is unknown. The formula is:
\[ \bar{x} \pm t^* \left(\frac{s}{\sqrt{n}}\right) \]
where:
- \(\bar{x}\) = sample mean
- \(s\) = sample standard deviation
- \(n\) = sample size
- \(t^*\) = critical t-value that corresponds to the desired confidence level and degrees of freedom (\(df = n - 1\))
Step 1: Identify the parameters
Given:
- \(n = 28\)
- \(\bar{x} = 32\)
- \(s = 4\)
Step 2: Calculate the degrees of freedom
\[ df = n - 1 = 28 - 1 = 27 \]
Step 3: Find the critical t-value
For a 95% confidence level and \(df = 27\), we look up the critical t-value in a t-distribution table or use a calculator:
Using a t-table or calculator, the critical t-value (\(t^*\)) for \(df = 27\) at a 95% confidence level is approximately:
\[ t^* \approx 2.051 \]
Step 4: Calculate the standard error (SE)
The standard error (SE) is calculated as:
\[ SE = \frac{s}{\sqrt{n}} = \frac{4}{\sqrt{28}} \approx \frac{4}{5.2915} \approx 0.756 \]
Step 5: Calculate the margin of error
Now, calculate the margin of error (ME):
\[ ME = t^* \cdot SE = 2.051 \cdot 0.756 \approx 1.550 \]
Step 6: Construct the confidence interval
Now, we can construct the confidence interval:
\[ \bar{x} - ME \text{ to } \bar{x} + ME \]
Calculating the lower and upper bounds:
\[ 32 - 1.550 \approx 30.5 \] \[ 32 + 1.550 \approx 33.6 \]
Final Confidence Interval
Thus, the 95% confidence interval for the mean is:
\[ (30.5, 33.6) \]
The final confidence interval rounded to one decimal place is:
\[ \boxed{(30.5, 33.6)} \]
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