Explanation

Since the population variance is unknown and the population is normally distributed, and the sample size is less than 30, we use a t-statistic. The t-statistic for a 95% confidence interval and 27 degrees of freedom (df = n-1 = 28-1 = 27) is 2.052.

Step 1: Calculate the Standard Error

$$\text{Standard Error} = \frac{\text{Sample Standard Deviation}}{\sqrt{\text{Sample Size}}} = \frac{14}{\sqrt{28}} = \frac{14}{5.2915} \approx 2.646$$

Step 2: Calculate the Margin of Error

$$\text{Margin of Error} = t_{\alpha/2, df} \times \text{Standard Error} = 2.052 \times 2.646 \approx 5.43$$

Step 3: Calculate the Confidence Interval

$$\text{Lower Bound} = \text{Sample Mean} - \text{Margin of Error} = 7.5 - 5.43 = 2.07\%$$ $$\text{Upper Bound} = \text{Sample Mean} + \text{Margin of Error} = 7.5 + 5.43 = 12.93\%$$

Therefore, the 95% confidence interval is [2.07%; 12.93%].

Key Points:

• t-distribution is used when population variance is unknown and sample size is small (n ≤ 30)
• Degrees of freedom = n - 1 = 27
• t-critical value for 95% confidence with 27 df is 2.052
• Using z-statistic instead would give incorrect results for small samples with unknown population variance