Questions: The commute times for the workers in a city are normally distributed with an unknown population mean and standard deviation. If a random sample of 27 workers is taken and results in a sample mean of 22 minutes and sample deviation of 3 minutes, find a 95% confidence interval estimate for the population mean using the Student's t-distribution.

Solution Steps

We are given the following information about the commute times for workers in a city:

• Sample size (\(n\)) = 27
• Sample mean (\(\bar{x}\)) = 22 minutes
• Sample standard deviation (\(s\)) = 3 minutes
• Confidence level = 95%

For a 95% confidence interval with \(n - 1 = 26\) degrees of freedom, we refer to the t-distribution table. The t-value corresponding to \(df = 26\) at a 95% confidence level is approximately \(t = 2.06\).

The standard error (SE) of the sample mean is calculated using the formula: \[ SE = \frac{s}{\sqrt{n}} = \frac{3}{\sqrt{27}} \approx 0.5774 \]

The margin of error (ME) is calculated as: \[ ME = t \cdot SE = 2.06 \cdot 0.5774 \approx 1.188 \]

The confidence interval is given by: \[ \bar{x} \pm ME = 22 \pm 1.188 \] Calculating the lower and upper bounds:

• Lower bound: \(22 - 1.188 \approx 20.812\)
• Upper bound: \(22 + 1.188 \approx 23.188\)

Thus, the confidence interval is approximately: \[ (20.81, 23.19) \]

Final Answer