Questions: In a random sample of 26 people, the mean commute time to work was 32.6 minutes and the standard deviation was 7.2 minutes. Assume the population is normally distributed and use a t-distribution to construct a 95% confidence interval for the population mean μ. What is the margin of error of μ? Interpret the results. The confidence interval for the population mean μ is ) (Round to one decimal place as needed.)

Solution Steps

To calculate the margin of error \( E \) for the population mean, we use the formula:

\[ E = Z \cdot \frac{\sigma}{\sqrt{n}} \]

where:

• \( Z = 1.96 \) (Z-score for a 95% confidence level),
• \( \sigma = 7.2 \) (sample standard deviation),
• \( n = 26 \) (sample size).

Substituting the values:

\[ E = 1.96 \cdot \frac{7.2}{\sqrt{26}} \approx 2.7675 \]

Thus, the margin of error is:

\[ \text{Margin of Error} \approx 2.7675 \]

The confidence interval for the population mean \( \mu \) is calculated using the formula:

\[ \bar{x} \pm t \cdot \frac{s}{\sqrt{n}} \]

where:

• \( \bar{x} = 32.6 \) (sample mean),
• \( t \approx 2.1 \) (t-score for a 95% confidence level with \( n-1 = 25 \) degrees of freedom),
• \( s = 7.2 \) (sample standard deviation),
• \( n = 26 \) (sample size).

Calculating the confidence interval:

\[ 32.6 \pm 2.1 \cdot \frac{7.2}{\sqrt{26}} \]

Calculating the lower and upper bounds:

\[ \text{Lower Bound} = 32.6 - 2.1 \cdot \frac{7.2}{\sqrt{26}} \approx 29.7 \] \[ \text{Upper Bound} = 32.6 + 2.1 \cdot \frac{7.2}{\sqrt{26}} \approx 35.5 \]

Thus, the confidence interval is:

\[ \text{Confidence Interval} \approx (29.7, 35.5) \]

The interpretation of the results is as follows:

The 95% confidence interval for the population mean commute time is between \( 29.7 \) and \( 35.5 \) minutes. This means we are 95% confident that the true mean commute time for the population falls within this interval.

Final Answer