Let's break down the solution step-by-step to determine the standard deviation of the sampling distribution for the difference in means \(\bar{x}_A - \bar{x}_C\) for Alex and Chris.
Given Data:
1. Alex's mean time \(\mu_A = 5.28\) minutes, standard deviation \(\sigma_A = 0.38\) minutes.
2. Chris's mean time \(\mu_C = 5.45\) minutes, standard deviation \(\sigma_C = 0.20\) minutes.
3. Sample size for Alex \(n_A = 10\).
4. Sample size for Chris \(n_C = 15\).
To find the standard deviation of the difference in sample means \(\bar{x}_A - \bar{x}_C\), we use the formula for the standard deviation of the difference between two independent sample means:
[tex]\[
\sigma_{\bar{x}_A - \bar{x}_C} = \sqrt{\left(\frac{\sigma_A^2}{n_A}\right) + \left(\frac{\sigma_C^2}{n_C}\right)}
\][/tex]
Following the steps:
1. Compute \(\frac{\sigma_A^2}{n_A}\):
[tex]\[
\frac{0.38^2}{10} = \frac{0.1444}{10} = 0.01444
\][/tex]
2. Compute \(\frac{\sigma_C^2}{n_C}\):
[tex]\[
\frac{0.20^2}{15} = \frac{0.04}{15} = 0.002667
\][/tex]
3. Add these two values together:
[tex]\[
0.01444 + 0.002667 = 0.017107
\][/tex]
4. Finally, take the square root of the sum to find the standard deviation:
[tex]\[
\sqrt{0.017107} \approx 0.130792
\][/tex]
Hence, the standard deviation of the sampling distribution for \(\bar{x}_A - \bar{x}_C\) is approximately \(0.130792\).
Given the options:
1. 0.09
2. 0.13
3. 0.17
4. 0.18
The closest value to our calculated result \(0.130792\) is \(0.13\).
Therefore, the correct answer is:
[tex]\[ \boxed{0.13} \][/tex]
