To determine the standard deviation of the sampling distribution for [tex]\(\bar{x}_v - \bar{x}_c\)[/tex], we must consider the properties of the means and standard deviations for each sample. Here's how you can approach this problem step-by-step:
1. Identify the parameters:
- For Virginia:
- Mean [tex]\( \mu_V = 59 \)[/tex]
- Standard deviation [tex]\( \sigma_V = 10 \)[/tex]
- Sample size [tex]\( n_V = 10 \)[/tex]
- For California:
- Mean [tex]\( \mu_C = 64 \)[/tex]
- Standard deviation [tex]\( \sigma_C = 12 \)[/tex]
- Sample size [tex]\( n_C = 10 \)[/tex]
2. Understand the formula:
The standard deviation of the sampling distribution of the difference between two means, [tex]\(\bar{x}_v - \bar{x}_c\)[/tex], is found using the formula:
[tex]\[
\sigma_{\bar{x}_v - \bar{x}_c} = \sqrt{\left(\frac{\sigma_V^2}{n_V}\right) + \left(\frac{\sigma_C^2}{n_C}\right)}
\][/tex]
3. Substitute the values:
- Compute the variance for each state:
- For Virginia: [tex]\(\sigma_V^2 = (10)^2 = 100\)[/tex]
- For California: [tex]\(\sigma_C^2 = (12)^2 = 144\)[/tex]
- Now, calculate the standard error for each sample:
- For Virginia: [tex]\(\frac{\sigma_V^2}{n_V} = \frac{100}{10} = 10\)[/tex]
- For California: [tex]\(\frac{\sigma_C^2}{n_C} = \frac{144}{10} = 14.4\)[/tex]
4. Sum the variances and take the square root:
[tex]\[
\sigma_{\bar{x}_v - \bar{x}_c} = \sqrt{10 + 14.4} = \sqrt{24.4}
\][/tex]
5. Calculate the result:
[tex]\[
\sigma_{\bar{x}_v - \bar{x}_c} = \sqrt{24.4} \approx 4.9396356140913875
\][/tex]
Hence, the standard deviation of the sampling distribution for [tex]\(\bar{x}_v - \bar{x}_c\)[/tex] is approximately [tex]\(4.9\)[/tex].
Considering the options provided:
1. 6
2. 2
3. 4.9
4. 7.0
The correct answer is:
[tex]\[ \boxed{4.9} \][/tex]
