To find \( \mu_{x} \) (the mean of the sampling distribution of the sample mean) and \( \sigma_{x} \) (the standard deviation of the sampling distribution of the sample mean) for samples of size \( n = 25 \), we can use the following formulas:
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The mean of the sampling distribution of the sample mean (\( \mu_{x} \)) is equal to the population mean (\( \mu \)): \[ \mu_{x} = \mu = 31 \]
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The standard deviation of the sampling distribution of the sample mean (\( \sigma_{x} \)) is calculated using the formula: \[ \sigma_{x} = \frac{\sigma}{\sqrt{n}} \] where \( \sigma \) is the standard deviation of the population, and \( n \) is the sample size. Plugging in the values: \[ \sigma_{x} = \frac{6}{\sqrt{25}} = \frac{6}{5} = 1.2 \]
Now we report our final results:
- \( \mu_{x} = 31 \)
- \( \sigma_{x} = 1.2 \)
Thus, rounded to one decimal place:
- \( \mu_{x} = 31.0 \)
- \( \sigma_{x} = 1.2 \)