To find the mean (μ_x) and standard deviation (σ_x) of the sampling distribution of the sample mean for samples of size \( n = 25 \), we use the following formulas:
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The mean of the sample means is the same as the population mean: \[ \mu_x = \mu = 23 \]
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The standard deviation of the sample means (also known as the standard error) is calculated using the formula: \[ \sigma_x = \frac{\sigma}{\sqrt{n}} \] where \( \sigma \) is the population standard deviation and \( n \) is the sample size.
In this case:
- \( \sigma = 3 \)
- \( n = 25 \)
Substituting the values: \[ \sigma_x = \frac{3}{\sqrt{25}} = \frac{3}{5} = 0.6 \]
Therefore, the results are:
- \( \mu_x = 23 \)
- \( \sigma_x = 0.6 \)
Thus, the answers are:
- \( \mu - x = 23 \)
- \( \sigma - x = 0.6 \)