To determine the distribution of the sample mean for a sample of size 100 drawn from a non-normal population with a known mean and standard deviation, we can use the Central Limit Theorem (CLT). The CLT states that, for sufficiently large sample sizes (generally n ≥ 30 is considered sufficient), the distribution of the sample mean will be approximately normal, regardless of the population's distribution.
Given:
- Population mean (μ) = 40
- Population standard deviation (σ) = 25
- Sample size (n) = 100
According to the Central Limit Theorem:
- The mean of the sample mean (μ_x̄) will equal the population mean (μ), which is 40.
- The standard deviation of the sample mean (standard error, σ_x̄) will equal the population standard deviation divided by the square root of the sample size (n):
\[ \sigma_x̄ = \frac{\sigma}{\sqrt{n}} = \frac{25}{\sqrt{100}} = \frac{25}{10} = 2.5 \]
Thus, the distribution of the sample mean will be approximately normal with:
- Mean = 40
- Standard deviation = 2.5
The correct choice based on this information is:
approximately normal with mean 40 and standard deviation 2.5
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