When drawing independent random samples from two normal populations, what is the distribution of the difference between the sample means $\mu =x_1-x_2$?

• A. It follows a chi-square distribution.
• B. It is normally distributed.
• C. It follows a uniform distribution.
• D. It is always skewed to the right.

Answer: (B)

Verified step by step guidance

1. Identify the populations and samples: We have two independent random samples drawn from two normal populations. Let the sample means be \( \overline{x}_1 \) and \( \overline{x}_2 \).
2. Recall the distribution of sample means: Since each population is normal, each sample mean \( \overline{x}_1 \) and \( \overline{x}_2 \) is also normally distributed due to the properties of the normal distribution.
3. Consider the difference of the sample means: The difference \( \mu = \overline{x}_1 - \overline{x}_2 \) is a linear combination of two independent normal random variables.
4. Use the property of normal distributions: The difference of two independent normal random variables is also normally distributed. Therefore, \( \mu \) follows a normal distribution.
5. Summarize the result: The distribution of \( \overline{x}_1 - \overline{x}_2 \) is normal, with mean equal to the difference of the population means and variance equal to the sum of the variances of the sample means.