Question

Consider two populations for which μ1 = 31, σ1 = 3, μ2 = 29, and σ2 = 2. Suppose that two independent random samples of sizes n1 = 4 and n2 = 53 are selected. Describe the approximate sampling distribution of x1 − x2 (center, spread, and shape).a. What is the mean of the distribution?b. What is the standard deviation of the distribution?

Answer 1

Answer:

Part (a): The mean of the distribution is 2.

Part (b): The standard deviation of the distribution is 1.525

Step-by-step explanation:

Consider the provided information.

Part (a) What is the mean of the distribution?

The mean of the sampling distribution of $(\bar x_1 -\bar x_2)$ is (µ₁ − µ₂).

It is given that μ₁ = 31, σ₁ = 3, μ₂ = 29, and σ₂ = 2.

Mean of distribution = (x₁ - x₂)

Mean of distribution = (31 - 29) = 2

Hence, the mean of the distribution is 2.

Part (b) What is the standard deviation of the distribution?

If the two samples are independent, the standard deviation of the sampling distribution is: $\sigma_{(x_1-x_2)}=\sqrt{\left(\dfrac{\sigma^2_1}{n_1}+\dfrac{\sigma^2_2}{n_2}\right)}}$

Substitute the respective values in the above formula.

$\sigma_{(x_1-x_2)}=\sqrt{\left(\dfrac{3^2}{4}+\dfrac{2^2}{53}\right)}}$

$\sigma_{(x_1-x_2)}=\sqrt{\left(\dfrac{9}{4}+\dfrac{4}{53}\right)}}$

$\sigma_{(x_1-x_2)}\approx1.525$

Hence, the standard deviation of the distribution is 1.525