Question

Question:A random sample of 40 observations is to be drawn from a large population of measurements. It is known that 30% of the measurements in the population are 1s, 20% are 2s, 20% are 3s, and 30% are 4s.

a. Give the mean and standard deviation of the (repeated) sampling distribution of$\overline{x}$, the sample mean of the 40 observations.

b. Describe the shape of the sampling distribution of$\overline{x}$. Does youranswer depend on the sample size?

Short answer

1. The mean and standard deviation of$\overline{x}$ are 2.5 and 0.19.
2. The shape of the sampling distribution of$\overline{x}$ is normal.

Step-by-step solution

Given Information

The sample size is 40.

According to the question, the probability distribution of x is given by

(a) Compute the mean and standard deviation

$$\begin{gathered} \backslash \frac{\sigma }{\sqrt{n}}=\frac{1.20}{\sqrt{40}} \\ =0.19 \end{gathered}$$The mean is computed as

$$\begin{gathered} E\left(X\right)=1\times 0.3+2\times 0.2+3\times 0.2+4\times 0.3 \\ =0.3+0.4+0.6+1.2 \\ =2.5 \end{gathered}$$

The standard deviation is computed as

$$\begin{gathered} d=\sqrt{\sum {\left(x-E\left(x\right)\right)}^{2}p\left(x\right)} \\ =\sqrt{{\left(1-2.5\right)}^2\times 0.3+{\left(2-2.5\right)}^2\times 0.2+{\left(3-2.5\right)}^2\times 0.2+{\left(4-2.5\right)}^2\times 0.3} \\ =\sqrt{1.45} \\ =1.20 \end{gathered}$$

The standard deviation of $\overline{x}$is

$$\begin{gathered} \frac{\sigma }{\sqrt{n}}=\frac{1.20}{\sqrt{40}} \\ =0.19 \end{gathered}$$

Therefore, the mean and standard deviation of $\overline{x}$are 2.5 and 0.19.

(b) Shape of the distribution of x¯

If n is large, then the distribution of $\overline{x}$follows the normal distribution with a mean E(X) and standard deviation$\sqrt{\frac{\sigma }{n}}$

Therefore, the shape of the sampling distribution isnormal.

Here, the number of samples is 40, which is greater than 30.

Therefore, the answer depends on the sample size.