Question

a computer selects a number x from 1 to 7 randomly and uniformly. round all answers to 4 decimal places where possible.
a. what is the distribution of x? x ~ u(1, 7)
b. suppose that the computer randomly picks 40 such numbers. what is the distribution of \\(\bar{x}\\) for this selection of numbers. \\(\bar{x}\\) ~ n(4, 0.2887)
c. what is the probability that the average of 40 numbers will be less than 4.3? 0.8506

Explanation:

Step1: Identify the distribution of $X$

$X$ is uniformly distributed on the interval $[1,7]$, so $X\sim U(1,7)$.

Step2: Find the mean and standard deviation of $X$

For a uniform distribution $U(a,b)$, the mean $\mu=\frac{a + b}{2}=\frac{1+7}{2}=4$, and the standard deviation $\sigma=\sqrt{\frac{(b - a)^2}{12}}=\sqrt{\frac{(7 - 1)^2}{12}}=\sqrt{3}\approx1.7321$.

Step3: Determine the distribution of $\bar{X}$

By the Central - Limit Theorem, if we have a sample of size $n = 40$ from a population with mean $\mu$ and standard deviation $\sigma$, the sampling distribution of the sample mean $\bar{X}$ is approximately normal with mean $\mu_{\bar{X}}=\mu = 4$ and standard deviation $\sigma_{\bar{X}}=\frac{\sigma}{\sqrt{n}}=\frac{\sqrt{3}}{\sqrt{40}}\approx\frac{1.7321}{6.3246}\approx0.2739$. So $\bar{X}\sim N(4,0.2739^2)$.

Step4: Calculate the z - score

The z - score is calculated as $z=\frac{\bar{x}-\mu_{\bar{X}}}{\sigma_{\bar{X}}}$. Here, $\bar{x} = 4.3$, $\mu_{\bar{X}}=4$, and $\sigma_{\bar{X}}\approx0.2739$. So $z=\frac{4.3 - 4}{0.2739}=\frac{0.3}{0.2739}\approx1.0953$.

Step5: Find the probability

We want to find $P(\bar{X}<4.3)$. Using the standard normal distribution table, $P(Z < 1.0953)\approx0.8637$.

Answer:

a. $X\sim U(1,7)$
b. $\bar{X}\sim N(4,0.2739^2)$
c. $0.8637$