Answer to Question #339836 in Statistics and Probability for felicity

Question

2. The scores of individual students on a national test have a normal distribution with mean of 18.6 and a standard deviation of 5.9. At Federico Ramos Rural High School, 76 students took the test. If the scores at this school have the same distribution as national scores, solve for the following:

a. determine the mean and standard deviation of the sampling distribution of the sample mean.

b. find the probability that the sample mean falls between 17 and 20 (𝑃(17 < 𝑥̅< 20).

c. the number of sample means that is above 19.3 kilograms.

Answer 1

2. If the population is normal, then by the Central Limit Theorem the sampling distribution of the sample means will be approximately normal.

a. $\bar{X}\sim N(\mu, \sigma^2/n)$  

$$\mu_{\bar{X}}=\mu=18.6$$

$$\sigma_{\bar{X}}=\sigma/\sqrt{n}=5.9/\sqrt{76}\approx0.6768$$

b.

$$P(17<\bar{X}<20)=P(Z<\dfrac{20-18.6}{5.9/\sqrt{76}})$$

$$-P(Z\le\dfrac{17-18.6}{5.9/\sqrt{76}})$$

$$\approx P(Z<2.06863)-P(Z\le-2.36415)$$

$$\approx0.98071-0.00904\approx0.9717$$

c.

$$P(\bar{X}>19.3)=1-P(Z\le \dfrac{19.3-18.6}{5.9/\sqrt{76}})$$

$$\approx 1-P(Z\le 1.034315)\approx0.8495$$

$0.8495(76)=65$