Answer to Question #340369 in Statistics and Probability for Ally

Question

Present complete solutions for each item. Illustrate the normal curve if possible.

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.

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.

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

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

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

$$\approx0.9717$$

c.

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

$$=1-P(Z\le1.034315)\approx0.1505$$

$76(0.1505)=12$ sample means are above 19.3 kilograms.