Question

ACTIVITY: Problem: The scores of individual students on a national achievement test have a normal distribution with a mean of 20.8 and a standard deviation of 1.9. At Feliciano Delos Santos High School, 84 students took the test. If the scores of the students at this school have the same distribution as the national scores, how will you describe the sampling distribution of the scores of these students? Assume that the population is infinite. Use the given steps above

Answer 1

Let $X_1, X_2, ..., X_n$ be a random sample from a distribution with mean $\mu$ and variance $\sigma^2.$

Then for  samples of any size drawn from a normally distributed population the sample mean $\bar{X}$ is normally distributed (with mean $\mu_{\bar{X}}=\mu$ and standard deviation $\sigma_{\bar{X}}=\sigma/\sqrt{n}.$

Given $\mu=2.8, \sigma=1.9, n=84$

$$\bar{X}\sim N(\mu, \sigma/\sqrt{n})$$

$$\bar{X}\sim N(2.8, 1.9/\sqrt{84})$$