Answer to Question #295997 in Statistics and Probability for partha ray

Question

the dean of admissions in a large university has determined that the scores of the freshman class in a mathematics test are apporoximately normally distributed with a mean 72 and a standard deviation of 4 based of this information,what is the probability that the mean score of a sample of 64 students it at least 73

Answer 1

Let $X=$ the mean score of a sample: $X\sim N(\mu, \sigma^2/n).$

Given $\mu=72, \sigma=4, n=64.$

$$P(X\ge 73)=1-P(X<73)$$

$$=1-P(Z<\dfrac{73-72}{4/\sqrt{64}})=1-P(Z<2)$$

$$\approx 0.02275$$