Answer to Question #341265 in Statistics and Probability for Dixxel

Question

A population of Grade 11 Arts and Design Students of size N=140 has mean grade of u=83 in Arts Subject and a population standard deviation of 25.

1.) What is the z-score of the the sample mean 80 if the sample size n=30?

2.) What is the area or probability of the sample mean 75 of the sample size n=30?

3.) What is the area greater than the sample mean of 80 if the sample size n=30?

Answer 1

1)

$$z=\dfrac{x-\mu}{\sigma\sqrt{\dfrac{N-n}{n(N-1)}}}=\dfrac{80-83}{25\sqrt{\dfrac{140-30}{30(140-1)}}}=$$

$$=-0.738844$$

2)

$$P(X<75)=P(Z<\dfrac{75-83}{25\sqrt{\dfrac{140-30}{30(140-1)}}})$$

$$\approx P(Z<-1.970251)=0.0244$$

3)

$$P(X>80)=1-P(Z\le\dfrac{80-83}{25\sqrt{\dfrac{140-30}{30(140-1)}}})$$

$$\approx 1-P(Z<-0.738844)=0.7700$$