Question

The Graduate Record Examination (GRE) is a test required for admission to many U.S. graduate schools. Students' scores on the quantitative portion of the GRE follow a normal distribution with standard deviation 8.8. Suppose a random sample of 10 students took the test, and their scores are given below. 152, 126, 146, 149, 152, 164, 139, 134, 145, 136 a. Are the criteria for approximate normality satisfied? Why? b. Calculate the sample mean, x with bar on top. Round your answer to one decimal place. c. Calculate the Z test statistic for this sample. Round your answer to two decimal places. d. Does this sample appear to be unusual? Why?

Answer 1

Students' scores on the quantitative portion of the GRE follow a normal distribution with standard deviation 8.8.

Thus population standard deviation is known = 8.8

We have random sample of 10 students thus n = 10

scores of 10 samples are : 152, 126, 146, 149, 152, 164, 139, 134, 145, 136

a. Are the criteria for approximate normality satisfied?

To cheak this we will use QQ plot and also use Shapiro-Wilk normality test

We will R -software for above test , any other sofware can be used.

# import sample data in R

> x=c(152, 126, 146, 149, 152, 164, 139, 134, 145, 136)

> qqnorm(x);qqline(x, col = 2)         # To dwaw Nowmal Q Q plor

> shapiro.test(x)              # for Shapiro-Wilk normality test

        Shapiro-Wilk normality test

data: x
W = 0.98222, p-value = 0.9759

From QQ-plot we can observe al point are nearly close to line , and also in Shapiro-Wilk normality test we can see P-value = 0.9759 .

Since P-value = 0.9759 >.9 ,

we can say that the criteria for approximate normality is satisfied.

b. Calculate the sample mean, x with bar on top. Round your answer to one decimal place.

= = (152+ 126 +146+ 149+ 152+ 164+ 139+134 +145 +136)/10 = 144.3

Thus Sample mean = = 144.3

c. Calculate the Z test statistic for this sample.

Z =   

where is population mean , and is population standard deviation

n = 10   ; = 144.3

Also = 8.8 but problem is that to calculated Z test statistic population mean is not given

So we can not compute Z test statistic as is not known

If you find population mean , then put it in above equation , and we will get Z test statistic for this sample.

d. Does this sample appear to be unusual? Why?

Cant say much about these sample as we dont have anu idea about Population mean and Z test statistic.

But given population standard deviation is 8.8

and sample standard deviation is 10.90413

{

> sd(x)         # standard deviation of above given sample
[1] 10.90413

}

So we can see there sample standard deviation is far greater than gven population standard deviation , this may be the only unusual think appeared in sample.