Question

A simple random sample of size

n equals 16

is drawn from a population that is normally distributed. The sample variance is found to be

13.7

Test whether the population variance is greater than

10

at the

alpha equals 0.05

level of significance.

I only need to find the test statistic and the p-value. Would you go through it step by step please.

Answer 1

Solution:

Given:

Sample size = n = 16

Sample variance = s² = 13.7

Level of significance =

We have to test if  the population variance is greater than 10

Step 1) State H0 and H1:

Step 2) Find test statistic:

We use Chi square test statistic for variance:

Step 3) Find P-value:

To get interval of P-value, we use Chi-square critical value table:

df = n - 1 = 16 - 1 = 15

Look in df = 15 row and find the interval in which fall, then find correspoding right tail area.

From above table, we can see   ,fall between 8.547and 22.307.

Corresponding right tail area is between 0.900 and 0.100

that is : between 0.100 and 0.900

Thus P-value is between 0.100 and 0.900 > 0.05 level of significance , hence we fail to reject null hypothesis H0.

To get Exact P-value , we need to use Excel or TI 84 plus calculator:

Excel:

=CHISQ.DIST.RT( x , df )

=CHISQ.DIST.RT( 20.55 , 15 )

=0.151833

Thus P-value= 0.1518

TI 84 plus calculator:

Step 1) Press 2ND and VARS

Step 2) Select

Step 3) Enter numbers:

Lower: 20.55

Upper : 999999999999 ( assume infinity )

df : 15

Paste

click on Paste press Enter two times.

=0.151833

Thus P-value = 0.1518

Since P-value = 0.1518 > 0.05 level of significance , hence we fail to reject null hypothesis H0.

Thus there is not sufficient evidence to conclude that: the population variance is greater than 10.