Question

A random sample is taken from a normal population. Do the hypothesis test to determine if there is evidence that the population standard deviation is greater than 5. Use a level of significance of α = 0.05.

n = 25

xbar = 68.7

s = 6.2

1. Use the appropriate notation to show the hypotheses.

H₀:

H₁:

2. The critical value is _________ = _______________________
3. The test statistic is __________= ________________________
4. The p value is __________________________________

Answer 1

Given :

$n=25$

$\bar{x}=68.7$

$s=6.2$

claim : the population standard deviation is greater than 5 .

This is the hypothesis test for population standard deviation so we need to use chi-square test .

a.

$H_0:\sigma \le 5$

H, :σ >5

a.

degrees of freedom = n-1 = 25-1 =24

df = 24

g= 0,05

Using excel function ,    =CHIINV( $\alpha$ , df )

= CHIINV( 0.05 , 24 )

= 36.415

The critical value is $\chi^2_c =36.415$

b.

The test statistic formula is ,

$\chi^2= \frac{(n-1)s^2}{\sigma^2}$

$\chi^2= \frac{(25-1)(6.2)^2}{5^2}$

$\chi^2=36.902$

The test statistic is $\chi^2 =36.902$

c.

The p value :

Using excel function ,   $=CHIDIST(\chi^2 , df )$

=CHIDIST( 36.902 , 24 )

= 0.045

P-value = 0.045

The p value is 0.045

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Decision : As p-value ( 0.045 ) is less than significance level $\alpha$ (0.05) , we reject Ho .

Conclusion : There is sufficient evidence to support the claim that the population standard deviation is greater than 5.