Question

A well known pharmaceutical manufacturer is manufacturing a newly developed vaccine, and is concerned about variability in the immune response of the recipients. They claim the variance in the immune response is 1.9. It is known that the immune responses follow a normal distribution. A random sample of 30 individuals were given the vaccine and their immune response recorded. It was found that they had a sample variance of 0.9. At the 5% significance level, test the hypothesis that the population variance of the immune response is equal to 1.9 against the two-sided alternative, that it is not. a. State the null and alternative hypothesis. b. Perform the hypothesis test. (You may use any of the methods we discussed that are appropriate for the problem.) c. State your decision. d. State your conclusion in full sentences.

Answer 1

a

The null hypothesis is
`H_o : \ \sigma^2 = 1.9`

The alternative hypothesis is
`H_a : \ \sigma^2 \\e 1.9`

b

`X^2 = 13.74`

c

The decision rule is

Reject the null hypothesis

d

The conclusion is

There is no sufficient evidence to conclude that the variance of the immune response is equal to 1.9.

Explanation:

From the question we are are told that

The variance is
`\sigma ^2 = 1.9`

The sample size is n = 30

The sample variance is
`s^2 = 0.9`

The level of significance is
`\alpha = 0.05`

The null hypothesis is
`H_o : \ \sigma^2 = 1.9`

The alternative hypothesis is
`H_a : \ \sigma^2 \\e 1.9`

Generally the test statistics is mathematically represented as

`X^2 = ( (n- 1 ) * s^2 )/( \sigma^2 )`

=>
`X^2 = ( (30 - 1 ) * 0.9 )/(1.9 )`

=>
`X^2 = 13.74`

Generally from the degree of freedom is mathematically represented as

`df = n- 1`

=>
`df = 30 - 1`

=>
`df = 29`

Generally from the chi distribution table the critical value of
`(\alpha)/(2) \ and \ 1 - ( \alpha )/(2)` at a degree of freedom of
`df = 29` is

`X^2 _{ (\alpha )/(2) , df } = X^2 _{ (0.05 )/(2) , 29 } = 45.7`

=>
`X^2 _{ 1 - (\alpha )/(2) , df } = X^2 _{1 - (0.05 )/(2) , 29 } = 16.0 5`

Gnerally from the values obtained we see that

`X^2 < X^2 _{ 1 - (\alpha )/(2) , df }` hence

The decision rule is

Reject the null hypothesis

The conclusion is

There is no sufficient evidence to conclude that the variance of the immune response is equal to 1.9.