Question

Suppose a local manufacturing company claims their production line has a variance of less than 9.0. A quality control engineer decides to test this claim by sampling 35 parts. She finds that the standard deviation of the sample is 2.12. Is this enough evidence at the 1% level of significance, to accept the manufacturing companies claim? Calculate the appropriate test statistic (round to 2 decimal places as needed)

Answer 1

The decision rule is

Fail to reject the null hypothesis

The conclusion is

There is no sufficient evidence to accept the manufacturing company claims

Explanation:

From the question we are told that

The sample size is n = 35

The sample standard deviation is
`s =  2.12`

The level of significance is
`\alpha = 0.01`

The null hypothesis is
`H_o  :  \sigma ^2  = 9.0`

The alternative hypothesis is
`H_a  :  \sigma ^2  <  9.0`

Gnerally the test statistics is mathematically represented as

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

`X^2 _(stat) =  ((35  -1) * 2.12^2 )/(9 )`

=>
`X^2 _(stat) =  16.98`

Generally the degree of freedom is mathematically represented as

`df =  n - 1`

=>
`df =  35 - 1`

=>
`df =  34`

From the chi - distribution table the critical value of
`\alpha` at a degree of freedom of
`df =  34` is

`X^2_(\alpha, 34 ) =56.060`

From the value obtained we see that the test statistics does not lie within the region of rejection (1.e 56.060 ,
`\infty` )

Then

The decision rule is

Fail to reject the null hypothesis

The conclusion is

There is no sufficient evidence to accept the manufacturing company claims