Question

Recently a sample of 36 pieces of thread showed a mean breaking strength of 8.93 oz. Can one conclude at a significance level of (a) 0.05, (b) 0.01 that the thread has become inferior?

Answer 1

Complete Question

It has been found from experience that the mean breaking strength of a particular brand of thread is 9.72 oz with a standard deviation of 1.4 oz. Recently a sample of 36 pieces of thread showed a mean breaking strength of 8.93 oz. Can one conclude at a significance level of (a) 0.05, (b) 0.01 that the thread has become inferior?

Answer:

At both
`\alpha = 0.05` and
`\alpha = 0.01` the conclusion is that the thread has become inferior

Explanation:

From the question we are told that

The population mean is
`\mu = 9.72 \ oz`

The standard deviation is
`\sigma = 1.40\ oz`

The sample size is n = 36

The sample mean is
`\= x = 8.93 \ oz`

The null hypothesis is
`H_o : \mu = 9.72 \ oz`

The alternative hypothesis is
`H_a : \mu < 9.72 \ oz`

Generally the test statistics is mathematically represented as

`t = ( \= x - \mu )/( (\sigma )/( √(n) ) )`

=>
`t = ( 8.93 -9.72)/( ( 1.4 )/( √(36) ) )`

=>
`t = -3.33`

So

The p-value obtained from the z- table is

`p-value = P( Z < -3.39) = 0.00034946`

So at
`\alpha = 0.0 5`

`p-value < \alpha`

So we reject the null hypothesis,hence we conclude that the thread has become inferior

So at
`\alpha = 0.0 1`

`p-value < \alpha`

So we reject the null hypothesis,hence we conclude that the thread has become inferior