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

Step-by-step 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

             $t = \frac{ \= x - \mu }{ \frac{\sigma }{ \sqrt{n} } }$

=>        $t = \frac{ 8.93 -9.72}{ \frac{ 1.4 }{ \sqrt{36} } }$

=>      $t = -3.33$

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