Question

An engineer designed a valve that will regulate water pressure on an automobile engine. The engineer designed the valve such that it would produce a mean pressure of 5.6 pounds/square inch. It is believed that the valve performs above the specifications. The valve was tested on 160 engines and the mean pressure was 5.7 pounds/square inch. Assume the variance is known to be 0.25. A level of significance of 0.01 will be used. Make a decision to reject or fail to reject the null hypothesis.

Answer 1

Answer:

The decision rule is  

Reject the null hypothesis  

Step-by-step explanation:

From the question we are told that

    The population mean is $\mu = 5.6 \ pounds/inch^2$

     The  sample size is  n =  160

      The sample mean is  $\= x = 5.7 \ pounds/ inch^2$

     The variance is $\sigma ^2 = 0.25$

       The level of significance is  $\alpha = 0.01$

The null hypothesis is  $H_o : \mu = 5.6$

The alternative hypothesis is $H_a : \mu > 5.6$

Generally the standard deviation is mathematically represented as

           $\sigma = \sqrt{\sigma ^2 }$

=>         $\sigma = \sqrt{0.25 }$  

=>         $\sigma = 0.5$  

Generally the test statistics is mathematically represented as

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

=>     $z = \frac{5.7 - 5.6 }{ \frac{0.5 }{\sqrt{ 160 } } }$  

=>     $z = 2.53$

From the z table  the area under the normal curve to the left corresponding to  2.53   is

        $p-value = P(Z > 2.53 ) =0.0057$

From the value obtained we see that the $p-value < \alpha$ hence

The decision rule is  

Reject the null hypothesis  

The conclusion is  

There is sufficient evidence to conclude that the believe  that the valve performs above the specifications is true