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.

Respuesta :

Answer:

The decision rule is  

Reject the null hypothesis  

Step-by-step explanation:

From the question we are told that

    The population mean is [tex]\mu = 5.6 \ pounds/inch^2[/tex]

     The  sample size is  n =  160

      The sample mean is  [tex]\= x = 5.7 \ pounds/ inch^2[/tex]

     The variance is [tex]\sigma ^2 = 0.25[/tex]

       The level of significance is  [tex]\alpha = 0.01[/tex]

The null hypothesis is  [tex]H_o : \mu = 5.6[/tex]

The alternative hypothesis is [tex]H_a : \mu > 5.6[/tex]

Generally the standard deviation is mathematically represented as

           [tex]\sigma = \sqrt{\sigma ^2 }[/tex]

=>         [tex]\sigma = \sqrt{0.25 }[/tex]  

=>         [tex]\sigma = 0.5[/tex]  

Generally the test statistics is mathematically represented as

        [tex]z = \frac{\= x - \mu }{ \frac{\sigma}{\sqrt{n} } }[/tex]

=>     [tex]z = \frac{5.7 - 5.6 }{ \frac{0.5 }{\sqrt{ 160 } } }[/tex]  

=>     [tex]z = 2.53[/tex]

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

        [tex]p-value = P(Z > 2.53 ) =0.0057[/tex]

From the value obtained we see that the [tex]p-value < \alpha[/tex] 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

 

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