A major car manufacturer wants to test a new engine to determine whether it meet

Question

A major car manufacturer wants to test a new engine to determine whether it meets new air-pollution standards. The mean emission mu of all engines of this type must be less than 20 parts per million of carbon. Thirty five new engines were tested, and the emission level of each is determined. A mean and standard deviation of 19.7 and 1.18, respectively, are computed. State the null and alternative hypothesis in a test to determine if the data provide sufficient evidence to conclude that this type of engine meets the pollution standard. Find the value of the test statistic. Find the observed significance level (p-value) of the test. What would you conclude at the alpha =.05 level of significance ? What would you conclude at the alpha =. 1 level of significance ? What is the probability of a type II error of the test procedure in (d) when the actual value of mu = 19.5 ?

Explanation / Answer

a)

Formulating the null and alternative hypotheses,              
              
Ho:   u   >=   20  
Ha:    u   <   20   [answer]

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B)
              
As we can see, this is a    left   tailed test.      
              
              
Getting the test statistic, as              
              
X = sample mean =    19.7          
uo = hypothesized mean =    20          
n = sample size =    35          
s = standard deviation =    1.18          
              
Thus, z = (X - uo) * sqrt(n) / s =    -1.50408808 [ANSWER, TEST STATISTIC]

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c)          
              
Also, the p value is              
              
p =    0.066279344   [ANSWER, P VALUE]

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d)      
              
As P > 0.05, we   FAIL TO REJECT THE NULL HYPOTHESIS.  

There is no significant evidence that the true mean emission is less than 20 parts per million of carbon. [CONCLUSION]

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e)

As P < 0.1, we   REJECT THE NULL HYPOTHESIS.  

There is significant evidence that the true mean emission is less than 20 parts per million of carbon. [CONCLUSION]
      

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