A particular type of gasoline is supposed to have a mean oct

A particular type of gasoline is supposed to have a mean octane rating greater than 85 (out of 100). Assume the octane ratings follow the normal distribution. 10 measurements are taken of the octane rating, as follows: 90.1, 88.8, 89.5, 91.0, 86.0, 87.9, 92.1, 89.0, 88.5, 92.5

a) Is the true standard deviation of the octane rating significantly less than 2? Use rejection area criterion for this question. ( = 0.05).

b) Find the Pvalue of this test.

c) Use interval to test whether the true mean of octane rating is really greater than 85 based on the 10 collected measurements listed above. ( = 0.01)

Solution

a)

Getting the sample standard deviation,

s = 1.969320244

Formulating the null and alternative hypotheses,              
              
Ho:   sigma   >=   2  
Ha:    sigma   <   2  
              
As we can see, this is a    left   tailed test.      
              
Thus, getting the critical chi^2, as alpha =    0.05   ,      
alpha =    0.05          
df = N - 1 =    9          
chi^2 (crit) =    3.325112843        
              
Getting the test statistic, as              
s = sample standard deviation =    1.969320244          
sigmao = hypothesized standard deviation =    2          
n = sample size =    10          
              
              
Thus, chi^2 = (N - 1)(s/sigmao)^2 =    8.726000003          
As chi^2 > 3.325, then we FAIL TO REJECT THE NULL HYPOTHESIS.              

Thus, there is no significant evidence that the true standard deviation is less than 2. [CONCLUSION]

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B)

By technology, the P value of chi^2 = 8.726, df = 9, is

P = 0.537058379 [ANSWER]

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c)

Note that              
              
Lower Bound = X - z(alpha) * s / sqrt(n)              
      
              
where              
alpha = (1 - confidence level) =    0.01          
X = sample mean =    89.54          
z(alpha) = critical z for the confidence interval =    2.326347874          
s = sample standard deviation =    1.969320244          
n = sample size =    10          
              
Thus,              
              
Lower bound =    88.09125816          
              
Thus, the confidence interval is              
              
(   88.09125816   ,   infinity   )

As this whole interval is greater than 85, then there is significant evidence that true mean of octane rating is really greater than 85. [CONCLUSION]