1) ONE SAMPLE TEST OF HYPOTHESIS FOR THE MEAN ( UNKNOWN) The owner of a gasoline

Question

1) ONE SAMPLE TEST OF HYPOTHESIS FOR THE MEAN ( UNKNOWN)
The owner of a gasoline station wants to study gasoline-purchasing habits of motorists at his station. He selects a random sample of 60 motorists during a certain week, with the following results for the amounts purchased: X-bar = 11.3 gallons, S=3.1 gallons.
1a) if the owner is interested in whether the average gas purchase is different than 10 gallons, what would be the null and alternate hypotheses?
1b) what would be the Type I error?
1c) what would be the Type II error?
1d) for the owner of the gas station, which type of error would be most meaningful?
1e) at the 0.05 level of significance, is there evidence that the population mean purchase was different from 10 gallons?
1f) determine the p-value in e)

Explanation / Answer

1.

a)

Formulating the null and alternative hypotheses,              
              
Ho:   u   =   10  
Ha:    u   =/   10   [ANSWER]

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b)

A type I error is incorrectly rejecting a true null hypothesis. Hence, it is saying that the true mean purchase is different from 10 gallons, when in fact, it is not.

c)

A type II error is incorrectly failing to reject a false null hypothesis. Hence, it is saying that the true mean purchase is not different from 10 gallons, when in fact, it is.

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d)              

He would not want to commit a type I error, because this changes his decision for his supplies.

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e)

As we can see, this is a    two   tailed test.      
              
Thus, getting the critical z, as alpha =    0.05   ,      
alpha/2 =    0.025          
zcrit =    +/-   1.959963985      
              
Getting the test statistic, as              
              
X = sample mean =    11.3          
uo = hypothesized mean =    10          
n = sample size =    60          
s = standard deviation =    3.1          
              
Thus, z = (X - uo) * sqrt(n) / s =    3.248308613          
              
Also, the p value is, as this is two tailed,              
              
p =    0.001160933          
              
As |z| > 1.96, and P < 0.05, we   REJECT THE NULL HYPOTHESIS.          

Hence, there is significant evidence that the population mean purchase was different from 10 gallons. [CONCLUSION]

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f)

As said,

P = 0.001160933 [ANSWER]

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