A manufacturer of a new car claims the typical car will aver

A manufacturer of a new car claims the typical car will average 32 mpg of gasoline. An independent consumer group is skeptical of the claim and thinks the mean gas mileage is significantly different than the 32 claimed. A sample of 23 randomly selected cars produced a mean mpg of 30.15 with a standard deviation of 4.87. Assume that cars’ gas mileage is normally distributed in the population.(8 pts)

a. The manufactures want to determine if the mean mpg for the cars is significantly different from 32. Write the null and alternative hypotheses for this question.

b. Are the assumptions necessary for inference met?

c. Perform the appropriate test including the formula that you used, the calculated value of the t-statistic, and the t-critical you compared your t-statistic to using a significance level of .05. Then, state your conclusion to the test.

d. Now, suppose the researchers had wanted to find evidence specifically that the car obtained less than 32 mpg. Rewrite your null and alternative hypothesis, then repeat the test and state your conclusion.

Solution

Formulating the null and alternative hypotheses,              
              
Ho:   u   =   32  
Ha:    u   =/   32 [ANSWER, A]

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b)

Yes, becuse the mileage is assumed to be normally distributed. Also, the sample mean looks different from 32.

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c)
              
As we can see, this is a    two   tailed test.      
              
Thus, getting the critical t,              
df = n - 1 =    22          
tcrit =    +/-   2.073873068   [ANSWER, CRITICAL T]
              
Getting the test statistic, as              
              
X = sample mean =    30.15          
uo = hypothesized mean =    32          
n = sample size =    23          
s = standard deviation =    4.87          
              
Thus, t = (X - uo) * sqrt(n) / s =    -1.821825117 [ANSWER, T STATISTIC]

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As -2.074<t<2.074, we FAIL TO REJECT THE NULL HYPOTHESIS.

There is no significant evidence that the mean mpg is different to 32.

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d)

Formulating the null and alternative hypotheses,              
              
Ho:   u   >=   32  
Ha:    u   <   32  
              
As we can see, this is a    left   tailed test.      
              
Thus, getting the critical t,              
df = n - 1 =    22          
tcrit =    -1.717144374      
              
Getting the test statistic, as              
              
X = sample mean =    30.15          
uo = hypothesized mean =    32          
n = sample size =    23          
s = standard deviation =    4.87          
              
Thus, t = (X - uo) * sqrt(n) / s =    -1.821825117          
As t < -.1717, WE REJECT THE NULL HYPOTHESIS.

There is significant evidence that the mean mpg is less than 32.