Can someone help me with this statistic question?

Question

I have the following problem and can't figure it out.

A mileage test is conducted for a new car model, the “Pizzazz.” Thirty random
selected Pizzazzes are driven for a month and the mileage is carefully measured
in each. The mean mileage for the sample is 28.6 miles per gallon (mpg) and the
sample standard deviation is 2.2 mpg. Use a 0.05 significance level to test each
claim for the mean mpg in the entire population of Pizzazzes.

(a) The population mean is 20 mpg.
(b) The population mean is 25 mpg.
(c) The population mean is 30 mpg.

Answer 1

Okay, this is an application of the central limit Theorem.  Your sample SD is 2.2, and your sample size is 30.  

 

The Standard Error of the sampling distribution is going to be 2.2 / (square root of 30) = about 0.40.  

 

What this means is that, if you took a whole bunch of samples of size 30, they are expected to follow a normal distribution about the mean of the population, with a standard deviation of 0.40.  

 

So, now we compare to the claims, and see how unusual it might be to get that reading of 28.6 if the population means were as claimed.

 

(a) (28.6 - 20) = 8.6   8.6/0.40 = 21.5 SDs from the mean.  

(b) (28.6 - 25) = 3.6   3.6/.4    = 9 SDs from the mean.

(c) (28.6 - 30) = 1.4   1.4 / .4  = 3.5 SDs from the mean.

 

.05 significance is within +/- 1.96 SDs from the mean, so all of the above claims fail.