Question

2. Anna reads that the average price of regular gas in the country is $2.27 per gallon. To see if the average price of gas is the same in her state (North Carolina) as it is in the country overall, she randomly selects 32 gas stations throughout her state and records the price per gallon of regular gas at each station and finds a mean price of $2.21 and a standard deviation of $0.23.  

(a) Does the data provide convincing evidence to refute the claim that the average price of gas in Anna’s state is the same as it is in the country as a whole? Perform the appropriate statistical test at the 1% level of significance.

(b) Suppose Anna had used a sample twice the size but obtained the same summary statistics and used a 5% level of significance instead. Would her conclusion change, and if so, how? Explain.

Answer 1

To Test :-

H0 :- µ = 2.27
H1 :- µ ≠ 2.27

Test Statistic :-
t = ( X̅ - µ ) / ( S / √(n))
t = ( 2.21 - 2.27 ) / ( 0.23 / √(32) )
t = -1.4757

Test Criteria :-
Reject null hypothesis if | t | > t(α/2, n-1)
Critical value t(α/2, n-1) = t(0.01 /2, 32-1) = 2.744
| t | > t(α/2, n-1) = 1.4757 < 2.744
Result :- Fail to reject null hypothesis

Decision based on P value
P - value = P ( t > 1.4757 ) = 0.1501
Reject null hypothesis if P value < α = 0.01 level of significance
P - value = 0.1501 > 0.01 ,hence we fail to reject null hypothesis
Conclusion :- Fail to reject null hypothesis

There is sufficient evidnece to support the claim that the average price of gas in Anna’s state is the same as it is in the country as a whole.

Part b)

Test Criteria :-
Reject null hypothesis if | t | > t(α/2, n-1)
Critical value t(α/2, n-1) = t(0.05 /2, 64-1) = 1.998
| t | > t(α/2, n-1) = 2.087 > 1.998
Result :- Reject null hypothesis

Decision based on P value
P - value = P ( t > 2.087 ) = 0.0409
Reject null hypothesis if P value < α = 0.05 level of significance
P - value = 0.0409 < 0.05 ,hence we reject null hypothesis
Conclusion :- Reject null hypothesis

There is insufficient evidnece to support the claim that the average price of gas in Anna’s state is the same as it is in the country as a whole.

Decision changes as sample size increases and level of significance decreases that's causes critical value to decreas from 2.744 to 1.998, due to which test statistic falls in the critical region.