A recent gasoline survey shows that the national average price of gasoline was $3.498 a gallon. It was felt that gasoline in Texas was significantly lower than the national average. A survey of 10 different suburbs in Dallas, Texas found the average price of gasoline to be $3.394 a gallon with a sample standard deviation of $0.1526. Assume that gas prices follow a normal distribution. We will perform a hypothesis test for this claim at α = 0.01
Should you accept the claim that gas is cheaper in Texas?
Question 6 options:
1)
Yes
2)
No
To determine whether the claim that gas is cheaper in Texas is supported by the data, we need to perform a hypothesis test. The null hypothesis (H0) is that the average gas price in Texas is not significantly lower than the national average, while the alternative hypothesis (Ha) is that the average gas price in Texas is significantly lower.
H0: μ ≥ $3.498 (average gas price in Texas is not significantly lower than the national average)
Ha: μ < $3.498 (average gas price in Texas is significantly lower than the national average)
To conduct the hypothesis test, we will use a one-sample t-test since we have a sample mean and standard deviation.
The test statistic is calculated as follows:
t = (x̄ - μ) / (s / √n)
where x̄ is the sample mean, μ is the population mean (national average), s is the sample standard deviation, and n is the sample size.
In this case, x̄ = $3.394, μ = $3.498, s = $0.1526, and n = 10.
t = ($3.394 - $3.498) / ($0.1526 / √10)
t = (-0.104) / (0.048329) ≈ -2.150
To determine whether the test statistic is statistically significant, we compare it to the critical value at α = 0.01.
The degrees of freedom for this test is n-1 = 10-1 = 9. Using a t-distribution table or a t-distribution calculator, the critical value for a one-tailed test at α = 0.01 with 9 degrees of freedom is approximately -2.821.
Since our test statistic (-2.150) is not less extreme than the critical value (-2.821), we fail to reject the null hypothesis.
Therefore, we do not have enough evidence to support the claim that gas is significantly cheaper in Texas than the national average.
The correct option is:
2) No