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.
Do we reject the null hypothesis?
Question 5 options:
1)
Yes
2)
No
To test the claim that gasoline prices in Texas are significantly lower than the national average, we can set up the following null and alternative hypotheses:
Null hypothesis (H0): μ >= $3.498 (The population mean in Texas is greater than or equal to the national average price of gasoline)
Alternative hypothesis (H1): μ < $3.498 (The population mean in Texas is lower than the national average price of gasoline)
We will perform a one-sample t-test to test this claim, as we have the sample mean and sample standard deviation.
The test statistic for this hypothesis test is calculated as follows:
t = (x̄ - μ) / (s / sqrt(n))
where x̄ is the sample mean, μ is the hypothesized population mean, 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 / sqrt(10))
t = -0.104 / (0.1526 / sqrt(10))
t ≈ -0.104 / 0.0483
t ≈ -2.149
Next, we need to compare the test statistic with the critical value at α = 0.01. Since the alternative hypothesis is that the mean gas price in Texas is lower than the national average, we are performing a one-tailed test.
Looking up the critical value for a one-tailed test with α = 0.01 and degrees of freedom (df) = 9, we find that the critical value is approximately -2.821.
Since the test statistic (-2.149) is not less extreme than the critical value (-2.821), we do not reject the null hypothesis.
Therefore, the correct answer is:
2) No