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.
To test the claim that gasoline prices in Texas are lower than the national average, we will perform a one-sample z-test.
Null Hypothesis (H0): The average price of gasoline in Texas is not significantly lower than the national average.
Alternative Hypothesis (Ha): The average price of gasoline in Texas is significantly lower than the national average.
Significance level (α): 0.01
We will calculate the test statistic using the formula:
z = (x̄ - μ) / (σ / √n)
Where:
x̄ = sample mean ($3.394)
μ = population mean ($3.498)
σ = population standard deviation ($0.1526)
n = sample size (10)
Calculating the test statistic:
z = (3.394 - 3.498) / (0.1526 / √10)
z = -0.104 / (0.1526 / √10)
z = -0.104 / 0.04826
z ≈ -2.154
Next, we will find the z-critical value at a significance level of 0.01. Using a standard normal distribution table or a calculator, the critical value for a two-tailed test at α = 0.01 is approximately ±2.576.
Since the calculated test statistic of -2.154 is less than the critical value of -2.576, we have enough evidence to reject the null hypothesis. This means that the average price of gasoline in Texas is significantly lower than the national average at α = 0.01.
In conclusion, the survey supports the claim that gasoline prices in Texas are significantly lower than the national average.