The average gasoline price of one of the major oil companies has been
$3.20 per gallon. Because of cost reduction measures, it is believed that there has
been a significant reduction in the average price. In order to test this belief, we
randomly selected a sample of 36 of the company’s gas stations and determined
that the average price for the stations in the sample was $3.14. Assume that the
standard deviation of the population (σ) is $0.12.
a. State the null and the alternative hypotheses.
b. Compute the test statistic.
c. At 95% confidence, test the company’s claim.
Not sure exactly how to calculate. Help would be appreciated. Thank you.
To test the claim that there has been a significant reduction in the average price of gasoline for one of the major oil companies, you can perform a hypothesis test.
a. Null and Alternative Hypotheses:
The null hypothesis (H0) assumes that there is no significant reduction in the average price. The alternative hypothesis (Ha) assumes that there is a significant reduction.
H0: μ = $3.20 (average price has not reduced significantly)
Ha: μ < $3.20 (average price has reduced significantly)
b. Test Statistic:
To compute the test statistic, you can use the formula for the z-test:
z = (x̄ - μ) / (σ / √n)
Where:
x̄ = sample mean
μ = population mean (claimed average price)
σ = population standard deviation
n = sample size
Given:
x̄ = $3.14
μ = $3.20
σ = $0.12
n = 36
Substituting the given values, we get:
z = ($3.14 - $3.20) / ($0.12 / √36)
z = -0.06 / ($0.12 / 6)
z = -0.06 / 0.02
z = -3
Therefore, the test statistic (z) is -3.
c. Testing at 95% Confidence:
To test the company's claim at 95% confidence, you need to compare the test statistic (z) with the critical value from the standard normal distribution.
At a 95% confidence level, the critical value is -1.96 (left-tailed test).
Since -3 < -1.96, the test statistic falls within the rejection region. Thus, we reject the null hypothesis.
Conclusion: Based on the given sample, there is sufficient evidence to support the claim that there has been a significant reduction in the average gasoline price for the major oil company.
a. The null hypothesis (H0): The average gasoline price is still $3.20 per gallon.
The alternative hypothesis (Ha): The average gasoline price is less than $3.20 per gallon.
b. To compute the test statistic, we will use the formula for a one-sample t-test:
t = (x̄ - μ) / (σ / √n)
where:
x̄ is the sample mean ($3.14),
μ is the population mean ($3.20),
σ is the population standard deviation ($0.12),
n is the sample size (36).
Plugging in the values:
t = (3.14 - 3.20) / (0.12 / √36)
t = (-0.06) / (0.12 / 6)
t = -0.06 / 0.02
t = -3
So, the test statistic is -3.
c. To test the company's claim at a 95% confidence level, we need to compare the test statistic with the critical value from the t-distribution. Since the alternative hypothesis is that the average price is less than $3.20, we are conducting a one-tailed test.
For a one-tailed test with a 95% confidence level and 35 degrees of freedom (36 - 1 = 35), the critical value from the t-distribution is approximately -1.689.
Since the test statistic (-3) is less than the critical value (-1.689), we can reject the null hypothesis. There is evidence to suggest that there has been a significant reduction in the average gasoline price of the major oil company.