A consumer group is testing a manufacturer’s claim that a new car model will travel at least 30 miles per gallon of gasoline. Assume that ��0:��=30 and ����:��<30, and that the population standard deviation �� is known and is equal to 3.2 miles per gallon. An SRS of 40 cars will be used for this test, with a level of significance of ��=0.01.
Compute the power of this test assuming a proposed alternative value for �� of 27.5 miles per gallon (i.e. assuming ����=27.5).
To compute the power of the test, we need to first find the critical value and then calculate the probability of rejecting the null hypothesis when the alternative hypothesis is true.
1. Find the critical value:
The level of significance is given as α = 0.01. Since this is a two-tailed test (we're testing whether the new car model will travel at least 30 mpg or not), we divide the significance level by 2 to get α/2 = 0.005.
Using a Z-table or calculator, find the critical value associated with an area of 0.005 in the tails of the standard normal distribution. This critical value is denoted as z(α/2).
2. Calculate the test statistic:
The test statistic for this hypothesis test is calculated using the formula:
z = (x̄ - μ) / (σ / √n),
where x̄ is the sample mean, μ is the population mean (hypothesized value), σ is the population standard deviation, and n is the sample size.
In this case, x̄ = 30, μ = 27.5 (alternative value), σ = 3.2, and n = 40.
Calculate the test statistic using these values.
3. Compute the power of the test:
The power of a test is the probability of correctly rejecting the null hypothesis when the alternative hypothesis is true. It is calculated based on the test statistic and critical value.
Power = P(z > z(α/2) or z < -z(α/2)),
where z(α/2) is the critical value and P represents the probability.
Using a standard normal distribution table or calculator, find the probability associated with the critical value z(α/2) and its corresponding tail area. Subtract this probability from 1 to obtain the power of the test.
By following these steps, you will be able to compute the power of the test assuming a proposed alternative value of 27.5 miles per gallon.