Let "X=" the gas mileage in miles per gallon
Assume that "\\mu_0=50\\ \\text{miles per gallon}."
"H_0:\\mu=\\mu_0"
"H_1:\\mu\\not=\\mu_0"
Given that "\\sigma=5.5\\ \\text{miles per gallon}, n=30, \\bar{X}=47\\ \\text{miles per gallon}"
From the Central Limit Theorem, we know this sampling distribution is normal with a mean of 50 and standard error of
"\\sigma\/\\sqrt{n}=5.5\/\\sqrt{30}"
This corresponds to a two-tailed test, for which a z-test for one mean, with known population standard deviation will be used.
Let "\\alpha=0.05"
The critical value for a two-tailed test is "z_c=1.96"
The rejection region for this two-tailed test is "R=\\{{z:|z|>1.96}\\}"
The z-statistic is computed as follows:
"z={47-50 \\over5.5\/\\sqrt{30}}\\approx-2.9876"
Since it is observed that "|z|=|-2.9876|=2.9876>1.96=z_c," it is then concluded that the null hypothesis is rejected.
The p-value
Since "p=0.0028<0.05," it is concluded that the null hypothesis is rejected.
Therefore, there is enough evidence to claim that the population mean "\\mu" is different than "50," at the "0.05" significance level.
Confidence interval
"47\\pm 1.96{5.5 \\over \\sqrt{30}}"
"47\\pm 1.968"
"(45.032,48.968)"
Comments
Leave a comment