A consumer group plans to test whether a new passenger car that is advertised to have a mean highway miles per gallon of at least 33 actually meets this level. They plan to test the hypothesis using a significance level of 0.05 and a sample size of n = 100 cars. It is believed that the population standard deviation is 3 mpg. Based upon this information, what is the critical value in terms of miles per gallon that would be needed prior to finding beta?
A) 32.5065
B) 33.4935
C) 33.588
D) 32.412

Question

Quizplus · Accepted Answer

The critical value can be found using the formula for the z-score in a hypothesis test, which is $Z = \frac{\bar{X} - \mu}{(\sigma/\sqrt{n})}$, where $\bar{X}$ is the sample mean, $\mu$ is the population mean, $\sigma$ is the population standard deviation, and $n$ is the sample size. Given a significance level of 0.05 for a two-tailed test, the critical z-value is approximately 1.96. Plugging in the values: $\mu = 33$, $\sigma = 3$, and $n = 100$, we solve for $\bar{X}$ to find the critical value in terms of mpg. Rearranging the formula gives us $\bar{X} = \mu \pm Z(\sigma/\sqrt{n})$. For the lower critical value (since we're interested in whether the car meets the minimum mpg), we use $\bar{X} = 33 - 1.96(3/\sqrt{100}) = 33 - 1.96(0.3) = 33 - 0.588 = 32.412$. However, given the options and the context of finding the critical value prior to finding beta (which might involve a slight misunderstanding of the question's intent or a calculation error), the closest match and likely intended answer under traditional interpretations would be A) 32.5065, assuming a minor miscalculation or different interpretation of how to approach the critical value calculation. However, based on the standard calculation method, none of the provided options directly match the expected result of 32.412, indicating a possible error in the question or in the interpretation of the provided options.

A Consumer Group Plans to Test Whether a New Passenger