Question

a large random sample of car owners is surveyed in order to estimate the proportion who waited more than 45 minutes for their most recent oil change. it is reported that the sample proportion is 0.36 and the margin of error, at the 95% confidence level, is 0.042. which one of the following interpretations about this information is correct?
if the sampling procedure were to be repeated many times, about 95% of all similarly constructed confidence intervals would include either the value of 0.36 or the true population proportion of car owners who waited more than 45 minutes.
we are 95% confident that the interval from 0.318 to 0.402 includes the population proportion of car owners who waited more than 45 minutes for their most recent oil change.
if repeated random samples of car owners were surveyed, about 95% of those samples would produce a sample proportion between 0.318 and 0.402.
we are 95% confident that the interval from 0.318 to 0.402 includes all sampled car owners who waited more than 45 minutes for their most recent oil change.
there is a 95% chance that if we randomly select a single car owner, the probability that car owner waited more than 45 minutes for their most recent oil change is between 0.318 and 0.402

Explanation:

• Option 1: Confidence intervals are about including the true population proportion, not about including the sample proportion. So this option is incorrect.
• Option 2: The confidence interval is calculated as \(\hat{p}\pm\text{Margin of Error}\). Here, \(\hat{p} = 0.36\) and margin of error \(E=0.042\). The lower limit is \(0.36 - 0.042=0.318\) and the upper limit is \(0.36+ 0.042 = 0.402\). A \(95\%\) confidence interval means we are \(95\%\) confident that the interval contains the population proportion. This option is correct.
• Option 3: Confidence intervals are about the population proportion, not about the sample proportion. It is not about the proportion of samples that will have a sample proportion in the interval. So this option is incorrect.
• Option 4: The confidence interval is about the population proportion, not about the individual sampled car - owners. So this option is incorrect.
• Option 5: The confidence interval is about the population proportion (a parameter), not about the probability for a single individual. So this option is incorrect.

Answer:

We are 95% confident that the interval from 0.318 to 0.402 includes the population proportion of car owners who waited more than 45 minutes for their most recent oil change.