Question

A survey organization drew a simple random sample of 625 households from a city of 100,000 households. The sample mean of people in the 625 households was 2.3, and a 95% confidence interval for the mean number of people in the 100,000 households was 2.16 < µ < 2.44. Which interpretation below is the most accurate?

a. We are 95% confident that the mean number of people per household is between 2.16 and 2.44.
b. We are 95% confident that the mean number of people in the 100,000 households is between 2.16 and 2.44.
c. The probability is 0.95 that the mean number of people in the 100,000 households is between 2.16 and 2.44.
d. 95% of the households in the sample contain between 2.16 and 2.44 people.
e. Cannot be determined from the information given.

Answer 1

The most accurate interpretation of the 95% confidence interval statement is that we are 95% confident the true mean number of people per household in the city lies between 2.16 and 2.44. This reflects the range in which the population mean is likely to fall, not a probability about the mean itself or about individual households. Option number d is correct.

Step-by-step explanation:

The most accurate interpretation of the 95% confidence interval for the mean number of people in a city's households, which is between 2.16 and 2.44, is option a: 'We are 95% confident that the mean number of people per household is between 2.16 and 2.44.' This statement correctly reflects the understanding that the confidence interval provides a range within which the true population mean is likely to fall. A confidence interval is constructed to estimate the unknown population parameter like the population mean, μ. So, 95% of the time we expect the interval to contain the true mean if we were to take many samples and construct a confidence interval each time.

Incorrect interpretations would involve the probability of the mean falling within the interval for the population, which is not accurate, as the true population mean is a fixed value and not a random variable. Also, the confidence interval does not speak to the proportions of the sample itself, such as the number of households that have a certain number of people in them.