Administrators in a large school district asked a random sample of seventh-grade girls to take an IQ test. They calculated a 99% confidence interval for the mean IQ as (95.3, 109.2).
Which of the following is a correct interpretation of this interval?
(a) The mean IQ of all seventh-grade girls in the school district is between 95.3 and 109.2, with 99% confidence (b) Ninety-nine percent of the time, mean IQäs of seventh-grade girls will belong to the interval (95.3, 109.2).
(c) Ninety-nine percent of the IQås are between 95.3 and 109.2.
(d) There is a 1% probability that the mean IQ of all seventh-grade girls is not in this confidence interval.
The correct answer and explanation is :
The correct answer is (a): The mean IQ of all seventh-grade girls in the school district is between 95.3 and 109.2, with 99% confidence.
Explanation:
A confidence interval provides a range of values that is likely to contain the true population parameter (in this case, the mean IQ) with a specified level of confidence. The interval that the administrators calculated is a 99% confidence interval, which means that if they were to repeatedly take random samples of seventh-grade girls and compute confidence intervals for each sample, 99% of those intervals would contain the true mean IQ of all seventh-grade girls in the school district.
Let’s break down the options:
(a) The mean IQ of all seventh-grade girls in the school district is between 95.3 and 109.2, with 99% confidence.
- This is the correct interpretation of a confidence interval. The interval (95.3, 109.2) gives a range in which we are 99% confident that the true mean IQ of all seventh-grade girls in the district lies. It reflects the uncertainty about the population mean and the degree of confidence (99%) associated with the estimate.
(b) Ninety-nine percent of the time, mean IQs of seventh-grade girls will belong to the interval (95.3, 109.2).
- This interpretation is incorrect. The confidence interval does not predict that individual IQ scores fall within this range. Instead, it pertains to the population mean IQ of all seventh-grade girls in the school district. The interval is about the mean of the population, not about individual data points.
(c) Ninety-nine percent of the IQs are between 95.3 and 109.2.
- This is incorrect because it confuses individual IQ scores with the population mean. The confidence interval does not suggest that 99% of individual IQ scores fall between 95.3 and 109.2. It provides an estimate for the population mean IQ.
(d) There is a 1% probability that the mean IQ of all seventh-grade girls is not in this confidence interval.
- This interpretation is also incorrect. The interval does not imply a probability for the true mean falling within the interval. The true population mean is either inside the interval or outside of it, and we can be 99% confident that it lies inside the interval. The 1% refers to the confidence level, not a probability of the mean being outside the interval.
In summary, option (a) is the correct interpretation because it properly explains that the confidence interval estimates the range of the true population mean IQ with 99% confidence.