POSSIBLE POINTS: 8.33 22. A random sample of 50 students at a large high school resulted in a 95 percent confidence interval for the mean number of hours of sleep per day of \( (6.73,7.67) \). Which of the following statements best summarizes the meaning of this confidence interval? (A) About \( 95 \% \) of all random samples of 50 students from this population would result in a \( 95 \% \) confidence interval of ( \( 6.73,7.67 \) ). (B) About \( 95 \% \) of all random samples of 50 students from this population would result in a \( 95 \% \) confidence interval that covered the population mean number of hours of sleep per day. (C) \( 95 \% \) of the students in the survey reported sleeping between 6.73 and 7.67 hours per day. (D) \( 95 \% \) of the students in this high school sleep between 6.73 and 7.67 hours per day. (E) A student selected at random from this population sleeps between 6.73 and 7.67 hours per day for \( 95 \% \) of the time.

To interpret the confidence interval provided in the problem, we need to understand what a 95% confidence interval means in the context of statistical sampling. 1. **Understanding the Confidence Interval**: The interval \( (6.73, 7.67) \) indicates that we are 95% confident that the true population mean number of hours of sleep per day for all students at the high school lies within this range. 2. **Evaluating the Statements**: - **(A)**: This statement suggests that if we took many random samples of 50 students, about 95% of those samples would produce a confidence interval of exactly \( (6.73, 7.67) \). This is incorrect because the intervals will vary from sample to sample. - **(B)**: This statement correctly reflects the meaning of a confidence interval. It states that about 95% of all random samples of 50 students would result in a confidence interval that includes the true population mean. This is a correct interpretation of the confidence interval. - **(C)**: This statement incorrectly implies that 95% of the students surveyed sleep between 6.73 and 7.67 hours. The confidence interval does not provide information about individual students but rather about the population mean. - **(D)**: Similar to (C), this statement incorrectly suggests that 95% of all students in the high school sleep between 6.73 and 7.67 hours. The confidence interval pertains to the mean, not individual data points. - **(E)**: This statement implies that a randomly selected student sleeps between 6.73 and 7.67 hours for 95% of the time, which is misleading. The confidence interval does not apply to individual students but to the population mean. 3. **Conclusion**: The best summary of the meaning of the confidence interval is statement **(B)**. It accurately describes the relationship between the sample confidence interval and the population mean. Thus, the correct answer is **(B)**.