Question

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.

Explanation:

To determine the correct interpretation of the 95% confidence interval \((6.73, 7.67)\) for the mean number of hours of sleep per day:

• Option A: Incorrect. Different random samples of 50 students will likely produce different confidence intervals, not the exact interval \((6.73, 7.67)\) for 95% of samples.
• Option B: Correct. The definition of a 95% confidence interval is that if we were to take many random samples of the same size (50 students) from the population, about 95% of those samples would produce a confidence interval that contains the true population mean.
• Option C: Incorrect. The confidence interval is about the population mean, not the percentage of students in the survey (sample) with sleep hours in that range.
• Option D: Incorrect. The interval estimates the population mean, not the percentage of all students with sleep hours in that range.
• Option E: Incorrect. The interval is about the population mean, not an individual student’s sleep hours over time.

Answer:

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.