A simple random sample of 30 students was selected from a normally distributed population of high school students. The following confidence intervals were calculated at 90,95, and \( 99 \% \) confidence levels to estimate the average number of hours of sleep the students got each night. Determine which confidence level goes with each confidence interval. Between 7.3 and 8.7 hours of sleep: Between 7.0 and 9.0 hours of sleep: Between 7.2 and 8.8 hours of sleep: DoNE \( 90 \% \) confidence level \( 95 \% \) confidence level \( 99 \% \) confidence level

Recall that a confidence interval for the mean is computed as   sample mean ± (t* × SE) where t* is the critical value from the t‐distribution (with n – 1 degrees of freedom) corresponding to the desired level of confidence and SE is the standard error. Since our sample size is 30, we have 29 degrees of freedom. The three given confidence intervals all have the same midpoint:   (7.3 + 8.7)/2 = (7.0 + 9.0)/2 = (7.2 + 8.8)/2 = 8.0 hours. Thus, they all estimate that the average is 8.0 hours, but they differ in their margins of error: • For 7.3 to 8.7 hours, the margin of error is 8.0 − 7.3 = 0.7 hours. • For 7.2 to 8.8 hours, the margin of error is 8.8 − 8.0 = 0.8 hours. • For 7.0 to 9.0 hours, the margin of error is 9.0 − 8.0 = 1.0 hour. The margin of error increases as the confidence level increases because a higher confidence level requires a more “conservative” (wider) interval. For 29 degrees of freedom, the approximate critical t-values are:   • About 1.70 for a 90% confidence interval (CI)   • About 2.05 for a 95% CI   • About 2.76 for a 99% CI In our case, the interval with the smallest margin (0.7 hours) would use the smallest t* value (about 1.70) and hence corresponds to the 90% level; the one with the margin of 0.8 hours would use the t* from the 95% CI, and the widest interval (margin of 1.0 hour) must use the highest t* value, corresponding to the 99% CI. So, the matching is: • Between 7.3 and 8.7 hours of sleep → 90% confidence level • Between 7.2 and 8.8 hours of sleep → 95% confidence level • Between 7.0 and 9.0 hours of sleep → 99% confidence level