Compare the 95% and 99% confidence intervals for the hours of sleep a student gets. Explain the difference between these intervals and why this difference occurs.
95% confidence intervals: 6.214, 7.686
99% confidence intervals: 5.944, 7.956
95% = ± 1.96 SD
99% = ± 2.575 SD
To compare the 95% and 99% confidence intervals for the hours of sleep a student gets, let's start by understanding what confidence intervals represent.
A confidence interval is a range of values within which an estimated population parameter, such as the average hours of sleep, is likely to fall. The confidence level represents the likelihood that the true value of the parameter lies within the interval.
The difference between the 95% and 99% confidence intervals lies in their respective levels of confidence. The 95% confidence interval implies that if we were to repeat the study multiple times, we would expect 95% of those intervals to contain the true population parameter (average hours of sleep). The 99% confidence interval, on the other hand, means we would expect 99% of the intervals to contain the true population parameter.
Looking at the specific values you provided:
- For the 95% confidence interval: 6.214, 7.686
This interval estimates that with 95% confidence, the true average hours of sleep for students lies somewhere between 6.214 and 7.686 hours. In other words, if we were to repeat the study many times, roughly 95% of the resulting intervals would contain the true average hours of sleep.
- For the 99% confidence interval: 5.944, 7.956
This wider interval estimates that with 99% confidence, the true average hours of sleep for students lies somewhere between 5.944 and 7.956 hours. With a higher level of confidence (99%), we expect a wider range to account for the increased uncertainty. In other words, the 99% confidence interval provides a larger margin of error to capture a broader range of potential values for the population parameter.
The reason for the difference in these intervals lies in the concept of confidence level. When we increase the confidence level, such as from 95% to 99%, we require a greater level of certainty that the true parameter lies within the interval. This increased requirement leads to a wider interval to accommodate a larger range of potential values and, consequently, a higher level of confidence in capturing the true parameter value.
It is worth noting that while a higher confidence level provides a greater level of assurance, it also results in a wider interval, which implies a larger range of potential values. Therefore, the choice of a confidence level depends on the specific requirements of the study and the tolerance for error.