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
To understand the difference between the 95% and 99% confidence intervals, we first need to understand what a confidence interval is. A confidence interval is a range of values within which we can estimate the true population parameter with a certain level of confidence.
In this case, the confidence intervals represent the range of values we believe the average hours of sleep a student gets per day falls between, based on the sample data.
The 95% confidence interval (6.214, 7.686) means that we are 95% confident that the true average hours of sleep a student gets falls between 6.214 and 7.686 hours. This means that if we were to repeat this study multiple times and calculate the confidence interval each time, 95% of these intervals would include the true population mean.
Similarly, the 99% confidence interval (5.944, 7.956) means that we are 99% confident that the true average hours of sleep a student gets falls between 5.944 and 7.956 hours. This means that if we were to repeat the study multiple times and calculate the confidence interval each time, 99% of these intervals would include the true population mean.
The difference between the two confidence intervals is mainly due to the level of confidence. The wider the confidence interval, the higher the level of confidence we have in capturing the true population mean. Therefore, the 99% confidence interval is wider than the 95% confidence interval because it accounts for a higher level of confidence.
Choosing between a 95% or a 99% confidence interval depends on the trade-off between precision and certainty. A narrower confidence interval (such as the 95% interval) provides a more precise estimate, but with slightly less certainty, while a wider confidence interval (such as the 99% interval) provides a more certain estimate but with slightly lower precision. The decision on which level of confidence to use depends on the specific requirements of the study and the importance of precision versus certainty.