7. Give and interpret the 95% confidence interval for the hours of sleep a student gets.
95% = mean ± 1.96 SEm
SEm = SD/√n
To give and interpret the 95% confidence interval for the hours of sleep a student gets, you would need a sample of data on the number of hours of sleep for different students. With this data, you can calculate the mean and standard deviation of the hours of sleep.
Here are the steps to calculate the confidence interval:
1. Collect a sample of data on the number of hours of sleep for different students.
2. Calculate the sample mean (x̄) and sample standard deviation (s) of the hours of sleep.
3. Determine the sample size (n) of the data.
4. Calculate the standard error of the mean (SEM), which can be found by dividing the sample standard deviation by the square root of the sample size: SEM = s / √n.
5. Determine the critical value for a 95% confidence level. For a large sample size (typically n > 30), you can use the z-score. The z-score for a 95% confidence level is approximately 1.96. If you have a smaller sample size, you may need to use a t-distribution table.
6. Calculate the margin of error by multiplying the standard error of the mean by the critical value: Margin of Error = SEM * Critical Value.
7. Calculate the lower confidence limit by subtracting the margin of error from the sample mean: Lower Confidence Limit = x̄ - Margin of Error.
8. Calculate the upper confidence limit by adding the margin of error to the sample mean: Upper Confidence Limit = x̄ + Margin of Error.
9. Now you have the 95% confidence interval for the hours of sleep a student gets, which is given by [Lower Confidence Limit, Upper Confidence Limit].
For example, if the sample mean is 7 hours and the margin of error is 0.5, the 95% confidence interval would be [6.5, 7.5]. This means that we are 95% confident that the true average hours of sleep for all students falls within this range.
Interpreting the confidence interval, it means that if we were to take repeated samples and calculate the 95% confidence interval each time, we would expect the true average hours of sleep for all students to fall within this range in 95% of the cases. It provides a measure of the precision and uncertainty associated with estimating the population parameter (in this case, the average hours of sleep) based on a sample.