Give and interpret the 95% confidence interval for the hours of sleep a student gets. (7,8,4,5,8,10,7,8,5,6,7,4,7,8,9,8,7,6,7,8)
95% = mean ± 1.96 SEm
SEm = SD/√n
Find the mean first = sum of scores/number of scores
Subtract each of the scores from the mean and square each difference. Find the sum of these squares. Divide that by the number of scores to get variance.
Standard deviation = square root of variance
I'll let you do the calculations.
6.95 ± 1.96 (.352) =
To calculate the 95% confidence interval for the hours of sleep a student gets, you can follow these steps:
1. Calculate the sample mean: Add up all the hours of sleep and divide by the number of students. In this case, there are 20 students, so the sum of hours of sleep is 144. The mean would be 144/20 = 7.2 hours.
2. Calculate the sample standard deviation: Find the difference between each individual data point and the mean, square each difference, add up all the squared differences, divide by the number of data points minus 1, and finally take the square root. In this case, the sum of squared differences is 42.8. So, the standard deviation would be sqrt(42.8/19) ≈ 1.49 hours.
3. Calculate the standard error of the mean: Divide the sample standard deviation by the square root of the number of data points. In this case, the standard error of the mean would be 1.49/sqrt(20), which is approximately 0.333 hours.
4. Calculate the margin of error: Multiply the standard error of the mean by the critical value for a 95% confidence level. The critical value for a 95% confidence level is approximately 1.96. Therefore, the margin of error would be 0.333 x 1.96 ≈ 0.652 hours.
5. Finally, calculate the confidence interval: Subtract the margin of error from the sample mean to get the lower bound and add the margin of error to the sample mean to get the upper bound. In this case, the lower bound would be 7.2 - 0.652 ≈ 6.548 hours, and the upper bound would be 7.2 + 0.652 ≈ 7.852 hours.
Interpretation:
The 95% confidence interval for the hours of sleep a student gets is approximately 6.548 to 7.852 hours. This means that we are 95% confident that the population mean of sleep hours for all students falls within this range.