Final answer:
To construct the 95% confidence interval for the difference in sleep hours between male and female students, we use the given means and standard errors, together with a Z-value of 1.96, resulting in a confidence interval of approximately (-5.01, 3.81) hours.
Step-by-step explanation:
To construct a 95% confidence interval for the difference between two means (male and female students' sleep hours), we can use the formula for the confidence interval of the difference between two independent means when we have standard errors instead of standard deviations.
The formula is:
CI = (mean1 - mean2) ± Z*(√(SE1² + SE2²))
In this case, for male students (M) we have a mean (μ_M) of 7.6 hours and a standard error (SE_M) of 1.68. For female students (F), the mean (μ_F) is 8.2 hours with a standard error (SE_F) of 1.5 hours.
The 95% confidence interval Z-value for a two-tailed test is typically 1.96 (you can find this value in a Z-table or use a calculator with statistical functions).
So, the confidence interval for the difference in mean sleep hours is calculated as follows:
CI = (7.6 - 8.2) ± 1.96*(√(1.68² + 1.5²))
CI = (-0.6) ± 1.96*(√(2.8224 + 2.25))
CI = (-0.6) ± 1.96*(√5.0724)
CI = (-0.6) ± 1.96*(2.25)
CI = (-0.6) ± 4.41
Therefore, the 95% confidence interval for the difference between male and female students' sleep hours is approximately (-5.01, 3.81) hours.
Complete question:
Suppose a random sample of high school students is selected to determine if there is a difference between how long male and female students sleep at night. If 92 male students are randomly chosen and yield an average of 7.6 hours of sleep with a standard error of 1.68. And 65 females are randomly selected with an average of 8.2 hours of sleep with a standard error of 1.5. Construct a 95% confidence interval for the difference between the two mean sleep hours for males vs. females.