Question

a professor was interested in determining the amount of time students spend, per week, on their classes outside of class time, such as for studying or completing homework or other assignments. the professor randomly selects 60 students and finds that the mean time they spend (in one week) on their classes outside of class time is 25.3 hours with a standard deviation of 7.2 hours. identify the correct formula and/or calculator function that would be used to construct a confidence interval for the mean time spent, per week, on their classes outside of class time.
formula: \\( hat { p } pm z _ { \frac { alpha } { 2 } } sqrt { \frac { hat { p } ( 1 - hat { p } ) } { n } } \\)
calculator function: 1 - propzint
formula:
\\( l b = \frac { ( n - 1 ) s ^ { 2 } } { x ^ { 2 } _ { \frac { alpha } { 2 } } } \\)
and
\\( u b = \frac { ( n - 1 ) s ^ { 2 } } { x ^ { 2 } _ { 1 - \frac { alpha } { 2 } } } \\)
formula: \\( overline { x } pm t _ { \frac { alpha } { 2 } } \frac { s } { sqrt { n } } \\)
calculator function: t - interval
formula: \\( l b = sqrt { \frac { ( n - 1 ) s ^ { 2 } } { x ^ { 2 } _ { \frac { alpha } { 2 } } } } \\)
and
\\( u b = sqrt { \frac { ( n - 1 ) s ^ { 2 } } { x ^ { 2 } _ { 1 - \frac { alpha } { 2 } } } } \\)

Explanation:

Step1: Check the formula type

The problem is about constructing a confidence interval for the mean. The formula $\hat{p}\pm z_{\frac{\alpha}{2}}\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$ is for a proportion (not a mean), so the first option is wrong. The formulas involving $\chi^{2}$ (second and fourth options) are for variance - related confidence intervals (not mean), so they are wrong.

Step2: Recall the mean confidence interval formula

For a confidence interval of the mean when the population standard deviation $\sigma$ is unknown (we use the sample standard deviation $s$ instead), the formula is $\bar{x}\pm t_{\frac{\alpha}{2}}\frac{s}{\sqrt{n}}$ and the calculator function is T - Interval.

Answer:

Formula: $\bar{x}\pm t_{\frac{\alpha}{2}}\frac{s}{\sqrt{n}}$, Calculator Function: T - Interval (the third option)