Question

question 6
for a 4 - unit class like statistics, students should spend average of 12 hours studying for the class. a survey was done on 21 students, and the distribution of total study hours per week is bell - shaped with a mean of 12 hours and a standard deviation of 3 hours.
use the empirical rule to answer the following questions.
a) 68% of the students spend between hours and hours on statistics each week.
b) 95% of the students spend between hours and hours on statistics each week.
c) 99.7% of the students spend between hours and hours on statistics each week.

Explanation:

Step1: Recall Empirical Rule for normal distribution

For a normal - distribution, about 68% of the data lies within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations.
Let $\mu$ be the mean and $\sigma$ be the standard deviation. Here, $\mu = 12$ hours and $\sigma=3$ hours.

Step2: Calculate bounds for 68%

The lower - bound is $\mu-\sigma$ and the upper - bound is $\mu + \sigma$.
$\mu-\sigma=12 - 3=9$
$\mu+\sigma=12 + 3=15$

Step3: Calculate bounds for 95%

The lower - bound is $\mu - 2\sigma$ and the upper - bound is $\mu+2\sigma$.
$\mu - 2\sigma=12-2\times3=12 - 6 = 6$
$\mu+2\sigma=12 + 2\times3=12 + 6=18$

Step4: Calculate bounds for 99.7%

The lower - bound is $\mu - 3\sigma$ and the upper - bound is $\mu+3\sigma$.
$\mu - 3\sigma=12-3\times3=12 - 9 = 3$
$\mu+3\sigma=12 + 3\times3=12 + 9=21$

Answer:

a) 9 hours and 15 hours
b) 6 hours and 18 hours
c) 3 hours and 21 hours