Question

2. According to survey data collected from 500 students in a recent Statistics class, the histogram for the number of hours of sleep they typically got per night is close to the normal curve with an average of 6.5 hours and a SD of 1.4 hours.

a) Find percentage of students who sleep between 7 and 8 hours

b) A student who sleeps 7 hours per night is at the _______th percentile of the score distribution

c) Find the first quartile of the sleep time and explain your result.

Answer 1

Solution:
Given in the question
number of hours of sleep they typically got per night is close to the normal curve with
Mean () = 6.5
Standard deviation() = 1.4
Solution(a)
We need to calculate the percentage of students who sleep between 7 and 8 hours i.e.
P(7<X<8) = P(X<8) - P(X<7)
Here we will use the standard normal distribution table, First, we will calculate Z-score which can be calculated as
Z-score =(X-)/ = (8-6.5)/1.4 = 1.07
Z-score = (7-6.5)/1.4 = 0.36
From Z table we found a p-value
P(7<X<8) = P(X<8) - P(X<7) = 0.8577 - 0.6406 = 0.2171
So there is 21.71 percent of students who sleep between 7 and 8 hours.
Solution(b)
We need to calculate the percentile of the score distribution for a student who sleeps 7 hours per night
Z-score = (7-6.5)/1.4 = 0.36
From Z table we found a p-value
P(X<7) = 0.6406
So A student who sleeps 7 hours per night is at the 64th percentile of the score distribution.
Solution(c)
the first quartile of the sleep time i.e. 25th percentile
p-value = 0.25 so from Z table we found Z-score = -0.6745
So Sleep time can be calculated as
X = + Z-score * = 6.5 - 0.6745*1.4 = 5.6 hours
the first quartile of the sleep time is 5.6 hours.