Question

The average time spent sleeping​ (in hours) for a group of medical residents at a hospital can be approximated by a normal​ distribution, as shown in the graph to the right. Answer parts​ (a) and​ (b) below.

12345678910 Bold Sleeping Times ofSleeping Times of Bold Medical ResidentsMedical Residents mu equals 5.7 hoursμ=5.7 hours sigma equals 1.2 hourσ=1.2 hour Hours

Answer 1

Solution :

Given

Mean= 5.7

Standard deviation = 1.2

a)z = (X-Mean)/SD
Top 5% is represented by 0.0500 area in the extreme right tail of the normal curve.
The z value which separates this area from the remaining is + 1.645
1.645 = (X - 5.7)/1.2
X = 1.645 *1.2+ 5.7
X = 7.674 or 7.67 on rounding to 2 dps
shortest sleeping time that place a resident in the top 5% of is 7.67 hours

b) Middle 50% is represented by the middle 0.5000 area about mean. It implies that 0.5000/2 = 0.2500 is the area on the left side and 0.2500 is the area on the right side of mean.
The z values concerned are - 0.675 and + 0.675
- 0.675 = (X - 5.7)/1.2
X = 5.7 - 0.675*1.2
X = 4.89

+ 0.675 = (X - 5.7)/1.2
X = 5.7 + 0.675*1.2
X = 6.51
Middle 50% sleep times lie between 4.89 hours and 6.51 hours

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