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Normal Distributions Exercises
These exercises are for those operating at the A level of work.
1. The reading speed (in words per minute) of sixth graders is Normally distributed. The 90th
percentile is 156 words per minutes; the 5th percentile is 80 words per minute.
a. Units? Variable?
b. Find Z-scores for the 90th and 5th percentiles.
c. Determine the difference in reading speeds for students at the 90th and 5th percentiles
two ways:
 in words per minute
 in standard deviations from the mean (the difference between Z-scores)
d. Equate your two expressions from part c to obtain the standard deviation.
e. Now obtain the mean by solving for µ in the expression x =  + Z. You may use
either of the two given percentiles.
f. Check that
x

 z for the other given percentile.
g. What is the probability a student reads over 130 words per minute?
h. What is the 99th percentile of the distribution of reading scores?
Note: Future problems will not explicitly call for all of the steps above: Some or all of parts b – f
may be omitted. In solving this type of problem, recognize that you must obtain the mean and
standard deviation before computing probabilities or percentiles. It generally helps to have drawn
a picture of the situation.
2. An oil-change store guarantees that service will take no longer than 30 minutes, or else the
customer gets a free oil change. Only 0.5% of customers receive a free oil change.
Employees are given a small bonus if the oil change takes less than 12 minutes. This happens
12% of the time. Assume the time for an oil change is Normally distributed.
a. Units? Variable?
b. Determine the mean and standard deviation of the service times.
c. What is the probability a change takes longer than 25 minutes?
d. Identify the middle 95% of times for an oil change.
3. 1% of the reservations made by phone to a hotel take longer than 6 minutes. 10% take longer
than 5 minutes. How short would such a call be if it fell in the quickest 5% of all calls?
Answer to the nearest second. Assume the distribution of times is Normal.
4. A charter boat offers a tour to a remote Caribbean island where the peak wave height at noon
varies daily according to a Normal distribution. When the peak wave height is less than 3
feet, the boat can land at the beach, and tourists can wander about the island during lunchtime
(otherwise they eat lunch on the boat). When the peak wave height is over 7 feet, the boat is
detoured to a different location for the lunch. Over time the boat’s captain has learned that on
20% of days the boat can land; on 4% of days the alternate location is used.
a. Units? Variable?
b. What is the probability the peak wave height between 4 and 5 feet? For what percent
of days does a peak between 4 and 5 occur?
c. When the peak wave height is below 2 feet the tourists are allowed to swim. How
often does this occur?
d. Determine the interquartile range of the peak wave heights.
5. The serum cholesterol level in men aged 18-24 are Normally distributed. For a randomly
selected male of this age, the probability is 0.02 that his level is below 100. Cholesterol
levels in the top 3% are used to identify high risk individuals; men with level over 255 are at
high risk.
a. Obtain the 10th and 90th percentiles of this distribution.
b. Determine the probability of a male of this age having a level above 275. Take the
reciprocal of this probability. The value you obtain tells you how many men, on
average, need to be measured in order to find one who has level above 275.
6. A company manufactures cylindrical rods. The rods are discarded if they are too short –
under 1.764 m (meter), and shortened if they are too long – over 1.772 m. From employees’
records it’s learned that 0.12% of the rods are shortened, and 0.03% are discarded. Assume
rod lengths are Normally distributed.
a. Units? Variable?
b. Determine the mean and standard deviation of the rod lengths.
c. What percent of rods are between 1.769 and 1.771 m?
d. Find the length separating the longest 5% of rods from the remainder.
7. Diameters of #9 wine corks are Normally distributed. A cork collector has 1000s of these
corks and discovers that 2% of the corks are over 24.25 mm in diameter; 3% are less than
23.75 mm. If a cork is selected at random, what is the probability its diameter is within 0.1
mm of the mean?
Solutions
1.
a. The units are 6th graders; the variable is reading speed.
b. -1.6449 and 1.2816
c. 156 – 80 = 76 words per minute; the difference in Z-scores is 1.2816 – (–1.6449) =
2.9264 standard deviations.
d. 76 = 2.9264, so  = 76/2.9264 = 25.97043.
e. x =  + Z . For x = 156 this gives 156 =  + 1.2815(25.97043), so  = 122.7176.
f. For x = 80, Z 
80  122.7176
= –1.6449.
25.97043
g. 0.3896.
h. 183.134 words per minute.
2.
a. Oil changes (cars is OK); time to change oil.
b. The mean is 17.63871 with standard deviation 4.798956.
c. 0.0625
d. 8.233 to 27.044
3.
The mean is 3.7734 minutes with standard deviation 0.9571 minutes. Calls in the shortest
5% of all calls are no longer than 2.20 minutes – that’s 2:12 (2 minutes and 12 seconds).
4.
a. The units are days; the variable is peak wave height at noontime. Peak wave height at
noon varies from day to day.
The mean is 3.922 ft with standard deviation 1.758 feet.
b. 0.1513 (15.13% of days).
c. 0.1372.
d. 2.3717 feet = 28.46 inches.
5
The mean is 180.9068 with standard deviation 39.3947.
a. 10% of men fall below 130.4205; 10% of men above 231.3931. (The 10th percentile is
130.42; the 90th is 231.39.)
b. 0.00846 of men have level above 275. The reciprocal of this is 118 – for every 118
men, on average, there is one with level over 275.
6.
a. The units are the rods; the variable is length.
b. The mean is 1.767755 with standard deviation 0.001237.
c. 0.1528.
d. 1.76979.