Normal distribution Suraj paudel
Problems on Normal distribution:
1. Given a standardized normal distribution (with a mean of 0 and a standard deviation of 1)
what is the probability that
a. Z is less than 1.5? Ans 0.9418
b. Z is greater than 1.84? Ans 0.0329
c. Z is between 1.57 and 1.84? Ans 0.0253
d. Z is less than 1.57 or greater than 1.84? Ans0.9747
e. Z is between -1.57 and 1.84? Ans 0.9089
f. Z is less than -1.57 or greater than 1.84? Ans 0.0911
2. Given a normal distribution with µ = 100 and = 10, what is the probability that
a. X > 75 answer 0.9938
b. X < 70 answer 0.00135
c. 75 < X <85 answer 0.0606
d. X < 80 or X > 110. Answer 0.1815
e. 10% of the values are less than what Value? Answer 87.20
f. 80% of the values are between what two values of X (symmetrically
distributed around the mean) ans 87.20 and 112.80
3. Toby’s Trucking Company determined that on an annual basis, the distance traveled per
truck is normally distributed with a mean of 50 thousand miles and a standard deviation
of 12 thousand miles.
a. What proportion of trucks can be expected to travel between 34 and 50 thousand
miles in the year? (Ans 0.4082)
b. What percentage of trucks can be expected to travel either below 30 or above 60
thousand miles in the year? (Ans 25.18%)
c. How many of the 1000 trucks in the fleet are expected to travel between 30 and 60
thousand miles in the year?
d. How many miles will be traveled by at least 80% of the trucks?
4. The amount of time necessary for assembly line workers to complete a product is a
normal random variable with a mean of 15 minutes and a standard deviation of 2 minutes.
a. So, 17% of the product would be assembled within how many minutes? Answer
13.1
b. So, 25% of the product would be assembled within how many minutes? (P.U.
2005)
5. Given that X is a normally distributed random variable with a mean of 50 and a standard
deviation of 2. What is the probability that X is between 47 and 54? Answer 0.9104
6. The owner of a fish market determined that the average weight for a catfish is 3.2 pounds
with a standard deviation of 0.8 pounds. Assuming the weight of catfish are normally
distributed
a. What is the probability that a randomly selected catfish will weight between 3 and 5
pounds? (Ans 0.5865)
b. Above what weight (in pounds) do 89.80% of the weights occur? (Ans 2.184) (PU
2007)
c. What is the probability that a randomly selected catfish will weight more than 4.4
pounds?
7. The number of column inches of classified advertisements appearing on Monday in a
certain daily newspaper is normally distributed with a population mean of 320 inches and
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Normal distribution Suraj paudel
a population standard deviation of 20 inches. What is the probability that there will be
between 280 and 360 column inches of classified advertisements? Answer 0.9545
8. If we know that the length of time it takes a college students to find a parking spot in the
library parking lot follows a normal distribution with a mean of 3.5 minutes and a
standard deviation of 1 minutes,
a. Find the probability that a randomly selected college student will take between 2
and 4.5 minutes to find a parking spot in the library parking lot. Answer 0.7745
b. Find the point in the distribution that 75.8% of the college students exceed when
trying to find a parking spot in the library parking lot. Answer 2.8 minutes
9. A medical company is developing a compact kidney dialysis machine, but its chief
engineer, Rajesh Vaidya, is having trouble controlling the variability of the rate at which
fluid moves through the device. Medical standards require that the hourly flow be 4 liters
plus or minus 0.1 liter, 80% of the time. Mr. Vaidya, in testing the prototype has found
that mean hourly flow is 4.02 liters with standard deviation of 0.08 hours. Does the
prototype satisfy the medical standards? Answer P(3.9<X<4.1) = 0.7745; does not
satisfy medical standards. (2013 Fall)
10. The breaking strength of plastic bags used for packaging produce is normally distributed
with a mean of 5 pounds per square inch and a standard deviation of 1.5 pounds per inch.
What proportion of the bags have a breaking strength of (P.U.2006)
a. Less than 3.17 pounds per square inch? Answer 0.1112
b. At least 3.6 pounds per square inch? Answer 0.8238
c. Between 5 and 5.5 pounds per square inch? Answer 0.1293
d. Between what two values symmetrically distributed around the mean will 95% of
the breaking strengths fall? Answer 2.06 and 7.94
11. A set of final examination grades in an introductory statistics course was found to be
distributed with mean of 73 and standard deviation 0f 8(P.U. 2004,2010 Fall)
a. What is the percentage of students scored between 65 and 89? Ans 81.85%
b. What is the probability of getting a grade no higher than 91 on this exam?
c. Only 5% of the students taking the test scored higher that what grade? Ans 86.16
d. What percentage of students scored between 65 and 89?
e. What is the probability of getting a grade of higher than 91 on this exam? 0.9878
12. A statistical analysis of 1,000 long distance telephone calls made from the headquarters
of the Bricks and Clicks Computer Corporation indicates that the length of these calls is
normally distributed with µ = 240 seconds and = 40 seconds.
a) What percentage of these calls lasted less than 180 seconds? Answer 0.0668
b) What is the probability that a particular call lasted between 180 and 300 seconds?
Answer 0.8664
c) How many calls lasted less than 180 seconds or more than 300 seconds? Answer
133.6
d) What is the length of a particular call if only 1% of all calls are shorter? Answer
146.80 seconds
13. Unisys.com is one of the most frequented business to business Web sites, assume that the
length of a visit on the Unisys Web sites is distributed as a normal random variable with a
mean of 65.7 minutes and a standard deviation of 15 minutes.
a. What is the probability that a randomly selected visit will last more than 90
minutes?
b. Only 20% of the visits will last less than how many minutes?
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Normal distribution Suraj paudel
c. Between what two values (in minutes) symmetrically distributed around the mean
will 90% of the visits last?
14. Many manufacturing problems involve the accurate matching of machine parts such as
shafts that fit into a valve hole. A particular design requires a shaft with a diameter of 22
mm, but shafts with diameters between 21.9 and 22.01 are acceptable. Suppose that the
manufacturing process yields shafts with diameters normally distributed with a mean of
22.002mm and a standard deviation of 0.05 mm. for this process, what is
a. The proportion of shafts with a diameter between 21.90 and 22? Answer 0.3446
b. The probability of an acceptable shaft? Answer 0.9452
c. The diameter that will be exceeded by only 2% of the shafts? Answer 22.0123
15. A continuous random variable, X follows a normal distribution such that 80% of the
values are between two X values viz. 87.2 and 112.8 (symmetrically distributed around
the mean). Find mean (µ) and standard deviation ( ) of the distribution. (Ans µ=100,
=10) (P.U. 2003)
16. Given that a continuous random variable, X has a normal distribution with µ = 750 and
= 100. Find the probability that X lies between 554 and 946.(P.U.2003)
17. The manager of a small postal substation is trying to qualify variation in the weekly
demand for mailing bags. She has decided to assume that this demand is normally
distributed. She knows that on average 100 bags are purchased weekly and that 90% of
the time weekly demand is below 115. What is the standard deviation of this distribution?
(Ans 11.72)
18. The entrance score of BBA students in PU is normally distributed with mean 65 and
standard deviation of 10
a. Find the lowest score of top 20% students. Answer 73.4
b. Find the highest score of lowest 30% students. Answer 59.8
c. What are the limits within which the middle 50% of the scores lie? Answer 58.3
and 71.7
19. A batch of 5000 electric lamps has a mean life of 1000 hours and a s.d. of 75 hours.
Assume a normal distribution. (PU 2009)
a. What percentage of lamps will fail before 925 hours? Ans 15.87%
b. How many lamps will fail between 950 and 1000 hours? Ans 1243
20. For overseas flights, an airline has 3 different choices on its desert menu – ice cream,
apple pipe, and chocolate cake. Based on the past experience, the airline feels that each
dessert is equally chosen. If a random sample of 90 passengers is selected, what is the
probability that
a) At least 20 will choose ice cream for dessert? (Answer: 0.9887)
b) Exactly 20 will choose ice cream for dessert? (Answer: 0.0084)
c) Less than 20 will choose ice cream for dessert? (Answer: 0.0197)
21. You are a customer representative for Zeal. Customer calls arrive at the rate of 25 every
10 minutes. What is the approximate probability of getting 20 to 30 calls in 10 minutes?
(Answer: 0.6802)
22. 10% of the electric bulbs manufactured by a company are defective. Find the probability
that in a sample of 200 bulbs.
i. 40 or more bulbs will be defective
ii. Exactly 30 bulbs will be defective
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