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CHAPTER 6: NORMAL DISTRIBUTION
(Skip sections 3 and 7)
Statistical thinking will one day be as necessary for efficient citizenship as the ability to read and
write
- H. G. Wells
For probability tables, see Appendix E, Table 2a or 2b
Example 1: A computer company has two sales representatives, Smith and Brown. Ms Smith
sells to banks and Mr. Brown sells to insurance companies. Monthly bank sales are normally
distributed with an average of $200,000 and a standard deviation of $30,000, whereas monthly
insurance sales are normally distributed with an average of $260,000 and a standard deviation of
$40,000. If Ms. Smith sells $260,000 of equipment in a given month, to do an equally good job,
how much equipment must Mr. Brown sell?
The standard normal variable Z has mean 0 and variance 1.
Example 2: You have two investment possibilities A and B. Although the exact returns are
unpredictable, you are told that the average returns on A and B are 10.4% and 11% respectively.
Assuming that returns are normally distributed with standard deviations of 1.2% for A and 4% for
B, which investment is more likely to produce a return of at least 12%?
Example 3: You need to order a certain quantity of items for a store. Although there are space
constraints, you would not want to run short either. Given that the weekly demand for that item is
normally distributed with a mean of 30 and a standard deviation of 10, how many should you
order for next week if you only want a 1% chance of running out?
Central Limit Theorem: If a sample is drawn from a population with mean  and variance 2
then the sample mean will be normally distributed with the same mean but variance 2/n, as long
as the sample size is sufficiently large.
Example 4: There are a number of different subway lines in London, some of which run in
parallel under the same streets. The builders of the Underground arranged this by stacking the
subway tunnels for each line under each other, sometimes two or three deep. At many
Underground stations, to get to the deepest tunnels you take a long escalator ride. For instance, at
the Pimlico station the escalator down to the deepest tunnel is like a moving stairway with 96
steps. During rush hour every single escalator step typically has two people on it side by side, so
that the escalator had to be designed to carry 192 people without overloading.
The population of Underground riders at rush hour is almost exclusively made up of adult
men and women, whose weight averages about 150 pounds with a standard deviation of about 28
pounds. If the engineers who planned the Pimlico station designed the escalator to carry 29,700
pounds worth of people without breaking, what proportion of the time when it is fully loaded with
192 people would it break down? Given that the morning and evening peak traffic periods
together last about an hour and a half, and the turnover on the escalator is such that it's like
having a new trip with 192 new people about every minute during this period, do you regard this
failure rate as acceptably low? If the engineers had wanted the escalator to fail only about once in
every 10,000 trips, how much weight should they have designed it to carry?
The normal distribution may be used to approximate the Binomial if the variance is at least 10.
Z=
X - np
(when p is close to 0.5 the normal approximation is very good)
np(1-p)
Generally if np and n(1-p) are each at least 5, the normal approximation works well.
If you cannot use the normal approximation, check if n is large (at least 20) and p is small (at
most 0.05) to use the Poisson approximation to the Binomial distribution with  = np.
The normal distribution may be used to approximate the Poisson distribution if  is at least 5.
Z=
X-

Example 5: Sheldahl, Inc. produces flexible circuits for automobiles and computers. In the
manufacturing process, circuit connectors are etched on thin copper film that is deposited on a
plastic film. If a bubble occurs in the film, the circuit is defective. The manager of quality
control has implemented a number of improvements in the process to reduce the number of
bubbles in the film. In the past 6 months the average number of defective circuits per roll was 20.
With the new procedures, a roll was produced that only had 13 defective circuits. Has an
improvement in the process occurred?
Example 6: The sales department of a computer manufacturing company is experimenting with
the use of telephone contacts with potential customers instead of personal visits. Past data
indicate that 30% of personal visits have resulted in a sale. After 3 weeks, telephone calls to 50
potential customers have resulted in 8 sales. The manager wants to know if the use of phone
contacts is decreasing sales.
Homework: # 43, 70, and 74