Download MDM 4UI: 2015-2016 Unit 9 Day 3: Normal Approximation to the

MDM 4UI: 2015-2016
Unit 9
Day 3: Normal Approximation to the
Hypergeometric Distribution
Last class we learned that we can approximate a discrete binomial distribution with a continuous normal
distribution, as long as np > 5 and nq > 5.
Similarly, we can approximate a discrete hypergeometric distribution with a continuous normal distribution, as
long as the sample size is small compared to the size of the population, typically less than one-tenth.
That is if n < 0.1 NP, where n is the sample size and NP is the size of the population.
If this is the case, then we can approximate using a normal distribution where:
𝜇 = 𝑛𝑝
𝑁𝑃−𝑛
𝜎 = √𝑛𝑝𝑞 ( 𝑁𝑃−1)
Since this is a discrete distribution that we are approximating with a continuous distribution, we will still need
to use a continuity correction.
Example 1
Chris works at a local daycare on a co-op work term. Chris plays a game with the children that involves pulling
marbles from a bag. The bag contains 24 black marbles and 36 red marbles, well mixed. One of the children
reaches in a takes out 5 marbles without looking. Chris records the number of black marbles.
a. Is it reasonable to approximate this distribution with a normal distribution?
b. Determine the mean and standard deviation of the normal approximation.
c. What is the probability that exactly 3 of the marbles are black?
d. What is the probability that a child pulls out
 at least 3 black marbles?
 More than 3 black marbles?
 Fewer than 3 black marbles?
 At most 3 black marbles?
Practice Problems
1. A bag of jellybeans contains 200 beans, of which 30 are red. Susan reaches into the bag and pulls out 15
beans at random. Can you model this with a normal approximation? What mean and standard deviation
would you use?
2. A barrel at the Pro Shop contains 30 white golf balls, 20 yellow golf balls, and 10 orange golf balls. A
contest requires a contestant to blindly select several balls without replacement. The prize depends on
the number of orange golf balls obtained. What is the maximum number of balls that could be selected
to model the contest using a normal approximation?
3. Five cards are dealt form a deck of 52 cards. The number of hearts is counted.
a. Is it reasonable to model this distribution with a normal distribution? Explain.
b. What mean should you use?
c. What standard deviation should you use?
4. Honey jars from the farm where Doris works say they contain 500 g of honey. A technician measures a
sample of 30 jars. The mean content is 502.83 g with a standard deviation of 1.95 g. The technician can
adjust the machine that fills the jars to change the mean. Assume that the standard deviation remains
unchanged.
a. Determine the probability that a honey jar contains less than 500 g of honey.
b. Do you need to use a continuity correction? Explain.
c. The owners of the company would like to ensure that the probability that a jar contains less than
500 g is at most 0.005. What setting for the mean is required?
5. Allison has a drawer full of unmatched socks. The drawer contains 30 blue socks, 30 green socks, and
30 yellow socks. She pulls seven socks from the drawer and records the number of blue socks in the
sample.
a. Is it reasonable to approximate this distribution with a normal distribution? Give a reason for
your answer.
b. Determine the mean and standard deviation of the normal approximation.
c. What is the probability that 3, 4, or 5 of the socks are blue?