7.5 Connections to Discrete Random Variables with ink.notebook
December 16, 2016
Unit 8 (Chapter 7): Probability Distributions with Continuous Variables
7.5 ­ Connections to Discrete Random Variables
7.5 Connections to Discrete Random Variables with ink.notebook
December 16, 2016
Probability Distributions for Discrete Variables
• Recall: Binomial Distribution
– For repeated trials
– success or failure
– trials are independent
– P(x) = nCx px qn­x
• If n(# of trials) is large, the binomial distribution becomes difficult to work with, and the binomial distribution takes on the characteristics of the normal distribution.
• Recall: Hypergeometric Distributions
> repeated trials
> success or failure
> trials are dependent
> • If the sample size is small compared to the population size, a hypergeometric distribution takes on the characteristics of a normal distribution.
7.5 Connections to Discrete Random Variables with ink.notebook
7.5
December 16, 2016
Connections to Discrete Random Variables
When is a normal approximation reasonable?
Binomial Distribution ­ usually considered reasonable if np > 5 and nq > 5
Click to Reveal
Hypergeometric Distribution ­ usually considered reasonable if n < 0.1 NP, where n is the Click to Reveal
sample size and NP is the population size.
Why is it not considered reasonable to use a normal approximation outside these parameters?
7.5 Connections to Discrete Random Variables with ink.notebook
7.5
December 16, 2016
Connections to Discrete Random Variables
Continuity Correction
Suppose you want to use a normal approximation to determine the probability of flipping a coin 5 times and getting exactly 1 head. You cannot simply determine the area under the normal curve for that particular point, as there is no area under a point. (A point has a width of zero). We apply a continuity correction in cases like this to determine the probability of discrete outcomes using a (continuous) normal distribution.
This rectangular area can be approximated by considering the area under the normal curve, over the same interval. So, the approximate probability is given by P(1 head) = P(0.5 < X < 1.5).
7.5 Connections to Discrete Random Variables with ink.notebook
December 16, 2016
The Normal Approximation to the Binomial Distribution
Steps:
1.
2.
3.
Check if you can use the normal to approximate the binomial.
To use the normal distribution, we need and . For a binomial distribution, we can calculate and using: Since we are dealing with discrete data in a binomial distribution, we must use the continuity correction (i.e. add/subtract 0.5 to ends)
7.5 Connections to Discrete Random Variables with ink.notebook
7.5
December 16, 2016
Connections to Discrete Random Variables
Example 1
Normal Approximation for a Binomial Distribution
Charlie didn't study for his Data Management quiz. The quiz consists of 25 multiple choice questions with 4 choices per question. Charlie guesses an answer for each question.
b) What values need to be determined in order to use a normal approximation? Determine these values.
Hint
c) What is the probability that Charlie will get a passing grade (50% or more) on this quiz?
Hint
a) Is it reasonable to approximate this distribution with a normal distribution?
Click here for the solution.
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mean
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7.5 Connections to Discrete Random Variables with ink.notebook
December 16, 2016
Solutions
Example 1
Normal Approximation for a Binomial Distribution
a)
Is it reasonable to approximate this distribution with a normal distribution?
Yes. The probability of guessing correctly on each question is 25%, so p = 0.25 and q = 0.75. Since np = (25)(0.25) = 6.25 > 5 and nq = 18.75 > 5, it is reasonable to approximate this binomial distribution with a normal distribution. b) What values need to be determined in order to use a normal approximation? Determine these values.
In order to use a normal approximation, we need to determine the mean and standard deviation:
c) What is the probability that Charlie will get a passing grade (50% or more) on this quiz?
The lowest passing grade would be 13 out of 25. We should use a continuity correction and calculate P(X > 12.5).
P(X > 12.5) = 0.00195
The probability of Charlie passing this quiz is less than 0.2%
Go back to the question
7.5 Connections to Discrete Random Variables with ink.notebook
December 16, 2016
• Ex: According to a survey, 44% of Canadian students think that the postal service is "acceptable". What is the probability that 140 or more Canadians in a random sample of 300 will hold this opinion?
Solution:
7.5 Connections to Discrete Random Variables with ink.notebook
December 16, 2016
7.5 Connections to Discrete Random Variables with ink.notebook
December 16, 2016
7.5 Connections to Discrete Random Variables with ink.notebook
December 16, 2016
The Normal Approximation to the Hypergeometric Distribution
Notes:
• Check if you can use the normal to approximate the hypergeometric. 1. To use the normal distribution, we need and . For a hypergeometric distribution, we can calculate and using: and
2.
Since we are dealing with discrete data in a hypergeometric distribution, we must use the continuity correction (i.e. add/subtract 0.5 to ends)
7.5 Connections to Discrete Random Variables with ink.notebook
7.5
December 16, 2016
Connections to Discrete Random Variables
Example 2
Normal Approximation for a Hypergeometric Distribution
Hint
Lizzie deals 5 cards from a standard deck of 52 cards. She would like to deal as many face cards as possible.
c) What is the probability that Lizzie will deal 3 or more face cards?
Click here for the solution.
Hint
b) What values need to be determined in order to use a normal approximation? Determine these values.
Hint
a) Is it reasonable to approximate this distribution with a normal distribution?
To us
you n
mean
7.5 Connections to Discrete Random Variables with ink.notebook
December 16, 2016
Solutions
Example 2
Normal Approximation for a Hypergeometric Distribution
a) Is it reasonable to approximate this distribution with a normal distribution?
Yes. Since n = 5 is slightly less than 0.1NP = 0.1(52), it is considered reasonable to approximate this scenario with a normal distribution.
b) What values need to be determined in order to use a normal approximation? Determine these values.
c) What is the probability that Lizzie will deal 3 or more face cards?
We should use a continuity correction.
There is about a 5% probability that Lizzie will deal 3 or more face cards.
Go back to the question
7.5 Connections to Discrete Random Variables with ink.notebook
Work: December 16, 2016