# The Normal Approximation - PowerPoint PPT Presentation

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The Normal Approximation. Unit 5, Day 4. Learning Goals for Today. I can identify the relationship between the Binomial Distribution and the Normal Distribution. I can state and find the mean and standard deviations used to approximate binomial distributions .

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The Normal Approximation

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## The Normal Approximation

Unit 5, Day 4

### Learning Goals for Today

• I can identify the relationship between the Binomial Distribution and the Normal Distribution.

• I can state and find the mean and standard deviations used to approximate binomial distributions.

• I can solve problems using Normal Approximation using various aids(examples, notes, etc.).

### Binomial Distribution Recap

We have seen that probabilities can be calculated using the formula for a binomial distribution provided that:

• Each trial in the experiment is identical.

• The outcomes of each trial can be classified as success or failure

### Investigation

A regular die is rolled 100 times and the number of 5’s that appear is recorded. Determine the probability that at most twenty 5’s are rolled.

### An observation…

Above are two graphs that show the probability distributions for the number of 5’s rolled based on 5 and 50 trials. What do you notice?

### Even more trials…

Key Point: As the number of trials increase, the Binomial Distribution can be modeled using the Normal Distribution!!!!!

### The Normal Approximation

If X has a binomial distribution with n trials and probability of success p, then it can be approximated by a normal distribution with

and

provided that np>= 5 and nq>=5

This means that we can take those huge binomial distribution questions, convert over to a Normal Distribution, and use z-scores to solve it!

### The Continuity Correction

One issue is the fact that a Binomial Distribution is discrete (contains only countable outcomes) and a Normal Distribution is continuous.

To account for this, we use a continuity correction (a value of +/- 0.5):

For example:

is corrected to

is corrected to

### Back to the Investigation

A regular die is rolled 100 times and the number of 5’s that appear is recorded. Determine the probability that at most twenty 5’s are rolled.

### Example #2 – Coin tossing

A coin is tossed 60 times.

• Find the probability that you toss tails exactly 25 times.

• Estimate the probability you get tails 25 or more times.

### Homework

Page 311 #3, 4, 5, 6, 7, 8