binomial probabilities and normal approximations

AP STATS: Warm-Up

Agenda • Today: Binomial Probabilities • Wednesday: Open Notes Quiz (8.1)/Problem Set for the Half of the Class-Intro to geometric distributions • Thursday: Geometric Distributions • Friday: More on geometric distributions and review • Monday: Unit Test

The Binomial Coefficient

The number of ways of arranging k successes among n observations. i.e. think back to our “Pop Quiz” Example. How many ways are there to get exactly two answers correct? Or for 2 people out of 5 kids to have type o blood.

The Binomial Probability If X has the binomial distribution with n observations and probability p of success on each observation, the possible values of X are 0, 1, 2…, n. If k is any one of these values,

On your Formula Sheet

Example: Our “Pop Quiz”. What’s the Probability of getting EXACTLY 3 questions correct out of the ten. Remember that the questions had answers a-e.

• Example 8.6: INSPECTING SWITCHES – SRS of 10 switches from a LARGE shipment – 10% of the switches are “bad” – In other words: B(10, 0.1) – P(No more than 1 of the 10 switches are “bad”) – Draw a Probability histogram – Binompdf(n, p, X) and Binomcdf(n, p, X) Think of PDF as “Pointing” to a value and CDF as aCcumulating values from 0 through x. – Remember: NEVER PLAY XYLOPHONE!!

You Try!

Example 8.9: Pop Quiz • Let’s come back to our “pop quiz” example. The number of correct guesses has the B(10, 0.2) distribution. Say that we want to find the probability of getting exactly 4 questions correct.

• What if we want the probability that the person gets 4 or less questions correct? (now the cdf is more appropriate).

You Try! • CORRINE’S FREE THROWS – 75% lifetime free-thrower – 12 shots in a key game were takes, and ONLY 7 made … Is this “unusual”? – i.e. find the P(X≤ 7)

Mean and Standard Deviation of a Binomial Distribution. • If a count X has a binomial distribution based on n observations with probability p of success, what is its mean μ? • Take for example a free throw shooter who shoots 75%. What is the mean for 12 shots?

From Your Formula Sheet

Example 8.9: Pop Quiz – Mean and Standard Deviation • Let’s come back to our “pop quiz” example. The number of correct guesses has the B(10, 0.2) distribution. Say that we want to find the mean and standard deviation of the distribution?

Approximation of a Normal Distribution • As the number of trials n gets large, we can approximate the binomial distribution with a normal probability distribution! N(np, np(1- p)) As a rule of thumb , we will use the normal approximation when n and p satisfy: np ≥ 10

and

n(1-p) ≥ 10

Example 8.12: Attitudes Towards Shopping

By normalcdf() By the normal distribution

N(np, np(1- p))

HW #41: • Read the rest of Section 8.1 (until the end). • Complete exercises: 8.34, 8.35 (a-f), 8.36, 8.38 • Optional: 8.40