6.3 (Green)Approximate

Binomial Distributions

PERFORMANCE EXAM: PERFORMANCE EXAM: Friday!Friday!

(topics are posted online)(topics are posted online)

Binomial Distribution

The binomial distribution describes the behavior of a count variable X if the following conditions apply:

1: The number of observations n is fixed.

2: Each observation is independent.

3: Each observation represents one of two outcomes ("success" or "failure").

4: The probability of "success" p is the same for each outcome.

Consider the binomial distribution consisting of n trials with probability p of success on each trial.

If and , then the binomial distribution can be approximated by a normal distribution with the following mean and standard deviation.

5np (1 ) 5n p

Mean:

Standard Deviation:

x np

(1 )np p

Example 1 – Find a Binomial ProbabilityAccording to a survey conducted by the Harris Poll, 23% of adults in the U.S. favor abolishing the penny and making the nickel the lowest denomination coin. You are conducting a random survey of 500 adults. What is the probability that you will find at most 106 adults who favor abolishing the penny?

x np

(1 )np p

( 106) ?P x

You Try – Find a Binomial ProbabilityUse the fact that hyperopia, or farsightedness, is a condition that affects approximately 25% of the adult population in the United States. Consider a random sample of 280 U.S. adults. What is the probability that 63 or more people are farsighted? What is the probability that 84 or fewer people are farsighted?

x np

(1 )np p

( 63)P x

( 84)P x

Homework

P. 227 #1-10

P. 237 #11-19

Study for Performance ExamPossible topics are posted on my website.

To test a hypothesis about a statistical measure for a population, use the following steps:

State the hypothesis.

Collect data and calculate the statistical measure.

Assume the hypothesis is true and calculate the resulting probability P. If P < 0.05, then REJECT.

Example 2 – Test a hypothesisA recent Harris Poll claimed the 44% of adults cut back on their spending in order to pay the increased price of gasoline. To test this finding, you survey 60 adults and find that 21 of them cut back on their spending in order to pay the increased price of gas. Should you reject the Harris Poll’s findings? Explain.

Step 1 – Step 1 – State the hypothesis:State the hypothesis: 44% of adults cut back their spending in order to pay the increased price of gasoline.

Step 2 – Step 2 – Collect the data and calculate a statistical Collect the data and calculate a statistical measuremeasure Your survey showed 21 out of 60 cut back. That’s 35%.

Example 2 – Test a hypothesis (cont.)

Step 3 – Step 3 – Assume the hypothesis is true:Assume the hypothesis is true: Find the probability that you could randomly select 21 or fewer adults out of 60 who have cut back on their spending in order to pay the increased gasoline price.

x np(1 )np p

( 21) ?P x

Example 2 – Test a hypothesis (CONT)Same poll that claimed the 44% of adults cut back on their spending in order to pay the increased price of gasoline. This time your survey found 18 out of 60 cut back.

Step 1 – Step 1 – State the hypothesis:State the hypothesis: 44% of adults cut back their spending in order to pay the increased price of gasoline.

Step 2 – Step 2 – Collect the data and calculate a statistical Collect the data and calculate a statistical measuremeasure Your survey showed 18 out of 60 cut back. That’s 30%.

Example 2 – Test a hypothesis (cont.)

Step 3 – Step 3 – Assume the hypothesis is true:Assume the hypothesis is true: Find the probability that you could randomly select 18 or fewer adults out of 60 who have cut back on their spending in order to pay the increased gasoline price.

x np(1 )np p

( 18)P x