5.1 sampling distributions for counts and proportions (continued)

5.1 Sampling Distributions for Counts and Proportions (continued)

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5.1 Sampling Distributions for Counts

and Proportions (continued)

Sample Proportions

We ended last class with the following example:

A person has a 5% chance of winning a free ticket in a state lottery. If she plays the game 12 times, what is the probability she will win more than 1 free ticket?

What if we had asked for the probability that the sample proportion of wins was at most 25%?

Page 3: 5.1 Sampling Distributions for Counts and Proportions (continued)

Mean and Standard Deviation

We can also find the mean and standard deviation of a proportion.

pp ˆ

ppp

)1(ˆ

Page 4: 5.1 Sampling Distributions for Counts and Proportions (continued)

Example

The US Forestry Service reports that 20% of the oak trees in PA are infested with leaf rot. What is the probability that in a random sample of 20 trees, 10% or fewer have leaf rot? Find the mean and standard deviation of the sample proportion.

Page 5: 5.1 Sampling Distributions for Counts and Proportions (continued)

What do the mean and standard deviation of a proportion mean?

The mean of a proportion being equal to the probability of success means that the sample proportion is an unbiased estimator of the population proportion p.

As n gets larger, the standard deviation gets smaller.

Page 6: 5.1 Sampling Distributions for Counts and Proportions (continued)

Practice Problem with review

A Las Vegas casino has determined that 10% of the dice it purchases are unsuitable for game play. A random sample of 15 dice is taken.

1. Find the mean and standard deviation.

2. What is the probability this sample contains no unsuitable dice?

3. What is the probability this sample contains at most 20% unsuitable dice?

4. Find the mean and standard deviation of the sample proportion.

Page 7: 5.1 Sampling Distributions for Counts and Proportions (continued)

Normal Approximation

As we hinted at before, we may be able to approximate our discrete data with a normal curve.

Take a SRS of size n from a population. Let X be the count of successes and X/n the sample proportion of successes. When n is large, these statistics are approximately normal.

Page 8: 5.1 Sampling Distributions for Counts and Proportions (continued)

X is approximately

X/n is approximately

This is exactly the mean and standard deviation for a count and a proportion.

We use this if np≥10 and n(1-p)≥10.

))1(,( pnpnpN

))1(

,(n

pppN

Page 9: 5.1 Sampling Distributions for Counts and Proportions (continued)

Examples

Assume that we have a binomial experiment with n=40 and p=.3. Estimate the probability of

1. More than 10 successes.

2. 4 or less successes

3. Between 13 and 17 successes.

Page 10: 5.1 Sampling Distributions for Counts and Proportions (continued)

Practice Problem

A civil service multiple choice exam consist of 100 questions with four choices per question. Suppose an applicant randomly guesses the answer to every question. Estimate the probability that the applicant will answer 35 or more questions correctly. Estimate the probability that he will guess less than 28% correct.

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