Central limit TheoremSample Distribution Models for Means and Proportions

Distribution of sample proportions•We’re interested in finding a certain

proportion from a population. •We would need to take a sample from the

population. From this we’ll find the sample proportion,

•If we did this repeatedly keeping the same sample size and looked at the distribution of sample proportions:

•From the CLT, the distribution would be approximately Normal

( )p

( )p

•The mean of the sample distribution would be the population proportion,

•The standard deviation is

•There are two assumptions that must be met for this to be true

1.The sampled values must be independent2.The sample size, n, must be large enough

p̂

pq

n

p̂ p

,pq

N pn

Conditions to check for the assumptions1.Success/Failure: The expected number

of successes and failure is both greater than 10

2.10% Condition: Each sample is less than 10% of the population

3.Randomization: The sample was obtained through random sample techniques or we can at least assume that the sample is representative.

10 10np and nq

Example: Skittles•According to the manufacturer of the

candy Skittles, 20% of the candy produced is the color red. What is the probability that given a large bag of skittles with 58 candies that we get at least 17 red?

Conditions:1.10% condition: 58 skittles is less than

10% of all skittles produced.2.Success/Failure:

3.Randomization: Though not from a

random sample we can assume the bag is representative of the population.

We can use the Normal model for the distribution of sample proportions.

58 0.20 11.6 10np 58 0.80 46.4 10nq

•Mean: •Standard Deviation:

•So the model for becomes N(0.20,0.0525)

• Sample proportion:

0.20p

0.20 0.800.0525

58

pq

n

17ˆ 0.29358

p

p̂

•Then to find the probability that we get a sample proportion of 0.293 or higher:

0.293 0.201.771

0.0525z

ˆ( 0.293) ( 1.771) 0.0383P p P z

Central Limit Theorem•The mean of a random sample has a

sampling distribution whose shape can be approximated by a Normal model.

•The larger the sample, the better the approximation will be.

•This is regardless of the shape of the distribution of the population being sampled from or the shape of the distribution of the sample.

Distribution of Sample Means•A sample of size n is taken from a

population with mean, µ, and standard deviation, σ.

•From the CLT the distribution of the sample means, , will be approximately Normal.

•The mean is the mean of the population:

•The standard deviation is the population standard deviation divided by the square root of sample size, n.

x

x

xn

Assumptions

•There are two assumptions that must be met for this to be true

1.The sampled values must be independent2.The sample size, n, must be large enough

•Conditions:1.Randomization: The sample was obtained

through random sample techniques or we can at least assume that the sample is representative.

2.Independence(10% Condition)3.Large enough sample: •No set rule here. •If the population distribution is thought to

be unimodal and symmetric a small sample size will do.

•If it is more skewed, larger samples are needed. n > 30

Example: SAT Scores•Suppose that male seniors have a mean

score of 1200 with a standard deviation of 130.

•If we take a random sample of 100 male seniors what would be the distribution of sample means look like?

•What would be the probability that we would get an average of less than 1150?

Conditions1.Randomization: Stated as a random

sample2.Independent: 100 male seniors is less

than 10% of all male seniors3.A sample of 100 would be a large enough

sample

A Normal model would be appropriate for the distribution of sample means.

1200 130 100n

•Parameters:

•So we have a Normal model N(1200,13).

1200x 130

13100

xn

1150 12003.846

13z

5( 1150) ( 3.846) 6 10 0P X P z