SAMPLE PROPORTIONS

WHAT DO YOU THINK?

Are these parameters or statistics?

What specific type of parameter/statistic are they?

How do you think they were calculated? How can we be sure that method yielded a good estimate?

91% of teens have been bullied

74% of US teens carry a cell phone

57% of teens credit their cell phone with improving their life

42% of teens can text blindfolded

Source: stageoflife.com

SAMPLE PROPORTIONS

To figure out how we determine a proportion of interest in a large population.

So we can gain information about populations even when we can’t survey the entire population.

Objective Purpose

SAMPLING DISTRIBUTION OF A SAMPLE PROPORTION

The sample proportion is the statistic that we use to gain information about the unknown population parameter p.

How good is the statistic as an estimate of the parameter p?

To find out, we ask, “What would happen if we took many samples?” The sampling distribution of answers this question.

ˆ p

ˆ p

ˆ p

SAMPLING DISTRIBUTION OF A SAMPLE PROPORTION

The sample proportion is the statistic that we use to gain information about the unknown population parameter p.

How good is the statistic as an estimate of the parameter p?

To find out, we ask, “What would happen if we took many samples?” The sampling distribution of answers this question.

ˆ p

ˆ p

ˆ p

SAMPLING DISTRIBUTION OF A SAMPLE PROPORTION

The sample proportion is the statistic that we use to gain information about the unknown population parameter p.

How good is the statistic as an estimate of the parameter p?

To find out, we ask, “What would happen if we took many samples?” The sampling distribution of answers this question.

ˆ p

ˆ p

ˆ p

SAMPLING DISTRIBUTION OF A SAMPLE PROPORTION

The sample proportion is the statistic that we use to gain information about the unknown population parameter p.

How good is the statistic as an estimate of the parameter p?

To find out, we ask, “What would happen if we took many samples?” The sampling distribution of answers this question.

ˆ p

ˆ p

ˆ p

SAMPLING DISTRIBUTION OF A SAMPLE PROPORTION

The sample proportion is the statistic that we use to gain information about the unknown population parameter p.

How good is the statistic as an estimate of the parameter p?

To find out, we ask, “What would happen if we took many samples?” The sampling distribution of answers this question.

ˆ p

ˆ p

ˆ p

SAMPLING DISTRIBUTION OF A SAMPLE PROPORTION

Standard Deviation (

)

ˆ p

Mean ( )

ˆ p

Values of

ˆ p

SAMPLING DISTRIBUTION OF A SAMPLE PROPORTION

Standard Deviation (

)

ˆ p

Mean ( )

ˆ p

Values of

ˆ p

SAMPLING DISTRIBUTION OF A SAMPLE PROPORTION

Standard Deviation (

)

ˆ p

Mean ( )

ˆ p

Values of

ˆ p

SAMPLING DISTRIBUTION OF A SAMPLE PROPORTION

Chapter 7:

If

then

Chapter 8:

= count of “successes” in sample

size of sample

in ghosts.

Toolbox Mean

ˆ p

ˆ p X n (1 n)X

Y a bX

X np

Y a bX

X np(1 p)

ˆ p 0 (1/n)X

ˆ p 0 1

nnp p

Ex: Do You Believe in Ghosts?

160 / 515 = 0.31 said yes!

Y bX

SAMPLING DISTRIBUTIONOF A SAMPLE PROPORTION

Toolbox Standard Deviation

ˆ p 0 (1/n)X

ˆ p 1n

np(1 p)

np(1 p)

n2

p(1 p)n

Chapter 7:

If

then

Chapter 8:

Y a bX

X np

Y a bX

X np(1 p)

Y bX

IN ENGLISH, PLEASE!

The mean of the sampling distribution of a sample proportion is exactly p.

The standard deviation of the sampling distribution of a sample proportion is

ˆ p p(1 p)

n

ˆ p p

SO WHAT?

The sample proportion is an unbiased estimator of p!

The standard deviation of gets smaller as the sample size n increases.

That is, is less variable in larger samples.

ˆ p

ˆ p

ˆ p

SO WHAT?

The sample proportion is an unbiased estimator of p!

The standard deviation of gets smaller as the sample size n increases.

That is, is less variable in larger samples.

ˆ p

ˆ p

ˆ p

SO WHAT?

The sample proportion is an unbiased estimator of p!

The standard deviation of gets smaller as the sample size n increases.

That is, is less variable in larger samples.

ˆ p

ˆ p

ˆ p

SO WHAT?

The sample proportion is an unbiased estimator of p!

The standard deviation of gets smaller as the sample size n increases.

That is, is less variable in larger samples.

ˆ p

ˆ p

ˆ p

WARNING!

The formula for the standard deviation of doesn’t apply when the sample is a large part of the population. (In that case we could just examine the entire population!)

Rule of Thumb 1: Use the formula

for the standard deviation of only

when the population is at least 10

times as large as the sample.

ˆ p

ˆ p

USING THE NORMAL APPROXIMATION

The sampling distribution of is approximately normal

The larger the sample size n, the closer the sampling distribution is to a normal distribution

Rule of Thumb 2: We will use the

normal approximation to the sampling

distribution of for values of n and p

that satisfy np ≥ 10 and n(1-p) ≥ 10.

ˆ p

ˆ p

EXAMPLE: DO YOU JOG? (HOMEWORK PROBLEM #25)

The Gallup Poll once asked a random sample of 1540 adults, “Do you happen to jog?” Suppose that in fact 15% of all adults jog.

a) Find the mean and standard deviation of the proportion of the sample who jog. (Assume the sample is an SRS.)

b) Explain why you can use the formula for the standard deviation of in this setting.

c) Check that you can use the normal approximation for the distribution of .

d) Find the probability that between 13% and 17% of the sample jog.

e) What sample size would be required to reduce the standard deviation of the sample proportion to one-third the value you found in (a)?

ˆ p

p̂

ˆ p

EXAMPLE: RULES OF THUMB(HOMEWORK PROBLEM #30)

Explain why you cannot use the methods of this section to find the following probabilities.

a) A factory employs 3000 unionized workers, of whom 30% are Hispanic. The 15-member union executive committee contains 3 Hispanics. What would be the probability of 3 or fewer Hispanics if the executive committee were chosen at random from all the workers?

b) A university is concerned about the academic standing of its intercollegiate athletes. A study committee chooses an SRS of 50 of the 316 athletes to interview in detail. Suppose that in fact 40% of the athletes have been told by coaches to neglect their studies on at least one occasion. What is the probability that at least 15 in the sample are among this group?

c) Use what you learned in Chapter 8 to find the probability described in part (a).

REESE’S PIECES ACTIVITY!

Learning Goal:

To understand the effect of sample size on the sampling distribution.

www.Rossmanchance.com/applets/Reeses/ReesesPieces.html

CLOSING SUMMARY

Today we continued learning about sampling distributions.

We looked at one particular type of sampling distribution – the sampling distribution of the sampling proportion.

We learned that as long as the population (Rule of Thumb 1) and sample size (Rule of Thumb 2) are large enough, the sampling distribution is approximately normal with mean p and standard deviation

p(1 p)

n

EXIT TICKET

Suppose I want to know what proportion of U.S. teenagers play a sport. How could I come up with an answer to my question? Why does that method work? How could I improve my results?

EXAMPLE: APPLYING TO COLLEGE

A polling organization asks an SRS of 1500 first-year college students whether they applied for admission to any other college. In fact, 35% of all first-year students applied to colleges besides the one they are attending. What is the probability that the random sample of 1500 students will give a result within 2 percentage points of this true value?

What are we given? What are we looking for? What does our sampling distribution look like?

Is our population at least 10 times the size of our sample? Can we use the normal approximation?