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Sample Proportions Target Goal: I can FIND the mean and standard deviation of the sampling distribution of a sample proportion. DETERMINE whether or not it is appropriate to use the Normal approximation to calculate probabilities involving the sample proportion. CALCULATE probabilities involving the sample proportion. EVALUATE a claim about a population proportion using the sampling distribution of the sample proportion. 7.2a HW: pg 431: 21 – 24; pg 439: 27, 29, 33, 35, 37, 41

Sample Proportions Target Goal: I can FIND the mean and standard deviation of the sampling distribution of a sample proportion. DETERMINE whether or not

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Page 1: Sample Proportions Target Goal: I can FIND the mean and standard deviation of the sampling distribution of a sample proportion. DETERMINE whether or not

Sample Proportions

Target Goal: I canFIND the mean and standard deviation of the sampling distribution of a sample proportion.DETERMINE whether or not it is appropriate to use the Normal approximation to calculate probabilities involving the sample proportion.CALCULATE probabilities involving the sample proportion.EVALUATE a claim about a population proportion using the sampling distribution of the sample proportion.

7.2aHW: pg 431: 21 – 24; pg 439: 27, 29, 33, 35, 37, 41

Page 2: Sample Proportions Target Goal: I can FIND the mean and standard deviation of the sampling distribution of a sample proportion. DETERMINE whether or not

What is the proportion of teens that know that Columbus discovered America in 1492?

Gallop poll found that 210 out of a random sample of 501 teens knew this.

= 210/501 = 0.42

Page 3: Sample Proportions Target Goal: I can FIND the mean and standard deviation of the sampling distribution of a sample proportion. DETERMINE whether or not

Recall:

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

• The parameter is the population proportion. In practice, this value is always unknown.

(If we know the population proportion, then there is no need for a sample.)

• The statistic is the sample proportion. • We use to estimate the value of .

p

p̂p̂ p

Page 4: Sample Proportions Target Goal: I can FIND the mean and standard deviation of the sampling distribution of a sample proportion. DETERMINE whether or not

• The value of the statistic changes as the sample changes.

We describe the sampling model for by Shape Outliers Center Spread

Page 5: Sample Proportions Target Goal: I can FIND the mean and standard deviation of the sampling distribution of a sample proportion. DETERMINE whether or not

If our sample is an SRS of size n, then the following statements could describe the sampling model for :

1. The Shape is approximately normal.

ASSUMPTION: Sample size is sufficiently large.CONDITION: “Rule of Thumb 2” – Use the normal approximation when

10 10np and nq

Page 6: Sample Proportions Target Goal: I can FIND the mean and standard deviation of the sampling distribution of a sample proportion. DETERMINE whether or not

2. The center is described by the mean. Is

2. The spread is described by standard deviation.

is or

ASSUMPTION: Sample size is sufficiently large.

CONDITION: “Rule of Thumb 1” – Use

only when the population is at least 10 times as

large as the sample.

p̂ p

p̂ 1p p

n

pq

n

10population n

ˆ is an unbiased estimator of p p

As sample size increases, the spread decreases.As sample size increases, the spread decreases.

Page 7: Sample Proportions Target Goal: I can FIND the mean and standard deviation of the sampling distribution of a sample proportion. DETERMINE whether or not

Ex: Applying to Colleges

A SRS of 1500 first year college students were asked if they applied to any other college.

35% of all first year students said they applied to other colleges.

Determine the probability that the SRS of 1500 will give a result with in 2 percentage points of this true value.

Page 8: Sample Proportions Target Goal: I can FIND the mean and standard deviation of the sampling distribution of a sample proportion. DETERMINE whether or not

What does this mean “with in 2 percentage points of its

true value” ?

We want find the probability that falls between 0.33 and 0.37.

n = 1500, = 0.35

The sampling distribution has a mean = = 0.35p

p

Page 9: Sample Proportions Target Goal: I can FIND the mean and standard deviation of the sampling distribution of a sample proportion. DETERMINE whether or not

Find : “rule of thumb 1” says we must have;

Test: Is the population is at least 10 times as large as the sample?

10(1500) = 15,000

Yes, the population of first year students is over 1.7 million

Page 10: Sample Proportions Target Goal: I can FIND the mean and standard deviation of the sampling distribution of a sample proportion. DETERMINE whether or not

is =

= 0.0123

p̂ 1p p

n

(.35) .65

1500

Page 11: Sample Proportions Target Goal: I can FIND the mean and standard deviation of the sampling distribution of a sample proportion. DETERMINE whether or not

Is the distribution approximately normal?

“Rule of Thumb 2” – Use the normal approximation when

Test:

= 525 = 975

Yes, approximately normal.

110 10andnp n p

11500(.35) 1500 .650 10and

Page 12: Sample Proportions Target Goal: I can FIND the mean and standard deviation of the sampling distribution of a sample proportion. DETERMINE whether or not

Step 1: Standardize (create new statistic)

ˆ

ˆ

ˆ,

p

p

zp

0.35

0.

ˆ

0123z

p

P(0.33 ≤ ≤ 0.37)p̂

Page 13: Sample Proportions Target Goal: I can FIND the mean and standard deviation of the sampling distribution of a sample proportion. DETERMINE whether or not

Step2: Find z-scores of = 0.33 and = 0.37p̂ p̂

0.350.

0. 3

33

012z

1.63

0.350.

0. 3

37

012z

1.63

P(-1.63 ≤ z ≤ 1.63)

Page 14: Sample Proportions Target Goal: I can FIND the mean and standard deviation of the sampling distribution of a sample proportion. DETERMINE whether or not

Step3: Draw a picture. Use Table A or normcdf to find

desired area.

normcdf(-1.63,1.63) = .8968

Page 15: Sample Proportions Target Goal: I can FIND the mean and standard deviation of the sampling distribution of a sample proportion. DETERMINE whether or not

State results:

Almost 90% of all samples will give a result w/in 2 percentage points of the truth about the population.

In Context: 90% of samples of first year college students stated they applied to other colleges will give a result within 2 % points of the true proportion.

Page 16: Sample Proportions Target Goal: I can FIND the mean and standard deviation of the sampling distribution of a sample proportion. DETERMINE whether or not

Exercise:Do You Drink the Cereal Milk?

A USA Today poll asked a random sample of 1012 U.S. adults what they do with the milk in the bowl after they have eaten the cereal. Of the respondents, 67% said that they drink it. Suppose that 70% of U.S. adults actually drink the milk.

n = 1012p = 0.70

Page 17: Sample Proportions Target Goal: I can FIND the mean and standard deviation of the sampling distribution of a sample proportion. DETERMINE whether or not

a) Find the mean and standard deviation of the proportion of the sample that say they drink the cereal milk.

μ = p = 0.70

p̂ 1p p

n

(0.70) 0.30

1012

= 0.0144Why did we use .70 instead of .67?

This time we knew the population proportion.

Page 18: Sample Proportions Target Goal: I can FIND the mean and standard deviation of the sampling distribution of a sample proportion. DETERMINE whether or not

b) Explain why we can use the formula for the standard deviation of in this setting.(rule of thumb 1).

Clearly the U.S. population is 10 times greater than 1012.

c) Can we use the normal approximation? (rule of thumb 2)

Check:

np = 1012(0.70) ≥ 10?

n(1-p) = 1012(0.3) ≥ 10?

Yes.

Page 19: Sample Proportions Target Goal: I can FIND the mean and standard deviation of the sampling distribution of a sample proportion. DETERMINE whether or not

d) Find the probability of obtaining a sample of 1012 adults in which 67% or fewer say they drink the milk. Do we have any doubts about the result of this poll?

1) Find P( ≤ 0.67)

Draw a picture

2) Find the z-score

0.67 0.70

.0144p z

( 2.08)p z

= 0.0188

This is a fairly unusual result if 70% of the population actually drinks the milk.

Page 20: Sample Proportions Target Goal: I can FIND the mean and standard deviation of the sampling distribution of a sample proportion. DETERMINE whether or not

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

Just multiply the sample size by 4 or

p̂ 1p p

n

1

4n

p p

11

2

p p

n

ˆ

1

2p

Page 21: Sample Proportions Target Goal: I can FIND the mean and standard deviation of the sampling distribution of a sample proportion. DETERMINE whether or not

½ = and solve for np̂ p 1-p

n

If multiplying sample size by 4:4n = 4(1012)

= 4048 adults

As the sample size increases, the standard deviation of the sampling distribution decreases.

as n x

Page 22: Sample Proportions Target Goal: I can FIND the mean and standard deviation of the sampling distribution of a sample proportion. DETERMINE whether or not

f) If the pollsters had polled teenagers instead of adults, do you think the sample proportion greater than , equal to, or less than 0.67?

Explain:

It should be greater than since children tend to drink more milk than adults.