BPS - 3rd Ed. Chapter 181 Inference about a Population Proportion.

BPS - 3rd Ed. Chapter 18 1

Chapter 18

Inference about a Population Proportion


The proportion of a population that has some outcome (“success”) is p.

The proportion of successes in a sample is measured by the sample proportion:

Proportions

sample the in nsobservatio of number totalsample the in successes of numberp̂

“p-hat”


Inference about a ProportionSimple Conditions


Inference about a ProportionSampling Distribution


Case Study

Science News, Jan. 27, 1995, p. 451.

Comparing Fingerprint Patterns


Case Study: Fingerprints Fingerprints are a “sexually dimorphic trait…

which means they are among traits that may be influenced by prenatal hormones.”

It is known…– Most people have more ridges in the fingerprints

of the right hand. (People with more ridges in the left hand have “leftward asymmetry.”)

– Women are more likely than men to have leftward asymmetry.

Compare fingerprint patterns of heterosexual and homosexual men.


Case Study: FingerprintsStudy Results

66 homosexual men were studied.• 20 (30%) of the homosexual men showed

leftward asymmetry.

186 heterosexual men were also studied• 26 (14%) of the heterosexual men showed

leftward asymmetry.


Case Study: FingerprintsA Question

Assume that the proportion of all men

who have leftward asymmetry is 15%.

Is it unusual to observe a sample of 66 men with a sample

proportion ( ) of 30% if the true population proportion (p) is 15%?

p̂


Case Study: FingerprintsSampling Distribution

s.d.) ( 0.044 66

0.15)0.15(11

66 mean); ( 0.15

n

p)p(

np


Case Study: FingerprintsAnswer to Question

Where should about 95% of the sample proportions lie?– mean plus or minus two standard deviations

0.15 2(0.044) = 0.0620.15 + 2(0.044) = 0.238

– 95% should fall between 0.062 & 0.238 It would be unusual to see 30% with leftward

asymmetry (30% is not between 6.2% & 23.8%).


Inference about a population proportion p is based on the z statistic that results from standardizing :

Standardized Sample Proportion

ˆ p

np)p(

ppz

1

ˆ

– z has approximately the standard normal distribution as long as the sample is not too small and the sample is not a large part of the entire population.


Summary of Conditions


Building a Confidence IntervalPopulation Proportion


Since the population proportion p is unknown, the standard deviation of the sample proportion will need to be estimated by substituting for p.

Standard Error

n

ppSE

ˆˆ

1

ˆ p


Confidence Interval


Case Study: Soft Drinks

A certain soft drink bottler wants to estimate the proportion of its customers that drink another brand of soft drink on a regular basis. A random sample of 100 customers yielded 18 who did in fact drink another brand of soft drink on a regular basis. Compute a 95% confidence interval (z* = 1.960) to estimate the proportion of interest.



n

ppzp

ˆˆˆ 1

We are 95% confident that between 10.5% and 25.5% of the soft drink bottler’s customers drink another brand of soft drink on a regular basis.

10010018

110018

1.960100

18

0.255 to 0.105

0.0750.18


The standard confidence interval approach yields unstable or erratic inferences.

By adding four imaginary observations (two successes & two failures), the inferences can be stabilized.

This leads to more accurate inference of a population proportion.

Adjustment to Confidence IntervalMore Accurate Confidence Intervals for a Proportion


Adjustment to Confidence IntervalMore Accurate Confidence Intervals for a Proportion


Case Study: Soft Drinks“Plus Four” Confidence Interval

We are 95% confident that between 12.0% and 27.2% of the soft drink bottler’s customers drink another brand of soft drink on a regular basis. (This is more accurate.)

0.272 to 0.120

0.0760.192104

10420

110420

1.960104

20

4

1

n

ppzp

~~~

104

20

4100

218

p~


Choosing the Sample Size

Use this procedure even if you plan to use the “plus four” method.



Suppose a certain soft drink bottler wants to estimate the proportion of its customers that drink another brand of soft drink on a regular basis using a 99% confidence interval, and we are instructed to do so such that the margin of error does not exceed 1 percent (0.01).

What sample size will be required to enable us to create such an interval?



Thus, we will need to sample at least 16589.44 of the soft drink bottler’s customers.

Note that since we cannot sample a fraction of an individual and using 16589 customers will yield a margin of error slightly more than 1% (0.01), our sample size should be n = 16590 customers .

16589.440.5)(0.5)(10.01

2.5761

2

p*)(*pm

*zn

2

Since no preliminary results exist, use p* = 0.5.


The Hypotheses for Proportions

Null: H0: p = p0

One sided alternatives

Ha: p > p0

Ha: p < p0

Two sided alternative

Ha: p p0


Test Statistic for Proportions Start with the z statistic that results from

standardizing :

np)p(

pp̂z

1

Assuming that the null hypothesis is true(H0: p = p0), we use p0 in the place of p:

ˆ p

n)p(p

pp̂z

00

0

1


P-value for Testing Proportions Ha: p > p0

P-value is the probability of getting a value as large or larger than the observed test statistic (z) value.

Ha: p < p0 P-value is the probability of getting a value as small or

smaller than the observed test statistic (z) value.

Ha: p ≠ p0 P-value is two times the probability of getting a value as

large or larger than the absolute value of the observed test statistic (z) value.


Case Study

Brown, C. S., (1994) “To spank or not to spank.” USA Weekend, April 22-24, pp. 4-7.

Parental Discipline

What are parents’ attitudes and practices on discipline?


Nationwide random telephone survey of 1,250 adults that covered many topics

474 respondents had children under 18 living at home– results on parental discipline are based on

the smaller sample reported margin of error

– 5% for this smaller sample

Case Study: DisciplineScenario


“The 1994 survey marks the first time a majority of parents reported not having physically disciplined their children in the previous year. Figures over the past six years show a steady decline in physical punishment, from a peak of 64 percent in 1988”

– The 1994 sample proportion who did not spank or hit was 51% !

– Is this evidence that a majority of the population did not spank or hit? (Perform a test of significance.)

Case Study: DisciplineReported Results


Case Study: Discipline

Null: The proportion of parents who physically disciplined their children in 1993 is the same as the proportion [p] of parents who did not physically discipline their children. [H0: p = 0.50]

Alt: A majority (more than 50%) of parents did not physically discipline their children in 1993. [Ha: p > 0.50]

The Hypotheses



Based on the sample n = 474 (large, so proportions follow Normal distribution)

no physical discipline: 51%– – standard error of p-hat: (where .50 is p0 from the null hypothesis)

standardized score (test statistic)z = (0.51 - 0.50) / 0.023 = 0.43

0.023474.50).50(1

Test Statistic



P-value= 0.3336

From Table A, z = 0.43 is the 66.64th percentile.z = 0.43

0.500 0.5230.477 0.5460.454 0.5690.431:p̂

0 1-1 2-2 3-3z:

P-value


1. Hypotheses: H0: p = 0.50Ha: p > 0.50

2. Test Statistic:

3. P-value: P-value = P(Z > 0.43) = 1 – 0.6664 = 0.3336

4. Conclusion:

Since the P-value is larger than = 0.10, there is no strong evidence that a majority of parents did not physically discipline their children during 1993.

0.43

0.023

0.01

4740.5010.50

0.500.51

1 00

0

npp

ppz

ˆ


BPS - 3rd Ed. Chapter 181 Inference about a Population Proportion.

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