BPS - 3rd Ed. Chapter 181 Inference about a Population Proportion.
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Transcript of BPS - 3rd Ed. Chapter 181 Inference about a Population Proportion.
BPS - 3rd Ed. Chapter 18 1
Chapter 18
Inference about a Population Proportion
BPS - 3rd Ed. Chapter 18 2
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”
BPS - 3rd Ed. Chapter 18 3
Inference about a ProportionSimple Conditions
BPS - 3rd Ed. Chapter 18 4
Inference about a ProportionSampling Distribution
BPS - 3rd Ed. Chapter 18 5
Case Study
Science News, Jan. 27, 1995, p. 451.
Comparing Fingerprint Patterns
BPS - 3rd Ed. Chapter 18 6
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.
BPS - 3rd Ed. Chapter 18 7
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.
BPS - 3rd Ed. Chapter 18 8
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
BPS - 3rd Ed. Chapter 18 9
Case Study: FingerprintsSampling Distribution
s.d.) ( 0.044 66
0.15)0.15(11
66 mean); ( 0.15
n
p)p(
np
BPS - 3rd Ed. Chapter 18 10
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%).
BPS - 3rd Ed. Chapter 18 11
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.
BPS - 3rd Ed. Chapter 18 12
Summary of Conditions
BPS - 3rd Ed. Chapter 18 13
Building a Confidence IntervalPopulation Proportion
BPS - 3rd Ed. Chapter 18 14
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
BPS - 3rd Ed. Chapter 18 15
Confidence Interval
BPS - 3rd Ed. Chapter 18 16
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.
BPS - 3rd Ed. Chapter 18 17
Case Study: Soft Drinks
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
BPS - 3rd Ed. Chapter 18 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
BPS - 3rd Ed. Chapter 18 19
Adjustment to Confidence IntervalMore Accurate Confidence Intervals for a Proportion
BPS - 3rd Ed. Chapter 18 20
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~
BPS - 3rd Ed. Chapter 18 21
Choosing the Sample Size
Use this procedure even if you plan to use the “plus four” method.
BPS - 3rd Ed. Chapter 18 22
Case Study: Soft Drinks
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?
BPS - 3rd Ed. Chapter 18 23
Case Study: Soft Drinks
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.
BPS - 3rd Ed. Chapter 18 24
The Hypotheses for Proportions
Null: H0: p = p0
One sided alternatives
Ha: p > p0
Ha: p < p0
Two sided alternative
Ha: p p0
BPS - 3rd Ed. Chapter 18 25
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
BPS - 3rd Ed. Chapter 18 26
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.
BPS - 3rd Ed. Chapter 18 27
BPS - 3rd Ed. Chapter 18 28
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?
BPS - 3rd Ed. Chapter 18 29
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
BPS - 3rd Ed. Chapter 18 30
“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
BPS - 3rd Ed. Chapter 18 31
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
BPS - 3rd Ed. Chapter 18 32
Case Study: Discipline
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
BPS - 3rd Ed. Chapter 18 33
Case Study: Discipline
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
BPS - 3rd Ed. Chapter 18 34
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
Case Study: Discipline