Department of Statistics and Probability
Chapter 22 Comparing two proportions
Two populations, two unknown proportions p1 and p2
Problems
• Estimate the difference p1 - p2
• Test HO: p1 = p2
Samples: Two independent, large samples of sizes n1, n2
Sample proportions: [pic] [pic]
• Point estimate of p1 - p2 is [pic]
• If n1, n2 are large then [pic]is approximately normal
with mean p1 - p
• Standard deviation of [pic] is
Two-proportion z-interval
Assumptions
1. Random samples, each with independent observations
2. Independent samples
3. If sampling without replacement, the sample size n should be no more than 10% of the population.
4. "Large" samples (n1p1 > 10, n1q1>10, n2p2 > 10, n2q2 >10)
Standard Error:
C% Margin of Error:
where z* is a critical value for standard normal distribution that corresponds to C% confidence level
A C% confidence interval for a difference p1 - p2 is
[pic]
Example: In 2000 researchers contacted 25,138 Americans aged 24 years to see if they had finished high school;
84.9% of the 12,460 males and
88.1% of the 12,678 females
indicated that they had high school diploma. Create a 95% confidence interval for the difference in graduation rate between males and females.
Data
[pic]
Standard Error:
Critical value: z* = 1.96
95% Margin of Error:
C% confidence interval for a population proportion p is
[pic]
Answer: -.032(0.008 or (-0.040, -0.024)
Two-proportion z-test
Assumptions
1. Random samples, each with independent observations
2. Independent samples
3. If sampling without replacement, the sample size n should be no more than 10% of the population.
4. "Large" samples (n1p1 > 10, n1p1>10, n2p2 > 10, n2q2 >10)
Hypotheses:
1. Null hypothesis HO: p1 = p2 that is HO: p1 - p2 =0
2. Alternative hypothesis
HA: p1 > p2 or HA: p1 < p2 or HA: p1 ≠ p2 that is
HA: p1- p2 > 0 or HA: p1 - p2 < 0 or HA: p1- p2 ≠ 0
Attitude: Assume that the null hypothesis HO is true and uphold it, unless data strongly speaks against it.
To estimate the common p = p1 = p2 we combine (pool) the two samples together
[pic]
and use it to estimate the standard deviation of [pic]
Pooled standard error of [pic]
Test statistic: [pic]
Distribution under H0: approximately standard normal
P-value: Let zo be the observed value of the test statistic. The way we compute it depends on HA
|HA |P-value | |
|HA: p1 > p2 |P(z > zo) | |
|HA: p1 < p2 |P(z |zo|) + P(z < -|zo|) | |
Example.
Of 995 respondents, 37% reported they snored at least a few night a week. Split into two age categories, 26% of the 184 people under 30 snored, compared with 39% of 811 in the older group. Is this difference real (statistically significant) or due only to natural fluctuations. Use (=0.05
Assumptions
1. Random samples, each with independent observations
2. Independent samples
3. If sampling without replacement, the sample size n should be no more than 10% of the population.
4. "Large" samples (n1p1 > 10, n1p1>10, n2p2 > 10, n2q2 >10)
Data: [pic]
Hypotheses: HO: p1 = p2 (HO: p1 - p2 =0)
HA: p1 < p2 (HA: p1 - p2 < 0)
Estimate of the common p = p1 = p2
[pic]
Pooled standard error of [pic]
Test statistic: [pic]
P-value: P(z (0 |P(t > to) | |
|HA: ( < (0 |P(t |to|) + P(t < -|to|) | |
Example - cont.
Below is the speed of vehicles recorded on Triphammer Road:
[pic]
Test whether the data provides evidence that the mean speed of vehicles on Triphammer Road exceeds 30 mph.
n = 23 (small),
Histogram is symmetric, we assume normal model.
We use one-sample t-test
Hypotheses: HO: ( = 30 vs. HA: ( > (0
Test: one sample t-test
Standard error:
Test statistic: [pic]
Degrees of freedom: df = n - 1 = 22
P-value: bigger that .10
TI-83 tcdf(1.13,1E99,22) = 0.14
Fail to reject H0 even at ( = .10
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p2
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