Chapters 22 and 24 Notes: Two-Sample z-Intervals and Two Sample t-Intervals
Part I: Confidence Interval comparing two proportions
1. Two-Proportion Confidence Interval (two-proportion z-interval)
(^p1 - ^p2) ± z* ∙ √((^p1^q1/n1) + (^p2^q2/n2))
2. Conditions:
a. Independent Groups: The two groups we’re comparing must by independent of each other.
b. Randomization: The two groups should be selected at random.
c. 10%: The total sample represents less than 10% of the entire population.
d. Success/Failure: The total number of successes and the total number of failures of both
groups must both be greater than 10 for the sample to be large enough (Must be tested for both
samples).
e. Confidence Values (z*): 90%: 1.645, 95%: 1.960
3.
Example: In October 2009 the U.W. Department of Commerce reported the results of a
large-scale survey on high school graduation. Researchers contacted more than 25,000
American 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 diplomas:
a. Are the conditions necessary for inference satisfied? Explain.
The sample of 24 year old men and the sample of 24 year old women are all independent of
each other.
The sample of 24 year old men and the sample of 24 year old women were chosen at random.
The 25,000 Americans aged 24 represent less than 10% of all 24 aged Americans.
Males: Success: pn = 0.849(12,460) = 10578.54
Failure: qn = 0.151(12,460) = 1881.46
Females: Success: pn = 0.881(12,678) = 11169.32
Failure: qn = 0.119(12,678) = 1508.68
*The Success/Failure Condition Must be tested for both samples.
All samples are greater than 10, the sample is large enough
Conditions for a Normal model have been satisfied.
Two-Proportion z-Interval:
b. Create a 95% confidence interval for the difference in graduation rates between males and
females.
C = 95%, z* = 1.960, (pF – pM) ± 1.960√(pq)F/nF + (pq)M/nM
(0.881 - 0.849) ± 1.960√(0.881(0.119)/12678) + (0.849(0.151)/12460) = 0.032 ± 0.0084 =
(0.024, 0.040)
(2.4%, 4.0%)
c. Interpret your confidence interval (remember to write in context):
We are 95% confident that the proportion of 24 year old American women who have graduated
from high school is between 2.4% and 4.0% higher than the proportion of American men the
same age who have graduated from high school.
d. Does this provide strong evidence that girls are more likely than boys to complete high
school? Explain.
Yes: Since the interval for the difference in proportions of high school graduates does not
contain 0, there is strong evidence that women are more likely than men to complete high
school.