Download 7-Proportions-Summary - Web4students

Chapter 9.2 - Categorical Data and Proportions – One population
Examples of Proportions
 What proportion (% as a decimal) of Montgomery College students take Math 116?
 What proportion of people have blood type B?
 What proportion of US adults favors stricter gun control legislation?
Vocabulary:
Success: category of interest
Failure: all other categories
n = sample size
x = number of successes in the sample
p = success probability = x/n
1 – p = failure probability (some books use q as failure probability)
Section 7.1 - The Sampling Distribution of the Sample Proportion p-hat
We select an SRS of size n from a large population with proportion p and determine the proportion in
the sample that has the specific characteristic, p (p-hat).
Note: population size is at least 10 times larger than the sample
p = p

the mean of the p distribution equals the population proportion:

the standard deviation of the p-hat distribution is  p 

Central Limit Theorem
If np  10 and n(1 – p)  10, then the shape of the sampling distribution of p-hat is approximately
normal. Counts of successes and failures are both at least 10
p (1 p )
(standard error of the proportion)
n
Test statistic for proportions
z = (score – mean) / std dev
z
p p
p (1 p )
n
Section 8.2 - Confidence Intervals for Proportions ( npo  15 and n(1  po )  15 )
p z*
p  (1  p)
n
Section 9.2 - Hypothesis Testing for proportions ( npo  10 and n(1  po )  10 )
We want to test H o : p  po where po is the hypothesized value for p; then the test statistic is
z
p  po
po (1 po )
n
Notice that it is possible that in some cases the p-value method may yield a different conclusion
than the confidence interval method. This is due to the fact that when constructing confidence
intervals, we use an estimated standard deviation based on the sample proportion p-hat.
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Section 8.4 - Choosing the sample Size
The sample size needed to obtain a confidence interval with approximate margin of error E for a
p (1  p ) z 2
n
population proportion is
(when we have a prior p-hat available)
E2
Or:
n
0.25 z 2
(when no prior p-hat is available)
E2
Using the calculator to test hypothesis or construct confidence intervals for one population
proportion
For hypothesis testing use item 5:1-Prop-ZTest from the STAT, TESTS menu
Use this test when there are 10 or more successes and 10 or more failures in the sample.
For confidence intervals use item A:1-Prop-ZInterval from the STAT, TESTS menu
Use this test when there are 15 or more successes and 15 or more failures in the sample.
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