Chapter 12 Inferences for Population Proportions


Definition: Population proportion, p
The proportion (percentage) of the entire population that has the specified attribute

Definition: Sample proportion,
The proportion (percentage) of a sample from the population that has the specified attribute.
Sample proportion is computed by the formula where x denotes the number of members in the sample that have specified attribute and n denotes sample size.

One-Sample z-Interval Procedure
Assumptions:
1. Simple random sample
2. The number of successes, x, and the number of failures, n - x, are both 5 or greater.
i. For a confidence level of 1-a, use Table II to find za/2.
ii. The confidence interval for p is from   where za/2 is found in i., n is the sample size, and is the sample proportion.
iii. Interpret the confidence interval.

Margin of Error for the Estimate of p


Sample Size for Estimating p



One-Sample z-Test
Purpose To perform a hypothesis test for a population proportion, p
Assumptions:
1. Simple random sample
2. Both npo and n(1-po) are 5 or greater.
i. The null hypothesis is H0: p = po, and the alternative hypothesis is:
H0 : p ¹ po     or     H0 : p < po     or      H0 : p > po
(Two Tailed)        (Left Tailed)             (Right Tailed)
ii. Decide on the significance level, α
iii. Compute the value of the test statistic z = and denote than value z0.
(Critical-Value Approach)
vi. The critical value(s) are ±z0 (Two tailed)  or  -z0 (Left tailed)   or +z0 (Right tailed)
v. If the value of the test statistic falls in the rejection region, reject H0; otherwise do not reject H0
vi. Interpret the results of the hypothesis test

Two-Sample z-Test
Purpose To perform a hypothesis test for two population proportions, p1 and p2
Assumptions:
1. Simple random samples
2. Independent samples
3. x1, n1 - x1, x2, and n2 - x2 are all 5 or greater
i. The null hypothesis is H0: p1 = p2, and the alternative hypothesis is:
H0 : p1 ¹ p2     or     H0 : p1 < p2     or     H0 : p1 > p2
(Two Tailed)         (Left Tailed)            (Right Tailed)
ii. Decide on the significance level, α
iii. Compute the value of the test statistic  z =    where pp = (x1 + x2)/(n1 + n2). (Call it zo.)
(Critical-Value Approach)
iv. The critical value(s) are ±za/2 or -za or +za (Two tailed) (Left tailed) (Right tailed)
v. If the value of the test statistic falls in the rejection region, reject H0; otherwise do not reject H0
vi. Interpret the results of the hypothesis test

Two-Sample z-Interval Procedure
Purpose To find a confidence interval for the
difference between two population proportions, p1 and p2
Assumptions:
1. Simple random samples
2. Independent samples
3. x1, n1-x1, x2, and n2-x2 are all 5 or greater
i. For a confidence level of 1 – α, use Table II to find zα/2.
ii. The endpoints of the confidence interval for p1 – p2 are
iii. Interpret the confidence interval.

Margin of Error for estimate of p1 - p2
E = zα/2

Sample Size for Estimating p1 - p2
A (1-α)-level confidence interval for the difference between two population proportions that has a margin of error of at most E :
 n1 = n2 = 0.5 , rounded up to the nearest whole number.
If you can make educated guesses, p1g and p2g, for the observed values of p1 and p2:
n1 = n2 = rounded up to the nearest whole number. 

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