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AP Statistics 3.2 Sampling Distributions for Proportions

Practise modelling sample proportions by calculating centre and spread, checking normality and independence conditions, and interpreting probabilities in context.

Syllabus
Effective Fall 2026
Course
AP Statistics

Exam points

  • Calculate the mean and standard deviation of a sample-proportion sampling distribution.
  • Check randomization, the 10% condition, and expected success and failure counts before using normal models.
  • Standardize a sample proportion and interpret tail probabilities in the population context.

Covered syllabus points

3.2.A—Calculate the mean and standard deviation of a sampling distribution for a sample proportion3.2.B—Justify the appropriateness of conditions for the sampling distribution of a sample proportion3.2.C—Interpret the mean, standard deviation, and probabilities for a sampling distribution of a sample proportion

3.2 Sampling Distributions for Sample Proportions question 1

[Maximum number: 4]

A large company produces an equal number of brand-name lightbulbs and generic lightbulbs. The director of

quality control sets guidelines that production will be stopped if there is evidence that the proportion of all

lightbulbs that are defective is greater than 0.10. The director also believes that the proportion of brand-name

lightbulbs that are defective is not equal to the proportion of generic lightbulbs that are defective. Therefore, the

director wants to estimate the average of the two proportions.

To estimate the proportion of brand-name lightbulbs that are defective, a simple random sample of

400 brand-name lightbulbs is taken and 44 are found to be defective. Let X represent the number of

brand-name lightbulbs that are defective in a sample of 400, and let pXp_{X} represent the proportion of all

brand-name lightbulbs that are defective. It is reasonable to assume that X is a binomial random variable.

Question (a)

(a)

One condition for obtaining an interval estimate for pXp_{X} is that the distribution of p^X\hat{p}_{X} is approximately

normal. Is it reasonable to assume that the condition is met? Justify your answer.

Question (b)

(b)

How many standard errors is the observed value of p^X\hat{p}_{X} from 0.10 ?

To estimate the proportion of generic lightbulbs that are defective, a simple random sample of 400 generic

lightbulbs is taken and 104 are found to be defective. Let Y represent the number of generic lightbulbs that are

defective in a sample of 400. It is reasonable to assume that Y is a binomial random variable and the distribution

of p^Y\hat{p}_{Y} is approximately normal, with an approximate standard error of 0.0219. It is also reasonable to assume

that X and Y are independent.

The parameter of interest for the manager of quality control is D, the average proportion of defective lightbulbs

for the brand-name and the generic lightbulbs. D is defined as D=pX+pY2D=\frac{p_{X}+p_{Y}}{2}.

3.2 Sampling Distributions for Sample Proportions question 2

[Maximum number: 3]

It was recently reported that in the United States, 40.3\% of all births are to unmarried mothers. A county health administrator is investigating whether births to unmarried women is higher in her county than in the national average. If so, she will propose additional funding to counsel unmarried mothers. A random sample of 100 births in the county will be looked at. Let p= the proportion of the population of women giving birth in the county who are unmarried. Consider the following hypotheses. The hypotheses are H0:p=0.403H_0:p=0.403 and Ha:p>0.403H_a:p>0.403.

Question (a)

(a)

For true p=0.403, calculate σp^\sigma_{\hat p}.

[ 1 ]

Question (b)

(b)

For true p=0.45, calculate σp^\sigma_{\hat p}.

[ 1 ]

Question (c)

(c)

Calculate the probability that the null hypothesis is rejected.

[ 1 ]
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