Proportions and Inference

1 and 2 – proportion z-tests

AP Statistics

One-Proportion Z-Test

A coin that is balanced should come up heads half the time in the long run. The French naturalist Count Buffon (1707-1788) tossed a coin 4040 times. He got 2048 heads. Is this evidence that Buffon’s coin was not balanced?

Did you include…

Parameters (listed as variables AND in context) Hypotheses Assumptions and Conditions Name of the test Test statistic (… my bad)

Formula without the numbers plugged in Formula with the numbers plugged in Value (check this with your calculator)

Obtained P-value Make decision Statement of conclusion in context

Assumptions and Conditions for a 1-proportion z-test

The data are an SRS This takes care of randomization

The population is at least 10 times the sample size

The sample size is large enough that np0 ≥ 10 and n(1-p0) ≥ 10

PHANTOMS

Parameters (listed as variables AND in context)

Hypotheses Assumptions and Conditions Name of the test Test statistic – formula and value Obtain P-value Make decision Statement of conclusion in context

Let’s look at an old problem…Some boxes of a certain brand of breakfast cereal include a

voucher for a free video rental inside the box. The company that makes the ceral claims that voucher can be found in 20 percent of the boxes. However, based on their experiences eating this ceral at home, a group of students believes that the proportion of boxes with vouchers is less than 0.2. This group of student purchased 65 boxes of the cereal to investigate the company’s claim. The students found a total of 11 vouchers for free video rentals in the 65 boxes.

Suppose it is reasonable to assume that the 65 boxes purchased by the students are a random sample of all boxes of this cereal. Based on this sample, is there support for the students’ belief that the proportion of boxes with vouchers is less than 0.2? Provide statistical evidence to support your answer.

2

4

3

What if we want to compare two proportions?

To study the long-term effects of preschool programs for poor children, the High/Scope Educational Research Foundation has followed two groups of Michigan children since early childhood. One group of 62 attended preschool as 3 and 4 year olds. This is a sample from population 2, poor children who attended preschool. A control group of 61 children from the same area and similar backgrounds represents population 1, poor children with no preschool. Thus the sample sizes are n1 = 61 and n2 = 62.

One response variable of interest is the need for social services as adults. In the past 10 years, 38 of the preschool sample and 49 of the control sample have needed social services (mainly welfare).

Is there significant evidence that preschool reduces the latter need for social services?

Differences from the 1-proportion test

We don’t know either population proportion, only the two sample proportions.

We have two n’s and two p-hats

We want to turn this into a single hypothesis

Look at some combination of the two

Hypotheses

What are we trying to test? What are our parameters / statistics?

Hypotheses H0: p1 = p2

▪ There’s no difference between the two proportions

Ha: p1 < p2

• The first proportion is less than the second

How do we combine the two proportions?

We’re trying to find if there is a difference.

If there is no difference then p1 – p2 =0

If p1 is greater than p2 then p1 – p2 >0

We’ll standardize this difference.

What about the standard deviation?

Standard deviation… sort of…

This is standard error (SE)

Assumptions and Conditions for a 2-proportion z-test

The data from each sample are an SRS This takes care of randomization

Independent samples

The population is at least 10 times the sample size for both samples and populations

The sample size is large enough that np1 ≥ 10 and n(1-p1) ≥ 10 AND np2 ≥ 10 and n(1-p2) ≥ 10

So, let’s solve the problem…

P – ParametersH – HypothesesA – Assumptions / ConditionsN – Name the testT – Test statisticO – Obtain P-valueM – Make decisionS – State result in context