Two-Proportions-Stating Hypothesis

Key Takeaways — Significance Tests for a Difference in Populations Proportions | How to State Hypotheses

Significance Tests for a Difference in Population Proportions

• We use a two-sample z test for a difference in population proportions to compare two groups.

• Hypotheses are stated using population proportions (p), not sample proportions (p̂).

• The null hypothesis usually assumes no difference:
  H₀: p₁ − p₂ = 0

• The alternative hypothesis depends on the question and can be:
  Hₐ: p₁ > p₂, Hₐ: p₁ < p₂, or Hₐ: p₁ ≠ p₂

• Parameters must be clearly defined in context (for example, p_M and p_W).

Significance Tests for a Difference in Population Proportions (AP Statistics 6F)

Hi everyone, Mr. Antonucci here. In this video, we’re going to cover Section 6F: Significance Tests for a Difference in Population Proportions. We’ll focus on the first learning target, and in subsequent videos we’ll look at the remaining targets.

In this lesson, we will:

• State appropriate hypotheses for performing a test about a difference between two population proportions

• Check the required conditions

• Calculate the standardized test statistic and P-value

• Perform a two-sample z test for a difference in proportions

Stating Hypotheses for a Difference in Population Proportions

When we talk about stating hypotheses for a difference in population proportions, the null hypothesis has the general form:

H₀: p₁ − p₂ = hypothesized value

In AP Statistics, we are going to focus on the case where the hypothesized difference is equal to 0. There are two equivalent ways you may see this written:

H₀: p₁ − p₂ = 0
or
H₀: p₁ = p₂

Both forms mean the same thing.

Alternative Hypotheses

The alternative hypothesis will be one of three forms, depending on the question:

• Hₐ: p₁ > p₂ (looking for a difference in one direction)

• Hₐ: p₁ < p₂ (looking for a difference in one direction)

• Hₐ: p₁ ≠ p₂ (looking for a difference in either direction)

Example: Drive-Through Order Accuracy

Consumers spend billions of dollars each year in the drive-through lanes of fast-food restaurants. Having quick, accurate, and friendly service at a drive-through window translates directly into restaurant revenue. As a result, industry executives, stockholders, and analysts closely follow the drive-through ratings that appear each year in QSR, a publication that reports on the quick-service restaurant industry.

In a recent QSR study, researchers placed orders that included:

• A modified main item, such as a hamburger with no pickles

• A side item

• A drink

If any item was not received as ordered, if the customer was charged incorrectly, or if no napkins or straw were provided, the order was considered inaccurate.

Research Question

Is there convincing evidence of a difference in the population proportions of accurate drive-through orders at McDonald’s and Wendy’s?

We want to state the hypotheses for performing a significance test and clearly define the parameters of interest.

Hypotheses

Because we are looking for a difference between McDonald’s and Wendy’s, our hypotheses are:

Null Hypothesis:
H₀: p_M − p_W = 0

Alternative Hypothesis:
Hₐ: p_M − p_W ≠ 0

In the setup for this hypothesis test, we assume there is no difference in the population proportions of accurate drive-through orders, and then we look for convincing evidence that a difference does exist.

Parameters of Interest

When stating hypotheses, it is important to define what each parameter represents in the context of the problem.

• p_M is the population proportion of accurate drive-through orders at McDonald’s

• p_W is the population proportion of accurate drive-through orders at Wendy’s

Notice that we use the population proportion p, not the sample proportion p̂, when stating hypotheses.

That’s it for this video. I hope this was helpful.