Setting Up a | AP Stats Unit 6 Study Guide

Quick Summary

This guide covers the foundational steps for conducting a significance test for a population proportion. You will learn how to translate a real-world question into a pair of statistical hypotheses, identify the correct inference procedure, and meticulously verify the necessary conditions that ensure your test results are valid and reliable. Mastering this setup is the critical first step for drawing sound conclusions from data.

Key Concepts

A significance test uses sample data to assess the plausibility of a claim about a population parameter. For a population proportion, this process begins with a formal setup.

1. Stating the Hypotheses

The first step is to define the two competing claims we are testing. These claims are always about the unknown population proportion (p), never the sample statistic (p̂).

The Null Hypothesis (H₀):
- This is the "status quo" or "no effect" hypothesis. It represents the claim that is assumed to be true unless the evidence from our sample data strongly suggests otherwise.
- It always contains a statement of equality.
- Form:H₀: p = p₀, where p₀ is the hypothesized value of the population proportion. This value comes from a previous belief, a historical value, or a product claim.
The Alternative Hypothesis (Hₐ):
- This is the claim we are trying to find evidence for. It's what we suspect might be true instead of the null hypothesis.
- It can take one of three forms, determined by the wording of the research question:
  - One-Sided (less than): Hₐ: p < p₀
    - Keywords: "less than," "decreased," "is lower," "has fallen."
  - One-Sided (greater than): Hₐ: p > p₀
    - Keywords: "greater than," "increased," "is higher," "has improved."
  - Two-Sided (not equal to): Hₐ: p \neq p₀
    - Keywords: "different from," "has changed," "is not the same as."

Example: A company claims that 80% of its customers are satisfied. A consumer group suspects the true proportion is lower.

The hypothesized value is p₀ = 0.80.
The null hypothesis is the company's claim: H₀: p = 0.80.
The consumer group's suspicion ("lower") dictates the alternative: Hₐ: p < 0.80.

2. Identifying the Correct Inference Procedure

When you are testing a claim about a single proportion from a single population, the appropriate statistical test is a one-sample z-test for a population proportion. You must state this procedure by its full name.

3. Verifying the Conditions for Inference

Before you can perform the test and trust the results, you must verify three conditions. These conditions ensure that the sampling distribution of the sample proportion (p̂) is approximately Normal, allowing us to calculate a valid p-value.

Random Condition:
- Purpose: To ensure the sample is representative of the population, which allows us to generalize our conclusions from the sample to the population.
- How to Check: The problem must state that the data comes from a random sample or a well-designed, randomized experiment.
10% Condition (Independence):
- Purpose: When sampling without replacement, this condition ensures that individual observations are reasonably independent. If the sample size is too large relative to the population, the act of sampling changes the remaining population proportions, violating independence.
- How to Check: The sample size $n$ must be no more than 10% of the population size $N$ .
- Formula:n \le (1/10)N
Large Counts Condition (Normality):
- Purpose: To ensure that the sampling distribution of p̂ is approximately Normal.
- How to Check: We check if the number of expected "successes" and "failures" in the sample are both sufficiently large. Crucially, we use the hypothesized proportion p₀ from the null hypothesis for this check, because the entire test operates under the assumption that H₀ is true.
- Formulas:np₀ \ge 10 and n(1-p₀) \ge 10

[Image: A bell-shaped curve labeled "Sampling Distribution of p-hat" centered at the hypothesized value p₀. The x-axis shows values of p-hat. A caption reads: "When the Large Counts Condition is met, the sampling distribution of p-hat is approximately Normal, allowing us to perform a z-test."]

Key Vocabulary

Null Hypothesis (H₀): The initial claim about a population parameter (e.g., p = 0.50) that is assumed to be true until evidence suggests otherwise.
Alternative Hypothesis (Hₐ): The competing claim that we are seeking evidence for; it can be one-sided (p < p₀ or p > p₀) or two-sided (p \neq p₀).
Population Proportion (p): The true proportion of a characteristic in the entire population. This is the parameter we make hypotheses about.
Hypothesized Value (p₀): The specific numerical value of the population proportion that is stated in the null hypothesis.
One-Sample z-test for a Population Proportion: The name of the significance test used to evaluate a claim about a single population proportion.
Significance Level (α): The threshold for "unusual" results. If a p-value is less than alpha, we reject the null hypothesis. It is typically set at 0.05 unless otherwise specified.

Calculator Tech (TI-84)

While topic 6.4 focuses on the setup, these are the inputs you will use when you perform the full test on your calculator. Knowing them now helps you see where the setup pieces fit.

Function: $1 - P ro pZT es t$

Path:STAT -> TESTS -> 5:1-PropZTest...

Inputs:

$p_{0}$ : The hypothesized value from your null hypothesis (H₀).
$x$ : The number of "successes" in your sample. This must be an integer. If you are given a percentage, you must calculate $x = n * \overset{p}{^}$ and round to the nearest whole number.
$n$ : The total sample size.
$p ro p$ : This is where you specify your alternative hypothesis (Hₐ). Select $\neqp_{0}$ (two-sided), $< p_{0}$ (left-tailed), or $> p_{0}$ (right-tailed).

After entering these values, selecting $C a l c u l a t e$ will perform the test, giving you the test statistic (z) and the p-value, which you will learn about in subsequent lessons.

How to Show Work on the FRQ

To earn full credit on an inference FRQ, you must follow a clear, four-step process. For this topic, we focus on the first two steps: State and Plan.

The STATE Step

Clearly define the parameter you are testing, state the hypotheses, and list the significance level.

Template:

Parameter: Let $p$ be the true proportion of [describe the population and characteristic in context].
Hypotheses:
- H₀: p = [p₀ value]
- Hₐ: p [ <, >, or \neq ] [p₀ value]
Significance Level: We will use a significance level of α = [value from problem, usually 0.05].

The PLAN Step

Name the procedure you are using and explicitly check the three conditions. Do not just state the condition; show the work to verify it.

Template:

Procedure: We will perform a one-sample z-test for a population proportion.
Conditions:
- Random: The problem states that the data came from a [random sample / randomized experiment].
- 10% Condition: The sample size $n$ = [value] is less than 10% of the total population of [describe the population]. We can assume there are at least 10 * [n value] = [10n value] [population members].
- Large Counts: We check that the number of expected successes and failures are both at least 10, assuming H₀ is true:
  - np₀ = [n] * [p₀] = [value] \ge 10
  - n(1-p₀) = [n] * (1 - [p₀]) = [value] \ge 10
- Since all conditions are met, we can proceed with the test.

Practice Problems

Problem 1:

A national polling agency reported last year that 35% of all U.S. adults actively volunteer in their communities. A sociologist believes this proportion has increased. To test this claim, she takes a random sample of 400 U.S. adults and finds that 152 of them actively volunteer. Set up a significance test to determine if there is convincing evidence that the proportion of U.S. adults who volunteer has increased, using a significance level of α = 0.05.

Solution:

This solution follows the State and Plan framework.

STATE:

Parameter: Let $p$ be the true proportion of all U.S. adults who actively volunteer in their communities.
Hypotheses:
- H₀: p = 0.35 (The proportion has not changed from last year's report)
- Hₐ: p > 0.35 (The proportion has increased)
Significance Level: We will use a significance level of α = 0.05.

PLAN:

Procedure: We will perform a one-sample z-test for a population proportion.
Conditions:
- Random: The problem states that the sociologist took a "random sample of 400 U.S. adults."
- 10% Condition: The sample size is n = 400. It is reasonable to assume there are more than 10 * 400 = 4,000 U.S. adults, so the condition is met.
- Large Counts: We check the expected counts using p₀ = 0.35:
  - np₀ = 400 * 0.35 = 140 \ge 10
  - n(1-p₀) = 400 * (1 - 0.35) = 400 * 0.65 = 260 \ge 10
- Since all conditions are met, we can proceed.

Problem 2:

A city's department of transportation claims that 70% of its buses run on time. A local watchdog group believes this figure is inaccurate and wants to check if the true proportion is different from what is claimed. They randomly select 150 bus routes and find that 96 of them ran on time. Set up the appropriate test to address their concern.