Topic 9.6 Notes – Skills Focus: Selecting an Appropriate Inference Procedure

The Big Picture Decision

Every inference question starts with one of two goals:

• Estimate a parameter → Confidence interval
• Test a claim about a parameter → Hypothesis test

That’s step one. Then you narrow it down.

There are three things you must identify:

1. Type of variable
   • Categorical → proportions or counts
   • Quantitative → means or slope
2. Number of samples
   • One sample
   • Two independent samples
   • Matched pairs
3. Parameter of interest
   • $p$
   • $\mu$
   • $p_1 - p_2$
   • $\mu_1 - \mu_2$
   • $\beta$ (true slope)

If you can state those three clearly, you’re done thinking. The name of the procedure follows automatically.

One-Sample Procedures

These happen when you have one group and one variable.

One Proportion z Test / Interval

Use when:

• Variable is categorical
• One sample
• Parameter is $p$

Example situation: “What proportion of students at this school prefer online homework?”

Conditions include:

• Random
• 10% condition
• Large counts $np \ge 10$ and $n(1-p) \ge 10$

If you see the word proportion and just one group, this is your move.

One Sample t Test / Interval

Use when:

• Variable is quantitative
• One sample
• Parameter is $\mu$

Example: “Is the mean commute time for teachers at this school different from 25 minutes?”

For means, you almost always use t, not z.
Check:

• Random
• 10% condition
• Nearly Normal or large sample (no strong skew/outliers or $n \ge 30$)

Students often grab z by habit. If it’s a mean and you’re using sample standard deviation, it’s t.

Two-Sample Procedures

Now you have two groups. The key question becomes: independent or paired?

Two Proportion z Test / Interval

• Two independent groups
• Categorical variable
• Parameter: $p_1 - p_2$

Example: Compare support for a policy between urban and rural voters.

You must check large counts for both groups.

Two Sample t Test / Interval

• Two independent groups
• Quantitative variable
• Parameter: $\mu_1 - \mu_2$

Example: Compare mean recovery time for patients on Drug A vs Drug B.

Make sure the groups are independent. If the same people are measured twice, this is wrong.

Matched Pairs t Test

This one trips people up.

Use when:

• One quantitative variable
• Same subjects measured twice (before/after)
  OR naturally paired subjects (twins, matched partners)

Parameter: mean difference $\mu_d$

You:

1. Compute differences
2. Reduce to a one-sample t test on those differences

Here’s the idea. You turn each pair of measurements into a single difference, then run a one-sample t test on those $d$ values.

Matched pairs reduced to a one-sample t test on differences

If you see “before and after,” slow down and think paired.

Chi-Square Tests (Categorical Counts Only)

These are about distributions of counts, not proportions with z formulas.

Goodness of Fit

• One categorical variable
• Compare observed counts to a claimed distribution

Example: Do these dice outcomes match a fair die?

Test for Independence

• Two categorical variables
• One population
• Is there an association?

Example: Is snack preference related to grade level within one school?

Test for Homogeneity

• Compare distributions across multiple populations

Example: Do brand preferences differ across three cities?

All chi-square tests require:

• Random
• 10% condition
• All expected counts ≥ 5

If you’re comparing just two proportions, do not use chi-square. That’s a two-proportion z test.

Linear Regression t Procedures

Use when:

• Two quantitative variables
• You’re studying a linear relationship

Parameter: $\beta$ (true slope)

The usual test:

$$H_0: \beta = 0$$

On the AP exam, you’ll usually be given computer output like this. Focus on the row for the explanatory variable and these columns:

• Slope estimate (coef)
• Standard error (std err)
• t statistic (t)
• p-value (P>|t|)

Typical regression output table

If the question asks about the “relationship between two quantitative variables,” think regression immediately.

Step-by-Step Decision Flow

1. Interval or test?
2. Categorical or quantitative variable?
3. One sample, two independent samples, or matched pairs?
4. If categorical:
   • Comparing proportions → z
   • Comparing distributions → chi-square
5. If quantitative:
   • One or two groups → t
   • Two variables with a linear relationship → regression t

When grading FRQs, they want the procedure name clearly stated, like “two-sample t test for a difference in means.” Vague answers lose points.