9.6 Skills Focus: Selecting an Appropriate Inference Procedure

Selecting the right inference procedure means matching the question to the correct test or interval based on the data type, the number of samples, and what is being asked. By Unit 9 you have a full toolbox of procedures, so the skill here is reading a problem carefully and choosing between proportion $z$ procedures, mean $t$ procedures, chi-square tests, and linear regression $t$ procedures.

Why This Matters for the AP Statistics Exam

Inference questions show up across both multiple-choice and free-response, and many of them start by asking you to identify the correct procedure. If you pick the wrong test, the rest of your work usually falls apart, so this skill protects your score on every inference problem.

This topic pulls together everything from Units 6 through 9. You need to recognize whether the data is categorical or quantitative, whether you have one sample or two, whether samples are paired or independent, and whether the question is about a slope. Getting comfortable with these decisions makes both the setup and the conclusion easier to write clearly.

more resources to help you study

practice multiple choiceFRQ practice & scoringcheatsheetsscore calculatorkey terms

Key Takeaways

• Identify the data type first: proportions point to z procedures, means point to t procedures, counts in categories point to chi-square, and a slope points to the linear regression t procedures.
• Count your samples and check the structure: one sample, two independent samples, or paired (matched) data each lead to different procedures.
• For regression inference, you usually read a computer output and focus on the slope row to build a t-interval or run a t-test.
• A confidence interval and a significance test answer related questions, so a question may accept either when it asks you to support a claim.
• Check conditions before you calculate, since the right procedure still needs random data, independence, and the correct shape or count requirements.
• Always confirm what the question is actually asking so you do not run a test when an interval is wanted, or vice versa.

The Full Toolbox of Procedures

By this point in AP Statistics you have learned all of the following inference procedures:

• One Proportion Z Test
• One Proportion Z Interval
• One Sample T Test
• One Sample T Interval
• Matched Pairs T Test
• Two Proportion Z Test
• Two Proportion Z Interval
• Two Sample T Test
• Two Sample T Interval
• Chi Squared Goodness of Fit Test
• Chi Squared Test for Independence
• Chi Squared Test for Homogeneity
• Linear Regression T Interval
• Linear Regression T Test

A quick way to narrow your choice:

• Proportions (categorical, asking about a percent or share): one or two proportion z procedures.
• Means (quantitative, asking about an average): one sample, two sample, or matched pairs t procedures.
• Counts across categories: chi-square goodness of fit (one categorical variable against an expected distribution), homogeneity (same variable across different groups), or independence (two categorical variables from one sample).
• Relationship between two quantitative variables (slope): linear regression t procedures.

When a problem involves the linear regression t procedures, you will most often be given a computer output and asked to make a conclusion or construct an interval.

How to Use This on the AP Statistics Exam

Problem Solving

Work through a quick decision process on every inference problem:

1. Is the variable categorical (proportions or counts) or quantitative (means or slope)?
2. How many samples or groups are involved?
3. Are the two groups independent, or is the data paired?
4. Does the question want an interval estimate or a yes/no decision from a test?

Free Response

For regression inference from computer output, focus only on the slope row. Here is a worked example based on a study with a sample size of 30.

Remember from two-variable data work that you focus on the inference values associated with the slope, which is the row labeled "Sick Days."

Confidence Interval

To construct a confidence interval for the slope, you need the point estimate (sample slope), the t critical value, and the standard error.

Everything except the t critical value is given in the computer output, so calculate the t value from the confidence level and sample size. First find the degrees of freedom, which is $n - 2 = 28$, then use the invT function to get a t critical value of about 2.05 for a 95% confidence level.

For this computer output, the confidence interval is:

$0.96 \pm 2.05(0.12)$

which comes out to about $(0.714, 1.206)$.

Because 0 is not contained in the interval, there is evidence that the two variables (sick days and wellness visits) have some linear relationship. This is consistent with a high correlation, which you could find from the $R^2$ value.

Hypothesis Test

The other option is to use the p-value to make a decision about the test. Here the p-value for the slope is 0.02, which is small enough to reject the null hypothesis at the usual 0.05 level.

A conclusion would read:

• Since the p-value $0.02 < 0.05$, we reject the null hypothesis. We have convincing evidence that the true slope of the regression model relating the number of sick days taken and the number of wellness visits is not 0.

Since there is evidence that the slope is not 0, this supports a linear relationship between the two variables, which is also reflected in the $R^2$ value and the resulting correlation coefficient.

Common Trap

Read the prompt before choosing. Students often run a significance test when the question asks for an interval, or report expected counts when asked for a probability. Identify the task first, then check that your final answer actually addresses it.

Pick a Test Practice

Decide which procedure (or procedures) fits each scenario.

(1) A marketing research firm wants to determine whether the proportion of U.S. adults who use a certain brand of toothpaste is significantly different from 50%. They survey a random sample of 500 adults and find that 270 use the toothpaste.

(2) A high school statistics teacher wants to determine whether the mean score on a certain exam is significantly different from 80. They give the exam to a random sample of 25 students and find a mean score of 78.

(3) A psychology researcher measures anxiety levels in the same people before and after an intervention and looks at the mean difference in anxiety levels.

(4) A pollster surveys a random sample of 1000 registered voters in one region (400 support a candidate) and a separate random sample of 1000 voters in another region (300 support the candidate) and wants to compare support.

(5) A nutritionist wants to determine whether the mean daily caloric intake of a population is significantly different from 2000 calories, using a random sample of 50 individuals with a mean of 1950 calories.

(6) A historian wants to determine whether the distribution of birth months among a group differs from a uniform distribution, using a random sample of 100 people.

(7) A sociologist wants to determine whether there is an association between the type of car a person drives and their political party affiliation, using one random sample of 100 people.

(8) A medical researcher randomly assigns patients to treatment A or treatment B and compares the percentage who improve in each group.

(9) A real estate agent wants to determine whether there is a relationship between the size of a house (in square feet) and its sale price, using a random sample of houses.

Answers

(1) One Proportion Z-Test, One Proportion Z-Interval

(2) One Sample T-Test, One Sample T-Interval

(3) Matched Pairs T-Test

(4) Two Proportion Z-Test, Two Proportion Z-Interval

(5) One Sample T-Test, One Sample T-Interval

(6) Chi Squared Goodness of Fit Test

(7) Chi-Squared Test for Independence

(8) Chi-Squared Test for Homogeneity

(9) Linear Regression T-Test, Linear Regression T-Interval

Common Misconceptions

• Confusing homogeneity and independence. Use homogeneity when you compare the same categorical variable across separate groups or samples, and independence when you have one sample measured on two categorical variables.
• Mixing up paired and two-sample t procedures. If each subject (or matched pair) gives two related measurements, use the matched pairs t-test on the differences. If the two groups are separate and unrelated, use a two-sample t procedure.
• Choosing z for means or t for proportions. Proportions use z procedures and means use t procedures. Do not swap them.
• Thinking a non-significant slope means no correlation. Failing to reject the null does not prove the variables have zero relationship; it only means you did not find convincing evidence of one in this sample.
• Skipping condition checks. Choosing the right procedure is only half the job. You still need random data, independence, and the correct shape or count conditions before your results are valid.
• Running a test when an interval is requested. A significance test and a confidence interval answer related but different questions, so match your procedure to what the prompt asks for.

Related AP Statistics Guides

• Unit 9 Overview: Slopes
• 9.2 Confidence Intervals for the Slope of a Regression Model
• 9.1 Introducing Statistics: Do Those Points Align?
• 9.4 Setting Up a Test for the Slope of a Regression Model
• 9.3 Justifying a Claim About the Slope of a Regression Model Based on a Confidence Interval
• 9.5 Carrying Out a Test for the Slope of a Regression Model