9.4 Setting Up a Test for the Slope of a Regression Model

Pick the t-test for a slope. Steps (concise): 1. Hypotheses: H0: β = β0 (often β0 = 0). Ha: β < β0, β > β0, or β ≠ β0 (use context). 2. Test statistic: t = (b − β0) / SE_b, where b is the sample slope and SE_b is its standard error from the regression output. Degrees of freedom = n − 2. Use the t-distribution to get a p-value. 3. Conditions (verify before trusting p-value): linear relationship, constant σy (homoscedasticity), independence (random sample or randomized experiment; n ≤ 10% N if sampling without replacement), and approximate normality of residuals (or n > 30 if skewed). Check residual plots for these (CED VAR-7.L). 4. Decision: compare p-value to α. If p ≤ α reject H0; interpret result in context (evidence for/against slope ≠ β0). For an AP-aligned walkthrough, see the Topic 9.4 study guide (https://library.fiveable.me/ap-statistics/unit-9/setting-up-test-for-slope-regression-model/study-guide/KEnR8FNAnXWsr8dFSAKG). For extra practice, try problems at (https://library.fiveable.me/practice/ap-statistics).

Testing a slope and testing a mean are similar in logic (both use t-tests) but differ in what you’re testing and how you check conditions. For a mean you test H₀: μ = μ₀ using t = (x̄ − μ₀)/(s/√n) with df = n−1 and you check random sampling, independence (10% rule), and near-normality of the data (or n>30). For a regression slope you test H₀: β = β₀ (usually β₀ = 0) with t = (b − β₀)/SE(b), where SE(b) comes from residual variability and df = n−2. Regression inference requires extra checks: linearity between x and y, constant variance (homoscedasticity), independence (random sample or experiment + 10% rule), and normality of residuals (or larger n). You inspect residual plots to verify those conditions. For AP Stats alignment see Topic 9.4 (t-test for slope, hypotheses H₀: β=β₀, Hₐ: β≠/β₀) and the conditions in the CED. Review the Fiveable study guide for Topic 9.4 (https://library.fiveable.me/ap-statistics/unit-9/setting-up-test-for-slope-regression-model/study-guide/KEnR8FNAnXWsr8dFSAKG) and practice problems (https://library.fiveable.me/practice/ap-statistics).

Short answer: no—H₀ is H₀: β = β₀, and most often β₀ = 0, but it doesn’t have to be. Explanation: AP’s CED (Topic 9.4) says the t-test for a slope has null H₀: β = β₀ and alternative Hₐ: β < β₀, > β₀, or ≠ β₀. We usually set β₀ = 0 when we’re asking “is there a linear association (nonzero slope) between x and y?”—that’s the common test on the exam. But if a problem gives a specific hypothesized slope (say β₀ = 2), you test against that value instead. Use the t-test for slope with df = n − 2, check the regression conditions (linearity, constant variance, independence, normal residuals) before making inferences (CED VAR-7.L). For the AP-aligned study guide see (https://library.fiveable.me/ap-statistics/unit-9/setting-up-test-for-slope-regression-model/study-guide/KEnR8FNAnXWsr8dFSAKG). For more review and practice, check the Unit 9 overview (https://library.fiveable.me/ap-statistics/unit-9) and practice problems (https://library.fiveable.me/practice/ap-statistics).

Use a t-test for the slope whenever you want to make an inference about the population slope β in a simple linear regression (that’s the AP CED rule: VAR-7.J.1). That means your null is H0: β = β0 (often β0 = 0) and Ha is one-sided or two-sided (VAR-7.K.1). Compute t = (b − β0)/SE(b) and compare to a t distribution with df = n − 2. Before you use it, verify the CED conditions (VAR-7.L.1): linear relationship (residual plot), constant σy (no funnel in residuals), independence (random sample or experiment and the 10% rule if sampling without replacement), and approximate normality of residuals (or n > 30 if skewed). If these fail, don’t trust the t-test—consider transforming, using a different model, or a nonparametric approach. For the AP review, see the Topic 9.4 study guide (https://library.fiveable.me/ap-statistics/unit-9/setting-up-test-for-slope-regression-model/study-guide/KEnR8FNAnXWsr8dFSAKG) and more unit resources (https://library.fiveable.me/ap-statistics/unit-9). For practice, try the AP problem sets (https://library.fiveable.me/practice/ap-statistics).

Before you run the t-test for a regression slope (H₀: β = β₀ vs Hₐ: β ≠/ β₀), check these CED conditions: - Linearity: the true relationship between x and y is linear. Use a residual plot (no pattern). - Constant variance (homoscedasticity): residuals have roughly equal spread for all x (no funnel shape). - Independence: data come from a random sample or randomized experiment; if sampling without replacement, ensure n ≤ 10% of the population. - Normality of responses at each x: residuals are approximately normal. If residuals look skewed, you want n > 30; if n < 30, you need no strong skewness or outliers. - Check for outliers/influential points (they can distort b and SE(b)). These let you use the t-distribution for the slope test and the SE for b. For a quick CED-aligned refresher, see the Topic 9.4 study guide (https://library.fiveable.me/ap-statistics/unit-9/setting-up-test-for-slope-regression-model/study-guide/KEnR8FNAnXWsr8dFSAKG). For broader unit review and practice questions, visit the Unit 9 page (https://library.fiveable.me/ap-statistics/unit-9) and the AP practice bank (https://library.fiveable.me/practice/ap-statistics).

To check the linear condition for a t-test on the slope, look at the residuals (observed y − predicted y): - Make a residual plot (residuals vs. x). For a linear relationship you want a random scatter around 0 with no clear pattern (no curve, no funnel, no clusters). A systematic pattern (U-shape, curve) means the true relationship isn’t linear. - Also check constant variance (homoscedasticity) from that same plot: the vertical spread of residuals should be roughly the same for all x. A fan/funnel shape means variance changes with x. - Check residual normality (required for inference about the slope): use a histogram or normal probability plot of residuals. Small samples (n < 30) need residuals free of strong skewness/outliers; larger samples are more forgiving. - Independence isn’t checked with residuals—verify random sampling/experiment and the 10% rule. These checks follow the CED VAR-7.L conditions. For a short study guide on Topic 9.4 see (https://library.fiveable.me/ap-statistics/unit-9/setting-up-test-for-slope-regression-model/study-guide/KEnR8FNAnXWsr8dFSAKG). For extra practice, try problems at (https://library.fiveable.me/practice/ap-statistics).

Step-by-step check for a t-test on the slope (β): 1. Independence—data come from a random sample or randomized experiment. If sampling without replacement, confirm n ≤ 10% of the population. If not met, you can’t trust the test. 2. Linearity—plot y vs. x and look at the residual plot (residuals vs. x). Residuals should scatter randomly around 0 (no curve). If you see a pattern, the true relationship isn’t linear. 3. Constant variance (homoscedasticity)—in the residual plot residual spread should be roughly the same for all x. Fan shape or increasing spread → violates assumption. 4. Normality of responses for a given x—check residual histogram or normal Q-Q plot. If distribution is roughly normal you’re good. If skewed, require n > 30; if n < 30, you need no strong skewness or outliers. 5. Outliers/influential points—check residuals and leverage (look for large standardized residuals or high leverage). One influential point can spoil the slope inference. If all hold, use t-test for slope (H0: β = β0; Ha: β <, >, or ≠ β0) with df = n − 2. For walkthroughs and examples see the Topic 9.4 study guide (https://library.fiveable.me/ap-statistics/unit-9/setting-up-test-for-slope-regression-model/study-guide/KEnR8FNAnXWsr8dFSAKG) and more practice at the Unit 9 page (https://library.fiveable.me/ap-statistics/unit-9) or practice problems (https://library.fiveable.me/practice/ap-statistics).

The test statistic for a t-test on the slope is t = (b − β0) / SE_b where b is the sample slope, β0 is the hypothesized slope in H0 (often 0), and SE_b is the standard error of b. Using the AP formula sheet, SE_b = s / (s_x · sqrt(n − 1)), where s = sqrt[Σ(y − ŷ)²/(n − 2)] and s_x is the sample SD of the x-values. The test follows a t-distribution with df = n − 2. Remember the hypotheses: H0: β = β0 and Ha: β < β0, β > β0, or β ≠ β0. Before you use the t-test, check linearity, equal spread (homoscedasticity), independence (random sample / 10% rule), and near-normal residuals (see Topic 9.4 CED). For more on setup and conditions, see the Topic 9.4 study guide (https://library.fiveable.me/ap-statistics/unit-9/setting-up-test-for-slope-regression-model/study-guide/KEnR8FNAnXWsr8dFSAKG). For practice, try problems at (https://library.fiveable.me/practice/ap-statistics).

β is the true (population) slope in the regression model. In the null H₀: β = β₀, β₀ is just whatever value you’re claiming about that true slope—usually 0 when you’re testing “no linear association.” So H₀: β = 0 says “there’s no linear relationship between x and y in the population.” On the AP exam you’ll most often do a t-test for the slope (VAR-7.J / VAR-7.K). The test statistic is t = (b − β₀) / SE_b with df = n − 2, and the alternatives can be β < β₀, β > β₀, or β ≠ β₀ depending on your question. Don’t forget to check the regression conditions (linearity, constant variance, independence, normality of residuals) before making inference (VAR-7.L). If you want a quick refresher tied to Topic 9.4, see the Fiveable study guide (https://library.fiveable.me/ap-statistics/unit-9/setting-up-test-for-slope-regression-model/study-guide/KEnR8FNAnXWsr8dFSAKG) and try practice problems at (https://library.fiveable.me/practice/ap-statistics).

Decide by the research question—your alternative hypothesis determines the tail. - If you just ask “Is there a relationship?” or “Is the slope different from 0?” use a two-tailed test: Ha: β ≠ 0. - If you have a directional claim (e.g., higher x leads to higher y), use a one-tailed test in that direction: Ha: β > 0 (or Ha: β < 0 if you expect a decrease). Remember AP requirements: the test for slope is a t-test (CED VAR-7.J), null H0: β = β0 (usually 0), alternative H­a: β < β0, > β0, or ≠ β0 (VAR-7.K). Check conditions before inference (linearity, constant variance, independence, normal residuals; df = n − 2). One-tailed tests give more power in the specified direction but don’t let data-driven direction switching inflate Type I error. For a quick refresher on setting up hypotheses and conditions, see the Topic 9.4 study guide (https://library.fiveable.me/ap-statistics/unit-9/setting-up-test-for-slope-regression-model/study-guide/KEnR8FNAnXWsr8dFSAKG). For more practice problems, try Fiveable’s practice page (https://library.fiveable.me/practice/ap-statistics).

That phrase means homoscedasticity: the variability (σy) of the y values is roughly the same for every x. For the t-test for a slope (VAR-7.L.1.b), you need roughly equal spread so your standard error for the slope is valid. How to check it: - Make a residual plot (residuals vs. x). If the points show a roughly constant vertical scatter around 0 for all x, variance doesn’t change. If you see a “funnel” or cone shape (spread increasing or decreasing with x), the variance changes (heteroscedasticity). - You can also split x into bins, compute the SD of residuals in each bin, and compare them—they should be similar. - Optional: use a scale-location (spread) plot or statistical tests in software, but visual residual plots are what AP expects. For practice and more examples on checking conditions, see the Topic 9.4 study guide (https://library.fiveable.me/ap-statistics/unit-9/setting-up-test-for-slope-regression-model/study-guide/KEnR8FNAnXWsr8dFSAKG) and the unit overview (https://library.fiveable.me/ap-statistics/unit-9).

Short answer: no—not reliably. A curved pattern in the residual plot means the linearity condition is violated, so the t-test for the slope (which assumes the true relationship is linear) isn’t appropriate (CED VAR-7.L.1a). What to do instead: don’t run the t-test yet. Try a transformation (log, square root), add a quadratic or higher-term to the model, or fit a different model type and then re-check residuals. If you transform or change the model and residuals look random with roughly constant spread, then you can do inference on that valid model. Also verify homoscedasticity, independence (random sample / 10% rule), and normality of residuals—if n ≤ 30 watch for strong skewness/outliers (CED VAR-7.L.1b–d). For AP review, the CED expects you to use residual analysis to check linearity before doing the t-test (see the Topic 9.4 study guide on Fiveable: (https://library.fiveable.me/ap-statistics/unit-9/setting-up-test-for-slope-regression-model/study-guide/KEnR8FNAnXWsr8dFSAKG) and the Unit 9 overview (https://library.fiveable.me/ap-statistics/unit-9)). For extra practice, try problems at (https://library.fiveable.me/practice/ap-statistics).

You always use a t-test for the slope, but the sample-size rule of thumb (n > 30) is only about the normality condition for the residuals. If residuals are roughly normal (no strong skewness or outliers) you can do the t-test with n < 30. If the residuals are noticeably skewed, then the CED says you should have n > 30 so the t-distribution approximation is safer. So practically: check the conditions first—linearity, constant variance (homoscedasticity), independence (random sample or 10% rule), and that residuals are approximately normal. If residual plots and a histogram/QQ plot of residuals look okay, n can be under 30. If residuals are skewed or have outliers, aim for n > 30. For AP alignment, this is exactly VAR-7.L in the CED. Review the Topic 9.4 study guide for examples (https://library.fiveable.me/ap-statistics/unit-9/setting-up-test-for-slope-regression-model/study-guide/KEnR8FNAnXWsr8dFSAKG) and try practice problems (https://library.fiveable.me/practice/ap-statistics) to get comfortable checking residuals.

Independence is mostly about how the data were collected—it isn’t something you check with a residual plot. For the t-test for a slope you need either (a) a random sample or (b) a randomized experiment (CED VAR-7.L.1.c.i). If you sampled without replacement, also verify n ≤ 10% of the population (CED VAR-7.L.1.c.ii). Quick practical checks: was the data gathered by simple random sampling, random assignment, or a designed experiment? If yes, you can usually assume independence. If it’s a time series or repeated measures (observations close in time or clustered), independence may fail—look for autocorrelation or dependence in the study design. If you’re unsure, mention the limitation when you report results. For more on setting up the slope test and the full list of conditions, see the Topic 9.4 study guide (https://library.fiveable.me/ap-statistics/unit-9/setting-up-test-for-slope-regression-model/study-guide/KEnR8FNAnXWsr8dFSAKG). Practice problems are at (https://library.fiveable.me/practice/ap-statistics).

Short answer: they're the same procedure but different hypotheses. "Is there a relationship?" usually means test H0: β = 0 vs Ha: β ≠ 0 (or one-sided), i.e. does the population slope equal zero (no linear relationship). Testing "slope = a specific value" is the general form H0: β = β0 vs Ha: β ≠ β0 (or <, >)—β0 can be 0 or any number you want to test. Mechanics (AP CED): you always use the t-test for a slope: t = (b − β0) / SE_b with df = n − 2, compare to t-distribution. Check the four inference conditions (linearity, equal spread/homoscedasticity, independence/random sampling or 10% rule, normality of residuals) before trusting p-values. On the exam you’ll present hypotheses in terms of β, report t, df, p-value, and condition checks (see the Topic 9.4 study guide (https://library.fiveable.me/ap-statistics/unit-9/setting-up-test-for-slope-regression-model/study-guide/KEnR8FNAnXWsr8dFSAKG) and Unit 9 overview (https://library.fiveable.me/ap-statistics/unit-9)). For extra practice, try problems at (https://library.fiveable.me/practice/ap-statistics).