#AP Statistics: Hypothesis Testing for Regression Slope 🚀

Hey there, future AP Stats superstar! Let's nail down hypothesis testing for regression slopes. You've got this! 💪

#T-Distribution for Slope Estimation

• This is because the slope estimate is a linear combination of the observations, and it's a result of the Central Limit Theorem. Think of it as the sampling distribution of the slope.

#Calculating the T-Score

• The t-score helps us understand how far our sample slope is from the null slope (usually zero). It's our way of measuring the evidence against the null hypothesis. ⛰️

• Formula:

$$t = \frac{b - \beta}{SE_{b}}$$

Where:

- *b* is the sample slope.
- *β* is the hypothesized population slope (usually 0).
- *SE<sub>b</sub>* is the standard error of the sample slope.

• Degrees of Freedom: n - 2 (sample size minus the number of parameters estimated, which is 2 for a slope and intercept)

#Understanding the P-Value

• The p-value is the probability of observing a sample slope as extreme as (or more extreme than) the one we got, assuming the null hypothesis is true. 🚗

• A small p-value means our sample is unlikely if the null is true, giving us evidence to reject it.

• Example:

• If your t-score is 2.0791 with 21 degrees of freedom, and you're doing a two-tailed test, the p-value might be around 0.05. This means there's about a 5% chance of seeing a sample like yours if the true slope was 0. - If our significance level is also 0.05, we would reject the null hypothesis.

Image: Visual representation of the t-distribution and how the p-value is related to the t-score.

#Concluding Your Hypothesis Test

• Your conclusion is based on comparing the p-value to your significance level (alpha, often 0.05). 🎆

• Decision Rule:

• If p-value < alpha: Reject the null hypothesis (H₀). There's significant evidence for the alternative hypothesis (H_a).

• If p-value >= alpha: Fail to reject the null hypothesis (H₀). There isn't sufficient evidence for the alternative hypothesis (H_a).

• Response Templates:

• Reject H₀: "Since our p-value is less than our significance level, we reject H₀. We have significant evidence that the true slope of the regression line between [variable 1] and [variable 2] is (value in alternate hypothesis, usually not 0)."

• Fail to Reject H₀: "Since our p-value is greater than or equal to our significance level, we fail to reject H₀. We do not have significant evidence that the true slope of the regression line between [variable 1] and [variable 2] is (value in alternate hypothesis, usually not 0)."

#Practice Problem

• Scenario: You're testing the relationship between income and happiness using a sample of 50 people.

• Hypotheses:

• H₀: There is no linear relationship (slope = 0).

• H_a: There is a linear relationship (slope ≠ 0).

• Results:

• t-statistic = 2.5

• Degrees of freedom = 48

• p-value = 0.01

• Significance level = 0.05

• Conclusion:

• Since 0.01 < 0.05, we reject the null hypothesis. There is significant evidence of a linear relationship between income and happiness. 😀

#Final Exam Focus

• High-Priority Topics:

• Understanding t-distributions and degrees of freedom

• Calculating and interpreting t-scores

• Understanding and using p-values to make conclusions

• Writing clear and contextual conclusions

• Common Question Types:

• Multiple-choice questions on the interpretation of p-values and t-scores.

• Free-response questions requiring you to conduct a full hypothesis test for a regression slope.

• Last-Minute Tips:

• Time Management: Don't spend too long on one question. If you're stuck, move on and come back later.

• Common Pitfalls: Be careful with degrees of freedom (n-2). Always state your conclusion in context.

• Strategies: Practice writing out your conclusions. This will save you time on the exam.

#Practice Questions

Practice Question

Multiple Choice Questions

1. A researcher is testing the hypothesis that there is a positive linear relationship between hours of study and exam scores. They calculate a t-statistic of 2.8 with 28 degrees of freedom. What is the correct interpretation of the p-value in this scenario?

   a) The probability of the null hypothesis being true. b) The probability of observing a t-statistic of 2.8 or greater, assuming there is no relationship between hours of study and exam scores. c) The probability of observing a t-statistic of 2.8 or greater, assuming there is a positive relationship between hours of study and exam scores. d) The probability of observing a t-statistic of 2.8 or less, assuming there is no relationship between hours of study and exam scores.

2. A regression analysis is performed to test the relationship between the number of hours of exercise per week and body fat percentage. The resulting t-statistic for the slope is -3.15, with 48 degrees of freedom. Which of the following is the most appropriate conclusion at a significance level of 0.05?

   a) There is a positive linear relationship between exercise and body fat percentage. b) There is a negative linear relationship between exercise and body fat percentage. c) There is no significant linear relationship between exercise and body fat percentage. d) The results are inconclusive.

Free Response Question

A study was conducted to investigate the relationship between the number of hours students spend on social media per day and their GPA. A random sample of 40 students was taken, and the following regression analysis results were obtained:

• Sample slope (b): -0.08
• Standard error of the slope (SEb): 0.03

a) State the null and alternative hypotheses for testing if there is a linear relationship between hours on social media and GPA. b) Calculate the t-statistic for the slope. c) Determine the degrees of freedom for this test. d) Based on the t-statistic and degrees of freedom, the p-value is 0.015. Using a significance level of 0.05, what is your conclusion? Be sure to state your conclusion in the context of the problem. e) Explain what a Type II error would mean in the context of this study.

Answer Key

Multiple Choice

1. b
2. b

Free Response

a) Null hypothesis (H0): There is no linear relationship between hours on social media and GPA (slope = 0). Alternative hypothesis (Ha): There is a linear relationship between hours on social media and GPA (slope ≠ 0). b) t = b / SEb = -0.08 / 0.03 = -2.67 c) Degrees of freedom = n - 2 = 40 - 2 = 38 d) Since the p-value (0.015) is less than the significance level (0.05), we reject the null hypothesis. There is significant evidence to suggest that there is a linear relationship between the number of hours students spend on social media per day and their GPA. Specifically, there is evidence that as the number of hours on social media increases, GPA tends to decrease. e) A Type II error would mean that we fail to reject the null hypothesis when it is false. In this context, it means that we would conclude there is no linear relationship between hours on social media and GPA, when in reality, there is a linear relationship.

Remember, you've got this! Keep practicing, and you'll be ready to rock the AP Stats exam. 🎉