AP Statistics: Confidence Intervals for Regression Slopes 📈

Hey there, future AP Stats superstar! Let's break down confidence intervals for regression slopes. This is a crucial topic, so let's make sure you're feeling super confident about it.

Review: Regression Basics

Before we dive in, let's quickly recap some key ideas:

A confidence interval gives us a range of plausible values for the true slope of a population's linear regression model.
The point estimate for the slope is the slope of the line of best fit, denoted by b.
The interval estimate for the slope is calculated as: *b ± t (SE of b)**.

Key Concept

Remember, we're using sample data to make inferences about the entire population.

Confidence Level: Setting the Stage

What Does Confidence Level Mean?

The confidence level is the percentage of confidence intervals that would contain the true slope if we took many samples. For example, a 95% confidence level means that if we took many random samples and constructed confidence intervals, about 95% of them would contain the true population slope. 👍

Example: 95% Confidence

If we construct a 95% confidence interval, it means that if we created several random samples of the same size, from the same population, 95% of the resulting confidence intervals would contain the true slope of the population regression model.

Confidence Intervals: What They Tell Us

Interpreting the Interval

A confidence interval provides us with a range of plausible values for the slope. For example, if our confidence interval for the slope is (1.35, 2.7), we can be pretty confident that the true slope is positive and somewhere within that range.

Interpretation Template ➕

Here's how you can interpret a confidence interval for the slope:

"We are [confidence level]% confident that the true slope of the regression line showing the correlation between variable A and variable B is somewhere between [lower bound] and [upper bound]."
"In repeated random sampling with the same sample size, approximately [confidence level]% of confidence intervals created will capture the true slope of the population regression model."

Interval Width

As sample size increases, the width of the confidence interval decreases. This is because a larger sample size reduces the standard error.
As the confidence level increases, the width of the interval increases. Think of it like casting a wider net – you're more likely to catch the true value, but the range of values becomes larger.

Justifying a Claim: The Key Question

Is Zero in the Interval?

When justifying a claim about correlation using a confidence interval for slopes, the key question is: Is 0 contained in our interval? 0️⃣

If 0 is contained in the interval: It's plausible that the true slope is 0, meaning there is no linear correlation.
If 0 is NOT contained in the interval: We have evidence of a linear correlation. The sign of the interval (positive or negative) tells us the direction of the correlation.

Example:

Using our interval from before (1.35, 2.7), we can say that the two variables ARE correlated because 0 is not contained in the interval. We can be 95% confident that our slope is positive and our variables have a positive correlation.

Understanding how to interpret confidence intervals for slopes is crucial for both multiple-choice and free-response questions.

Example: Computer Output Analysis 🖥️

Let's tackle a typical AP exam-style question involving computer output. Suppose we have a sample size of 40 and a 95% confidence level.

Steps:

Find the t-score: Calculate the t-score using invT with degrees of freedom (df) = n - 2. In our case, df = 40 - 2 = 38. For a 95% confidence interval, the t-score is approximately 2.02. 2. Identify the slope estimate (b) and standard error (SE of b): These values will be given in the computer output. Let's say b = 0.448 and SE of b = 0.6565. 3. Calculate the confidence interval: Use the formula: b ± t* (SE of b)
- Lower bound: 0.448 - 2.02 * 0.6565 = -0.87813
- Upper bound: 0.448 + 2.02 * 0.6565 = 1.77413
Interpret the interval: Our confidence interval is (-0.87813, 1.77413).

Interpretation

Since 0 is contained in this interval, we do NOT have evidence of a linear correlation. This also aligns with a low R² value and a low r value (0.176).

Common Mistake

Be careful not to use the t-score given in the table. That is the t-score for our sample, not for the desired confidence interval.

Final Exam Focus 🎯

Key Concepts: Confidence intervals, slope interpretation, confidence level, standard error, t-score, degrees of freedom.
Common Question Types: Interpreting computer output, determining if a correlation exists, explaining the effect of sample size and confidence level on interval width.
Time Management: Practice identifying the key values in computer output quickly. Focus on the interpretation, not just the calculation.
Common Pitfalls: Confusing the t-score from the table with the t-score needed for the confidence interval, misinterpreting the meaning of 0 in the interval.

Exam Tip

Always double-check your degrees of freedom (n-2) when calculating the t-score for regression.

Practice Questions

Practice Question

Multiple Choice Questions

A researcher is investigating the relationship between hours of study and exam scores. They calculate a 95% confidence interval for the slope of the regression line as (2.5, 4.1). Which of the following is a correct interpretation of this interval?

(A) We are 95% confident that the true slope of the population regression line is between 2.5 and 4.1. (B) 95% of sample slopes will fall between 2.5 and 4.1. (C) There is a 95% probability that the true slope is between 2.5 and 4.1. (D) We are 95% confident that the sample slope is between 2.5 and 4.1. (E) The probability that the true slope is between 2.5 and 4.1 is 0.95. 2. A study examines the relationship between the number of hours of sleep and reaction time. A 90% confidence interval for the slope is (-0.2, 0.1). Based on this interval, what conclusion can be made about the relationship between hours of sleep and reaction time?

(A) There is a positive linear relationship. (B) There is a negative linear relationship. (C) There is no linear relationship. (D) There is insufficient evidence to determine if a linear relationship exists. (E) The relationship is not linear.
A researcher collects data on the relationship between temperature and ice cream sales. They compute a confidence interval for the slope of the regression line and find that it does not contain zero. What can be concluded from this finding?

(A) There is no relationship between temperature and ice cream sales. (B) There is a linear relationship between temperature and ice cream sales. (C) The relationship between temperature and ice cream sales is non-linear. (D) The researcher made an error in their calculations. (E) The sample size is too small.

Free Response Question

A study was conducted to investigate the relationship between the number of hours a student studies per week and their GPA. The following data were collected from a random sample of 10 students:

Hours of Study (x)	GPA (y)
10	2.8
15	3.2
20	3.4
25	3.7
30	3.9
35	4.0
12	3.0
18	3.3
22	3.5
28	3.8

A linear regression analysis was performed, and the following results were obtained:

Slope (b): 0.045
Standard Error of the Slope (SE(b)): 0.008
Degrees of Freedom: 8

(a) Calculate a 95% confidence interval for the slope of the regression line. Assume the conditions for inference are met.

(b) Interpret the confidence interval calculated in part (a) in the context of the problem.

(c) Based on the confidence interval, is there evidence of a linear relationship between the number of hours a student studies per week and their GPA? Explain your reasoning.

FRQ Scoring Breakdown

(a) Calculate a 95% confidence interval for the slope of the regression line.

1 point: Correctly identifies the t-critical value (t*) for a 95% confidence interval with 8 degrees of freedom (approximately 2.306).
1 point: Correctly calculates the margin of error: t* × SE(b) = 2.306 × 0.008 = 0.018448. * 1 point: Correctly constructs the confidence interval: 0.045 ± 0.018448 = (0.026552, 0.063448).

(b) Interpret the confidence interval calculated in part (a) in the context of the problem.

1 point: Correctly interprets the confidence interval in context: "We are 95% confident that the true slope of the regression line relating hours of study to GPA is between 0.026552 and 0.063448."

(c) Based on the confidence interval, is there evidence of a linear relationship between the number of hours a student studies per week and their GPA? Explain your reasoning.

1 point: Correctly concludes that there is evidence of a linear relationship.
1 point: Provides a correct explanation: Since the confidence interval (0.026552, 0.063448) does not contain 0, we have evidence to suggest that there is a linear relationship between hours of study and GPA.

Memory Aid

Remember: "If zero's a hero, correlation's a zero!" (If 0 is in the interval, there's no significant correlation.)

Alright, you've got this! Remember to stay calm, think clearly, and trust your preparation. You're going to do great! 💪

Justifying a Claim About the Slope of a Regression Model Based on a Confidence Interval

Isabella Lopez