Carrying Out a Test for the Slope of a Regression Model Chapter Notes | AP Statistics - Grade 9 PDF Download

In AP Statistics, one of the most critical skills is selecting the correct inference procedure for conducting hypothesis tests or constructing confidence intervals. Below is a detailed guide to the inference procedures covered in the course, along with examples and key terms to help you master this skill.

Inference Procedures Overview

The following procedures are commonly used in AP Statistics for hypothesis testing and confidence intervals:

• 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 Test
• Linear Regression T Interval

For linear regression procedures, you’ll often work with computer output to draw conclusions or calculate intervals, particularly focusing on the slope of the regression line.

Illustrative Examples

Example 1: Linear Regression Analysis

Consider a study with a sample size of 30 examining the relationship between sick days and wellness visits. The computer output provides key values for constructing a confidence interval or conducting a hypothesis test, focusing on the slope (labeled “Sick Days”).
Confidence Interval
To create a 95% confidence interval for the slope, you need the sample slope (point estimate), standard error, and t-score. The computer output provides the sample slope (0.96) and standard error (0.12). To find the t-score, calculate the degrees of freedom (n - 2 = 30 - 2 = 28) and use the invT function for a 95% confidence level, yielding a t-score of 2.05.
The confidence interval is calculated as:
0.96 ± 2.05 × 0.12 = (0.714, 1.206)
Since the interval (0.714, 1.206) does not include 0, we conclude there is evidence of a correlation between sick days and wellness visits. This is further supported by a high R² value, indicating a strong linear relationship.
Hypothesis Test
For a hypothesis test, the computer output provides a p-value of 0.02 for the slope. Since 0.02 < 0.05, we reject the null hypothesis, concluding there is significant evidence that the true slope of the regression line between sick days and wellness visits is not zero, indicating a correlation. This is consistent with the high R² value and correlation coefficient.

Example 2: Selecting the Appropriate Test

Below are scenarios requiring you to choose the correct inference procedure:

• Scenario: A marketing firm surveys 500 adults to determine if the proportion using a specific toothpaste brand differs significantly from 50%, with 270 users identified.
  Appropriate Tests: One Proportion Z Test, One Proportion Z Interval
• Scenario: A statistics teacher tests 25 students and finds a mean exam score of 78, wanting to know if it differs significantly from 80.
  Appropriate Tests: One Sample T Test, One Sample T Interval
• Scenario: A researcher measures anxiety levels before and after an intervention in treatment and control groups, finding a mean difference of -5.
  Appropriate Test: Matched Pairs T Test
• Scenario: A pollster surveys 1000 voters in two regions, finding 400 and 300 supporting a candidate, respectively, to compare support against 40%.
  Appropriate Tests: Two Proportion Z Test, Two Proportion Z Interval
• Scenario: A nutritionist finds a mean caloric intake of 1950 calories in a sample of 50, testing if it differs from 2000 calories.
  Appropriate Tests: Two Sample T Test, Two Sample T Interval
• Scenario: A historian examines birth months of 100 people, noting more summer births, to test against a uniform distribution.
  Appropriate Test: Chi-Squared Goodness of Fit Test
• Scenario: A sociologist studies car types and political affiliations in 100 people, noting more Democrats drive sedans.
  Appropriate Test: Chi-Squared Test for Independence
• Scenario: A researcher compares improvement rates between two treatments, finding treatment A more effective.
  Appropriate Test: Chi-Squared Test for Homogeneity
• Scenario: A real estate agent finds a positive relationship between house size and sale price in a sample.
  Appropriate Tests: Linear Regression T Test, Linear Regression T Interval.

Key Terms