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AP Statistics
Unit 9 Overview: Slopes
9.1 Introducing Statistics: Do Those Points Align?
9.2 Confidence Intervals for the Slope of a Regression Model
9.3 Justifying a Claim About the Slope of a Regression Model Based on a Confidence Interval
9.4 Setting Up a Test for the Slope of a Regression Model
9.5 Carrying Out a Test for the Slope of a Regression Model
9.6 Skills Focus: Selecting an Appropriate Inference Procedure

📊ap statistics review

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

Verified for the 2025 AP Statistics exam•Last Updated on June 18, 2024

A couple reminders from earlier sections:

  • A confidence interval gives us a good prediction on what the slope of the true linear regression model for a population’s set of data by giving us a range of values to predict.
  • The point estimate for the slope of a regression model is the slope of the line of best fit, b.
  • For the slope of a regression model, the interval estimate is b + t (SE of b).*

In this section, we'll answer a burning question in this unit: how can we justify (or dispute) a claim about a linear regression model using this data? 🤔

Source: Analyst Prep

Confidence Level

One of the important things about a confidence interval that we must set is the confidence level. Remember that this confidence level reflects the percentage of confidence intervals that would contain the true value we are aiming towards (in this case slope) if we were to take several unique samples of our given sample size. 👍

For example, if we were to construct a 95% confidence interval to estimate the slope of a linear regression model, this means that if we were to create 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 Interval

A confidence interval is going to provide us with plausible values for our slope. For instance, if our confidence interval for the slope is (1.35, 2.7), we can be pretty certain that our correlation is positive and our slope is somewhere between 1.35 and 2.7.

Our interpretation of this would state something like: ➕

  • We are 95% confident that the true slope of the regression line showing the correlation between variable A and variable B is somewhere between 1.35 and 2.7.
  • In repeated random sampling with the same sample size, approximately 95% of confidence intervals created will capture the slope of the regression model, i.e., the true slope of the population regression model.

This is a very similar interpretation to what we used in Units 6 and 7, but altered to estimate the true slope instead of true mean or proportion.

Another thing worth noting is that the width of the confidence interval is going to decrease as the sample size increases. This is because an increased sample size decreases our standard error. Also, as the confidence level increases, the width of our interval will increase. 

Justifying a Claim

If we are seeking to justify a claim about correlation with our confidence interval for slopes, we should be seeking to determine if 0 is contained in our interval. 0️⃣

If 0 is contained in our confidence interval, it is definitely plausible that 0 is the slope of our least squares regression model. If 0 is the slope, there essentially is no linear correlation.

For example, if we use our interval from the previous example (1.35, 2.7), this tells us that the two variables of interest ARE correlated because there 0 is not contained in our interval so we can be 95% confident (or whatever confidence level) that our slope is positive and our variables have a positive correlation of some sort.

Example

The most likely type of question you would see on linear regression on the AP exam would involve a computer output. Using a computer output, we'll interpret what our confidence interval would look like. We also need a sample size to compute our t score, so let’s assume our sample size is 40 for our scatterplot and a 95% confidence level. 🖥️

First, we would need to compute our t score by doing invT based on 38 degrees of freedom (n - 2). The other aspects of our confidence interval are already in our problem. Our t-score for a 95% confidence interval comes out to be 2.02.

Our confidence interval would be 0.4482.02(0.6565), which is the slope estimate plus/minus (t score)(standard deviation/error). 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.

This would yield a final example of (-0.87813, 1.77413). Since 0 is contained in this interval, we do not have evidence that there is a linear correlation (which is also evident by the low R2 value and subsequent low r value (0.176).

🎥  Watch: AP Stats [object Object]

Key Terms to Review (15)

Confidence Level: Confidence Level is a statistical measure that represents the degree of certainty or probability that a parameter falls within a specified confidence interval. It indicates how confident we are in our estimate, often expressed as a percentage, such as 90%, 95%, or 99%. Higher confidence levels mean wider intervals, reflecting more certainty about capturing the true population parameter.
Confidence Interval: A confidence interval is a range of values derived from sample statistics that is likely to contain the true value of an unknown population parameter, with a specified level of confidence. This concept connects statistical inference to the estimation of parameters, allowing researchers to make informed claims about populations based on sample data.
Correlation: Correlation is a statistical measure that expresses the extent to which two variables are linearly related. It helps in understanding how changes in one variable may be associated with changes in another, making it a key concept in evaluating relationships in data analysis and regression models.
Degrees of Freedom: Degrees of freedom refer to the number of independent values or quantities that can vary in a statistical calculation without breaking any constraints. This concept is crucial when conducting hypothesis tests or constructing confidence intervals, as it impacts the distribution of the test statistic and influences the conclusions drawn from statistical analyses.
Interval Estimate: An interval estimate is a range of values, derived from sample data, that is used to estimate an unknown population parameter with a certain level of confidence. This method provides more information than a point estimate, as it reflects the uncertainty surrounding the estimation by including an upper and lower limit. In the context of justifying claims about the slope of a regression model, interval estimates can be crucial in determining whether the slope is statistically significant and helps to make informed decisions based on data analysis.
Justifying a Claim: Justifying a claim involves providing evidence or reasoning to support a specific assertion made about a statistical relationship, such as the slope of a regression model. It requires demonstrating that the observed effects or relationships are statistically significant and not due to random chance, typically through the use of confidence intervals or hypothesis testing. This process is crucial in determining the validity of conclusions drawn from data analysis.
Least Squares Regression Model: The Least Squares Regression Model is a statistical method used to find the best-fitting line through a set of data points by minimizing the sum of the squares of the vertical distances (residuals) between the observed values and the values predicted by the line. This model is foundational for understanding relationships between variables, as it provides insights into trends and can be used to make predictions based on data. The slope of this line is particularly important because it indicates how much one variable is expected to change when the other variable changes, which can be assessed using confidence intervals.
Linear Regression Model: A linear regression model is a statistical method used to model the relationship between a dependent variable and one or more independent variables by fitting a linear equation to observed data. This model helps in predicting the value of the dependent variable based on the values of independent variables, making it essential for understanding trends and making informed decisions based on data. Key components of this model include the slope, which indicates the strength and direction of the relationship, and residuals, which show the differences between observed and predicted values.
Point Estimate: A point estimate is a single value that serves as an approximation of a population parameter, such as a mean or proportion. It provides the best guess for the true value based on sample data, and it’s often used in statistics to infer properties about a larger group without having to measure every individual.
R-squared (R2): R-squared, also known as the coefficient of determination, measures the proportion of the variance in the dependent variable that can be explained by the independent variable(s) in a regression model. It provides insight into how well the regression model fits the data, indicating the strength and reliability of the relationship between the variables. A higher R-squared value suggests a better fit and more predictive power, while a lower value indicates that the model does not explain much of the variability in the dependent variable.
Sample Size: Sample size refers to the number of observations or data points collected from a population for the purpose of statistical analysis. It plays a critical role in determining the reliability and validity of the results, impacting the precision of estimates and the power of hypothesis tests.
Scatterplot: A scatterplot is a graphical representation that displays values for two quantitative variables using dots for individual data points. This type of plot helps visualize the relationship between the variables, allowing for the identification of patterns, trends, and correlations.
Slope of the Line of Best Fit: The slope of the line of best fit represents the relationship between the independent variable and the dependent variable in a regression model, indicating how much the dependent variable changes for each unit increase in the independent variable. This slope is essential for understanding the strength and direction of this relationship, and it plays a crucial role in making predictions based on the data. Additionally, it helps assess whether the relationship is statistically significant through confidence intervals.
Standard Error: Standard Error is a statistic that measures the accuracy with which a sample represents a population, specifically quantifying the variability of a sample mean from the population mean. It plays a critical role in constructing confidence intervals and conducting hypothesis tests, helping to assess how much sample means are expected to fluctuate around the true population mean. A smaller standard error indicates that the sample mean is a more precise estimate of the population mean.
T Score: A T Score is a standardized score that indicates how far a data point is from the mean in terms of standard deviations, specifically used in the context of small sample sizes. It helps assess the significance of regression coefficients, particularly the slope, by comparing the estimated slope to the standard error of that estimate. T Scores are crucial when creating confidence intervals and conducting hypothesis tests to justify claims about the slope of a regression model.
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9.4 Setting Up a Test for the Slope of a Regression Model