Confidence Interval for Slope of Regression Line

Meaning of Confidence Interval for Slope of Regression Line

By now you know that when there is a linear relationship between a variable \(x\) and a variable \(y\) – the linear correlation coefficient \(r\) is non-zero – you can model it with a linear regression. This regression consists of:

\[\hat{y}=\beta_0+\beta_1x\]

where:

• \(\beta_0\) is the y-intercept;

• \(\beta_1\) is the slope of the regression;

• \(x\) is the independent variable; and

• \(\hat{y}\) the predicted value of the dependent variable.

For a better reminder of this topic, see our article Least-Squares Regression. Remember that the correlation coefficient \(r\) tells you how much of a correlation there is between the two variables. If \(r\) is close to zero, then there is little to no correlation between the variables, while \(r\) values close to \(-1\) or \(1\) indicate that there is a strong correlation between the two variables.

On the other hand, the slope \(\beta_1\) represents how much \(\hat{y}\) changes to the changes in the \(x\)-values, that is, for each unit of increase of \(x\), \(\hat{y}\) increases \(\beta_1\) units.

By calculating a confidence interval with a high confidence level, say \(c\%\), for the slope \(\beta_1\), you get two values that define the limits of a range of values in which you can find the slope. You can say with \(c\%\) confidence that the value of the slope will be between those two values.

Furthermore, you can say that the method used to construct the interval is successful in capturing the actual slope of the linear regression model about \(c\%\) of the time.

Conditions for Confidence Interval for the Slope of a Regression Line

The conditions for constructing a confidence interval for the slope of a linear regression are the same as for constructing a linear regression. These conditions are:

1. Quantitative variable condition: Correlation only applies if both variables are quantitative.

2. Straight enough condition: Look at the scatter plot and make sure your data has an approximately linear relationship. Correlation only measures the strength in a linear association. This can also be done by looking at the correlation coefficient of the data.

3. Independence of Variables: Data should be collected randomly, and if sampling without replacement is done, the sample size is less than or equal to \(10\%\) of the total population.

4. Normal: The independent variable is normally distributed.

Formula of Confidence Interval for Slope of Regression Line

Like any confidence interval you have studied so far, a confidence interval for the slope \(\beta_1\) of the least squares regression line has the following structure:

sample statistic – margin of error \(\le \beta_1\le\) sample statistic + margin of error,

where margin of error = critical value \(\times\) standard error.

Now, you just have to understand what each of those three elements is for the slope \(\beta_1\):

• The sample statistic will be \(\hat{\beta}_1\), the point estimator of the slope \(\beta_1\);

• For the margin of error:

  • this time the critical value will be of a \(t\)-distribution with \(n-2\) degrees of freedom, i.e., \(t\) with \(df=n-2\);

  • the standard error for the slope, written \(SE_{\beta_1} \), will be:\[SE_{\beta_1}=\frac{s}{\sqrt{\sum_{i=1}^{n}(x_i-\bar{x})^2}}\]where \(s\) is the sample standard deviation calculated as:\[s={\sqrt{\frac{\sum_{i=1}^{n}(y_i-\hat{y}_i)^2}{n-2}}}\ \]

Thus, the formula for a confidence interval for the slope \(\beta_1\) is:

\[\hat{\beta}_1- t\cdot SE_{\beta_1}\le \beta_1\le \hat{\beta}_1+ t\cdot SE_{\beta_1}\]

or an even shorter version:

\[\hat{\beta}_1\pm t\cdot SE_{\beta_1}\\]

This confidence interval is for any confidence level, but confidence levels that you will see most often are \(90\%\), \(95\%\), and \(99\%\). These are the values you should consider when calculating the critical value \(t\).

Calculations for Confidence Interval for Slope of Regression Line

From what you have read so far, the formula for a confidence interval for the slope suggests a set of steps you should follow when you want to find it.

Step 1: Find the sample statistic \(\hat{\beta}_1\).

You get the value of the point estimator \(\hat{\beta}_1\) by constructing the regression line for the data set you are working with.

Step 2: Select a confidence level \(c\%\).

The confidence level describes the uncertainty of a sampling method. You will most often be asked for a confidence level of \(90\%\), \(95\%\), or \(99\%\).

The purpose of knowing the confidence level is to be able to find the critical value \(t\), by consulting a \(t\) table, with two bits of information:

1. the degrees of freedom, given by the:\[ \text{sample size } -2 = n-2\]where \(n\) is the sample size; and

2. the confidence level adjusted for the table you are using.

Step 3: Find the margin of error \(t\cdot SE_{\beta_1}\).

As you already know, the margin of error is the product of the critical value \(t\) with the value of the standard error. The formula for the standard error is:

\[SE_{\beta_1}=\frac{s}{\sqrt{\sum_{i=1}^{n}(x_i-\bar{x})^2}}\]

where \(s\) is the sample standard deviation.

Step 4: Find the confidence interval.

Here you just have to replace the values you got in the previous step in the formula:

\[\hat{\beta}_1\pm t\cdot SE_{\beta_1}\\]

Let's look at an example where you can apply the steps by hand.

Example of Confidence Interval for Slope of Regression Line

Let's look at an example of doing the calculations necessary for finding the confidence interval for the slope of a regression line.