Wize University Statistics Textbook > Inference for Linear Regression

Confidence Intervals for Regression

Confidence Interval for Linear Regression

The slope of the population line $\beta_1$ is the most important parameter in regression. The slope measures the change of $Y$ for every addition one unit increase in $X$ .

The population slope $\colorFour{\beta_1}$ is the parameter we are trying to estimate using the sample slope $\colorFour{b_1}$ , which is the statistic or point estimate.

$[point \ estimate]\pm[MOE]$

$\left[LCL\right]\leftarrow\left[point\ estimate\right]\rightarrow\ \left[UCL\right]$

A confidence interval is a range of values, based on the sample data, that is used to estimate an unknown population parameter – in this case, the population mean – at a given confidence level – usually 90%, 95%, or 99%.

Formulas

Image illustrating Confidence Interval for Linear Regression

df=n-k-1

Example

Image illustrating Confidence Interval for Linear Regression

Example: Confidence Interval for Regression

Image illustrating Example

(a) Construct a 95% confidence interval for the true unknown population intercept ${\beta}_o$ .

$df=n-k-1=18-1-1=16$

$t^{\ast}=2.12$ [Use t-table to find this value.]

$b_o\pm t^{\ast}SE\left(b_o\right)$

$7.247\pm2.12\left(4.515\right)$

$\left[-2.32,\ 16.82\right]$

We are 95% confident that the true population intercept is in that interval.

(b) Construct a 95% confidence interval for the true unknown population slope ${\beta}_1$ .

$b_1\pm t^{\ast}SE\left(b_1\right)$

$0.4709\pm2.12\left(0.1081\right)$

$\left[0.242,\ 0.700\right]$

We are 95% confident that the true population slope is in that interval.

You can see the confidence intervals for both intercept and slope at the bottom right part of the ANOVA table.

Illustration for practice question: Confidence Interval for Linear Regression

There are 42 observations for this linear regression model to estimate a movie's rating ( $Y$ ) based on its box office gross earnings ( $X$ ).

Construct a 95% confidence interval to estimate the population slope $\beta_1$ .

$\left[-19,51\right]$

$[0.6, 1.5]$

$[0.7, 1.4]$

$[1.1, 0.46]$

I don't know

Slope: The Significance of "Zero" (or Lack Thereof)

In statistics, we often hear the phrase "statistically significantly different from zero".

What happens if the confidence interval for the intercept $\colorOne{\beta_o}$ contains “0”?

No problem. The intercept can be positive or negative.

What happens if the confidence interval for the slope $\colorOne{\beta_1}$ contains “0”?

Since the confidence interval contains “0”, that means it is possible for the slope of the regression line to be zero!

When the slope $b_1=0$ , that means $y$ does not change when $x$ changes.
This means there is no significant (linear) relationship between $X$ and $Y$ .
Thus, we cannot express it as a linear function.

Example #1

Confidence Interval (Slope $\beta_1$ )

$\left[20,\ 30\right]$

Image illustrating Slope: The Significance of "Zero" (or Lack Thereof)

Example #2

Confidence Interval (Slope $\beta_1$ )

$\left[-16,-8\right]$

Example #3

Confidence Interval (Slope $\beta_1$ )

$\left[-10,\ 10\right]$

Practice: Confidence Interval for Slope with Zero

A linear regression model is used to estimate a person's net worth based on the number of cars they own. The confidence interval for the slope is $\left[-18660,29380\right]\$ . Which of the following is true?

The confidence interval does not contain 0; there is no evidence of a linear relationship between a person's net worth based on the number of cars they own.

The confidence interval does not contain 0; there is evidence of a linear relationship between a person's net worth based on the number of cars they own.

The confidence interval contains 0; there is no evidence of a linear relationship between a person's net worth based on the number of cars they own.

The confidence interval contains 0; there is evidence of a linear relationship between a movie’s rating and its box office gross earnings.

I don't know

Confidence Intervals for Regression

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Confidence Interval for Linear Regression

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Example: Confidence Interval for Regression

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Slope: The Significance of "Zero" (or Lack Thereof)

What happens if the confidence interval for the intercept $\colorOne{\beta_o}$ contains “0”?

What happens if the confidence interval for the slope $\colorOne{\beta_1}$ contains “0”?

Practice: Confidence Interval for Slope with Zero

Create a free account to get started

Confidence Intervals for Regression

Confidence Interval for Linear Regression

Example: Confidence Interval for Regression

Slope: The Significance of "Zero" (or Lack Thereof)

What happens if the confidence interval for the intercept βo\colorOne{\beta_o}βo​ contains “0”?

What happens if the confidence interval for the slope β1\colorOne{\beta_1}β1​ contains “0”?

Practice: Confidence Interval for Slope with Zero

What happens if the confidence interval for the intercept $\colorOne{\beta_o}$ contains “0”?

What happens if the confidence interval for the slope $\colorOne{\beta_1}$ contains “0”?