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Wize AP Statistics Textbook > Inference for Quantitative Data: Slopes

Confidence Intervals for Regression

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Confidence Interval for Linear Regression

The slope of the population line β1\beta_1 is the most important parameter in regression. The slope measures the change of YY for every addition one unit increase in XX.

The population slope β1\colorFour{\beta_1} is the parameter we are trying to estimate using the sample slope b1\colorFour{b_1}, which is the statistic or point estimate.

[point estimate]±[MOE][point \ estimate]\pm[MOE]

[LCL]←[point estimate]→ [UCL]\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%.


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Formulas
df=n−k−1df=n-k-1

Example



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Example: Confidence Interval for Regression


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

df=n−k−1=18−1−1=16df=n-k-1=18-1-1=16

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

bo±t∗SE(bo)b_o\pm t^{\ast}SE\left(b_o\right)

7.247±2.12(4.515)7.247\pm2.12\left(4.515\right)

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

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

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(b) Construct a 95% confidence interval for the true unknown population slope β1{\beta}_1.

b1±t∗SE(b1)b_1\pm t^{\ast}SE\left(b_1\right)

0.4709±2.12(0.1081)0.4709\pm2.12\left(0.1081\right)

[0.242, 0.700]\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.


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

Construct a 95% confidence interval to estimate the population slope β1\beta_1.
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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 βo\colorOne{\beta_o} contains “0”?

No problem. The intercept can be positive or negative.

What happens if the confidence interval for the slope β1\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 b1=0b_1=0 , that means yy does not change when xx changes.
  • This means there is no significant (linear) relationship between XX and YY .
  • Thus, we cannot express it as a linear function.

Wize Concept
When the confidence interval for the slope β1\colorOne{\beta_1} does not contain “0”, it means that the slope is statistically significantly different from zero.
This means there is a significant (linear) relationship between XX and YY.

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Example #1

Confidence Interval (Slope β1\beta_1)

[20, 30]\left[20,\ 30\right]


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Example #2


Confidence Interval (Slope β1\beta_1)

[−16,−8]\left[-16,-8\right]



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Example #3

Confidence Interval (Slope β1\beta_1)

[−10, 10]\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 [−18660,29380] \left[-18660,29380\right]\ . Which of the following is true?