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3.3.7: Prediction

  • Page ID
    28711
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    One valuable application of the regression model is to make predictions about the value of the dependent variable if the independent variable is known.  

    Consider the example about rainfall and sunglasses sales. Suppose we know that a city has 22 inches of rainfall. We can use the regression equation to predict the sales of sunglasses:

    \(\hat{Y}=45.647-.767 X\)

    \(\hat{Y}_{22}=45.647-.767(22)=28.7\)

    For a city with 22 inches of annual rainfall, the model predicts sales of 28.7 per 1000 population.

    To measure the reliability of this prediction, we can construct confidence intervals. However, we first have to decide what we are estimating. We could (1) be estimating the expected sales for a city with 22 inches of rainfall, or we could (2) be predicting the actual sales for a city with 22 inches of rainfall.

    clipboard_ea5774784cc2db2f01d11bda2df0da2cb.png

    In the graph shown, the green line represents \(Y=\beta_{0}+\beta_{1} X+\varepsilon\) the actual regression line which is unknown. The red line represents the least square equation, \(\hat{Y}=45.647-.767 X\), which is derived from the data. The black dot represents our prediction \(Y_{22}=28.7\). The green dot represents the correct population expected value of \(Y_{22}\), while the yellow dot represents a possible value for the actual predicted value of \(Y_{22}\). There is more uncertainty in predicting an actual value of \(Y_x\) than the expected value.

    Confidence interval and Prediction interval

    The confidence interval for the expected value of \(Y\) for a given value of \(X\) is given by:

    \[\hat{Y}_{X} \pm t \cdot s_{e} \sqrt{\dfrac{1}{n}+\dfrac{(X-\bar{X})^{2}}{S S X}} \nonumber \]

    Degrees of freedom for \(t =n‐2\)

    The prediction interval for the actual value of \(Y\) for a given value of \(X\) is given by:

    \[\hat{Y}_{X} \pm t \cdot S_{e} \sqrt{1+\frac{1}{n}+\frac{(X-\bar{X})^{2}}{S S X}} \nonumber \]

    Degrees of freedom for \(t =n‐2\)

    Example: Rainfall sunglasses sales
    1. Find a 95% confidence interval for the expected value of sales for a city with 22 inches of rainfall.
    2. Find a 95% prediction interval for the value of sales for a city with 22 inches of rainfall.

    Solution

    1. Confidence interval

    \(28.7 \pm 3.182 \cdot 3.586 \sqrt{\dfrac{1}{5}+\dfrac{(22-23)^{2}}{580}}=28.7 \pm 5.1 \rightarrow(23.6,33.8)\)

    We are 95% confident that the expected annual sales of sunglasses for a city with 22 inches of annual rainfall is between 23.6 and 33.8 sales per 1000 population.

    1. Prediction interval

    \(28.7 \pm 3.182 \cdot 3.586 \sqrt{1+\dfrac{1}{5}+\dfrac{(22-23)^{2}}{580}}=28.7 \pm 12.5 \rightarrow(16.2,41.2)\)

    We are 95% confident that the actual annual sales of sunglasses for a city with 22 inches of annual rainfall is between 16.2 and 41.2 sales per 1000 population.

     


    3.3.7: Prediction is shared under a CC BY-SA license and was authored, remixed, and/or curated by LibreTexts.

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