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3.3.4: Hypothesis Test for Simple Linear Regression

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    We will now describe a hypothesis test to determine if the regression model is meaningful; in other words, does the value of \(X\) in any way help predict the expected value of \(Y\)?

    Simple Linear Regression ANOVA Hypothesis Test

    Model Assumptions

    • The residual errors are random and are normally distributed.
    • The standard deviation of the residual error does not depend on \(X\)
    • A linear relationship exists between \(X\) and \(Y\)
    • The samples are randomly selected

    Test Hypotheses

    \(H_o\):  \(X\) and \(Y\) are not correlated   

    \(H_a\):  \(X\) and \(Y\) are correlated   

    \(H_o\):  \(\beta_1\) (slope) = 0   

    \(H_a\):  \(\beta_1\) (slope) ≠ 0

    Test Statistic

    \(F=\dfrac{M S_{\text {Regression }}}{M S_{\text {Error }}}\)

    \(d f_{\text {num }}=1\)

    \(d f_{\text {den }}=n-2\)

    Sum of Squares

    \(S S_{\text {Total }}=\sum(Y-\bar{Y})^{2}\)

    \(S S_{\text {Error }}=\sum(Y-\hat{Y})^{2}\)

    \(S S_{\text {Regression }}=S S_{\text {Total }}-S S_{\text {Error }}\)

    In simple linear regression, this is equivalent to saying “Are X an Y correlated?”

    In reviewing the model, \(Y=\beta_{0}+\beta_{1} X+\varepsilon\), as long as the slope (\(\beta_{1}\)) has any non‐zero value, \(X\) will add value in helping predict the expected value of \(Y\). However, if there is no correlation between X and Y, the value of the slope (\(\beta_{1}\)) will be zero. The model we can use is very similar to One Factor ANOVA.

    The Results of the test can be summarized in a special ANOVA table:

    Source of Variation Sum of Squares (SS) Degrees of freedom (df) Mean Square (MS) \(F\)
    Factor (due to X) \(\mathrm{SS}_{\text {Regression }}\) 1 \(\mathrm{MS}_{\text {Factor }}=\mathrm{SS}_{\text {Factor }} / 1\) \(\mathrm{F}=\mathrm{MS}_{\text {Factor }} / \mathrm{MS}_{\text {Error }}\)
    Error (Residual) \(\mathrm{SS}_{\text {Error }}\) \(n-2\) \(\mathrm{MS}_{\text {Error }}=\mathrm{SS}_{\text {Error }} / \mathrm{n}-2\)  
    Total \(\mathrm{SS}_{\text {Total }}\) \(n-1\)    
    Example: Rainfall and sales of sunglasses

    Design: Is there a significant correlation between rainfall and sales of sunglasses?

    Research Hypotheses:

    \(H_o\):  Sales and Rainfall are not correlated      \(H_o\):  1 (slope) = 0

    \(H_a\):  Sales and Rainfall are correlated      \(H_a\):  1 (slope) ≠ 0

    Type I error would be to reject the Null Hypothesis and \(t\) claim that rainfall is correlated with sales of sunglasses, when they are not correlated. The test will be run at a level of significance (\(\alpha\)) of 5%.

    The test statistic from the table will be \(\mathrm{F}=\dfrac{\text { MSRegression }}{\text { MSError }}\). The degrees of freedom for the numerator will be 1, and the degrees of freedom for denominator will be 5‐2=3.  

    Critical Value for \(F\) at  \(\alpha\)of 5% with \(df_{num}=1\) and \(df_{den}=3} is 10.13.  Reject \(H_o\) if \(F >10.13\). We will also run this test using the \(p\)‐value method with statistical software, such as Minitab.  



    \(F=341.422 / 12.859=26.551\), which is more than the critical value of 10.13, so Reject \(H_o\). Also, the \(p\)‐value = 0.0142 < 0.05 which also supports rejecting \(H_o\).  


    Sales of Sunglasses and Rainfall are negatively correlated.


    3.3.4: Hypothesis Test for Simple Linear Regression is shared under a CC BY-SA license and was authored, remixed, and/or curated by LibreTexts.