The data from exercise 3 follow. xi 2 6 9 13 20 yi 7

step 1 For exercise 25A, we have the following information. That's that S -S -E equals 1659 minus 7 .6

The data from exercise 3 follow. xi 2 ...

$$ \begin{aligned} &\text { The data from exercise } 3 \text { follow. }\\ &\begin{array}{r|rrrrr} \boldsymbol{x}_{i} & 2 & 6 & 9 & 13 & 20 \\ \hline \boldsymbol{y}_{\boldsymbol{i}} & 7 & 18 & 9 & 26 & 23 \end{array} \end{aligned} $$ a. What is the value of the standard error of the estimate? b. Test for a significant relationship by using the $t$ test. Use $\alpha=.05$. c. Use the $F$ test to test for a significant relationship. Use $\alpha=.05 .$ What is your conclusion?

$$
\begin{aligned}
&\text { The data from exercise } 3 \text { follow. }\\
&\begin{array}{r|rrrrr}
\boldsymbol{x}_{i} & 2 & 6 & 9 & 13 & 20 \\
\hline \boldsymbol{y}_{\boldsymbol{i}} & 7 & 18 & 9 & 26 & 23
\end{array}
\end{aligned}
$$
a. What is the value of the standard error of the estimate?
b. Test for a significant relationship by using the $t$ test. Use $\alpha=.05$.
c. Use the $F$ test to test for a significant relationship. Use $\alpha=.05 .$ What is your conclusion?

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Key Concepts

Standard Error of the Estimate

The standard error of the estimate quantifies the average distance that the observed values fall from the regression line. It represents the standard deviation of the residuals (errors) between observed and predicted values, providing an indication of the model’s predictive accuracy.

t Test for Regression Coefficient

The t test for a regression coefficient assesses whether the independent variable has a statistically significant linear relationship with the dependent variable. It does so by testing the null hypothesis that the slope (regression coefficient) is zero, indicating no linear association.

F Test for Overall Regression

The F test evaluates the overall significance of the regression model by comparing the model's explained variance to its unexplained variance. It tests the null hypothesis that all regression coefficients (except the intercept) are zero, determining whether the predictors, as a group, contribute significantly to the model.

Significance Level (alpha)

The significance level, often denoted as alpha (?), sets the threshold for rejecting the null hypothesis in hypothesis testing. Commonly set at 0.05, it defines the maximum allowed probability of incorrectly rejecting a true null hypothesis (Type I error), ensuring results are statistically meaningful.

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