The data from exercise 1 follow. xi 1 2 3 4 5 yi 3

step 1 For exercise 32, we have S sub Y0 sub p equal to 2 .033. Then we have in root one fifth plus in

The data from exercise 1 follow. xi 1 ...

$$ \begin{aligned} &\text { The data from exercise } 1 \text { follow. }\\ &\begin{array}{c|ccccc} \boldsymbol{x}_{i} & 1 & 2 & 3 & 4 & 5 \\ \hline \boldsymbol{y}_{i} & 3 & 7 & 5 & 11 & 14 \end{array} \end{aligned} $$ a. Use equation (14.23) to estimate the standard deviation of $\hat{y}^{\text {? }}$ when $x=4$. b. Use expression (14.24) to develop a $95 \%$ confidence interval for the expected value of $y$ when $x=4$ c. Use equation ( 14.26 ) to estimate the standard deviation of an individual value of $y$ when $x=4$ d. Use expression (14.27) to develop a $95 \%$ prediction interval for $y$ when $x=4$

$$
\begin{aligned}
&\text { The data from exercise } 1 \text { follow. }\\
&\begin{array}{c|ccccc}
\boldsymbol{x}_{i} & 1 & 2 & 3 & 4 & 5 \\
\hline \boldsymbol{y}_{i} & 3 & 7 & 5 & 11 & 14
\end{array}
\end{aligned}
$$
a. Use equation (14.23) to estimate the standard deviation of $\hat{y}^{\text {? }}$ when $x=4$.
b. Use expression (14.24) to develop a $95 \%$ confidence interval for the expected value of $y$ when $x=4$
c. Use equation ( 14.26 ) to estimate the standard deviation of an individual value of $y$ when $x=4$
d. Use expression (14.27) to develop a $95 \%$ prediction interval for $y$ when $x=4$

Key Concepts

t-Distribution in Regression Inference

The t-distribution is commonly used in regression analysis, particularly when sample sizes are small, because it accounts for the additional uncertainty in estimating the error variance from the sample. When constructing confidence intervals and prediction intervals for parameters or future observations, quantiles from the t-distribution are used to obtain the appropriate critical values, ensuring that the intervals have the correct level of confidence under the model assumptions.

Prediction Interval for an Individual Response

A prediction interval for an individual response provides a range within which a new observation is expected to fall with a specified probability (such as 95%). This interval accounts for both the uncertainty in predicting the mean response (through its standard error) and the additional variability due to individual random deviation from the mean. It is generally wider than a confidence interval for the mean response due to the extra uncertainty inherent in predicting individual outcomes.

Standard Error for an Individual Prediction

The standard error for an individual prediction measures the uncertainty associated with predicting a single new observation rather than the mean response. It is generally larger than the standard error of the predicted mean response because it incorporates not only the error in estimating the mean but also the intrinsic variability of individual observations around the regression line.

Standard Error of the Predicted Mean Response

The standard error of the predicted mean response estimates the variability or uncertainty in the prediction of the mean value of the dependent variable at a given value of the independent variable. It is derived from the residual variance of the regression model and accounts for the dispersion of the x-values around their mean, providing an indication of how precisely the regression line estimates the mean response.

Confidence Interval for the Mean Response

A confidence interval for the mean response provides a range of plausible values for the expected value of the dependent variable at a specified value of the predictor. This interval is typically constructed using the standard error of the predicted mean response and the t-distribution, reflecting the uncertainty in the parameter estimates and the sampling variability, with a specified level of confidence (e.g., 95%).

Simple Linear Regression

This is a statistical method used to model the relationship between a dependent variable and an independent variable by fitting a linear equation to observed data. The fitted line is typically obtained by the method of least squares, which minimizes the sum of the squared differences between the observed and predicted values. It forms the basis for making predictions and assessing the strength and form of the relationship between the variables.

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