The data from exercise 2 follow. xi 3 12 6 20 14 yi

step 1 For exercise 33A, we estimate the standard deviation of Y0 1, X equals 8. So we'll have S, sub Y

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

$$ \begin{aligned} &\text { The data from exercise } 2 \text { follow. }\\ &\begin{array}{r|rrrrr} \boldsymbol{x}_{i} & 3 & 12 & 6 & 20 & 14 \\ \hline \boldsymbol{y}_{i} & 55 & 40 & 55 & 10 & 15 \end{array} \end{aligned} $$ a. Estimate the standard deviation of $\hat{y}$ when $x=8$. b. Develop a $95 \%$ confidence interval for the expected value of $y$ when $x=8$. c. Estimate the standard deviation of an individual value of $y$ when $x=8$. d. Develop a $95 \%$ prediction interval for $y$ when $x=8$.

$$
\begin{aligned}
&\text { The data from exercise } 2 \text { follow. }\\
&\begin{array}{r|rrrrr}
\boldsymbol{x}_{i} & 3 & 12 & 6 & 20 & 14 \\
\hline \boldsymbol{y}_{i} & 55 & 40 & 55 & 10 & 15
\end{array}
\end{aligned}
$$
a. Estimate the standard deviation of $\hat{y}$ when $x=8$.
b. Develop a $95 \%$ confidence interval for the expected value of $y$ when $x=8$.
c. Estimate the standard deviation of an individual value of $y$ when $x=8$.
d. Develop a $95 \%$ prediction interval for $y$ when $x=8$.

Key Concepts

t-Distribution

The t-distribution is used in regression analysis to construct confidence and prediction intervals when the sample size is small and the population variance is unknown. It adjusts for additional uncertainty by taking into account the degrees of freedom (typically n - 2 in simple linear regression), leading to wider intervals compared to the normal distribution.

Prediction Interval for an Individual Response

A prediction interval estimates the range within which a new individual observation is likely to fall given a specific predictor value. It is wider than the confidence interval for the mean response because it accounts for both the uncertainty in estimating the mean and the additional random error inherent in individual observations.

Standard Error of the Predicted Mean Response

This is the estimated standard deviation of the predicted mean response for a given value of the predictor. It quantifies the uncertainty in the prediction of the average value of the response variable (y) at a particular predictor value (x), accounting for the variability in the estimated regression line.

Confidence Interval for the Mean Response

A confidence interval for the mean response provides a range of values within which the true average response is expected to lie with a specified level of confidence (e.g., 95%). This interval is constructed using the estimated regression line, the standard error of the predicted mean response, and critical values from the t-distribution based on the degrees of freedom from the data.

Linear Regression Model

This concept involves modeling the relationship between a predictor variable and a response variable through a linear equation. It estimates parameters that best fit the data, allowing predictions of the response for any given value of the predictor. The model also provides a basis for assessing variability and uncertainty in these predictions.

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