Given a simple regression analysis, suppose that we have...

Given a simple regression analysis, suppose that we have obtained a fitted regression model $$ \hat{y}_i=14+7 x_i $$ and also $$ s_e=7.45 \quad \bar{x}=8 \quad n=25 \sum_{i=1}^n\left(x_i-\bar{x}\right)^2=300 $$ Find the $95 \%$ confidence interval and $95 \%$ prediction interval for the point where $x=11$.

Given a simple regression analysis, suppose that we have obtained a fitted regression model
$$
\hat{y}_i=14+7 x_i
$$
and also
$$
s_e=7.45 \quad \bar{x}=8 \quad n=25 \sum_{i=1}^n\left(x_i-\bar{x}\right)^2=300
$$
Find the $95 \%$ confidence interval and $95 \%$ prediction interval for the point where $x=11$.

Key Concepts

Simple Linear Regression

This is a statistical method used to model the relationship between a dependent variable and a single independent variable by fitting a linear equation to observed data. The model includes an intercept and a slope and is useful for estimating and predicting responses.

Fitted Regression Model and Predictions

The fitted regression model provides an estimated mean response for any given value of the independent variable. It is used to predict the expected value of the dependent variable while acknowledging that individual observed values might deviate from this estimate.

Confidence Interval for the Mean Response

A confidence interval for the mean response at a specific predictor value provides a range within which the average value of the dependent variable is expected to lie with a specified level of confidence (typically 95%). It accounts for the uncertainty in estimating the regression parameters.

Prediction Interval for a New Observation

A prediction interval for a new observation at a given predictor value provides a range that is expected to contain the actual value of the dependent variable for a single new observation. It is wider than the confidence interval because it incorporates both the uncertainty in the mean response and the additional random error inherent in individual observations.

t-Distribution and Degrees of Freedom

The t-distribution is used to calculate confidence and prediction intervals when the sample size is small and the standard error is estimated from the data. Degrees of freedom, usually the sample size minus the number of estimated parameters, dictate the shape of the t-distribution and are critical in determining the critical values for the intervals.

Sum of Squares of the Predictor Deviations

This measures the variability in the independent variable around its mean and is used in determining the precision of the estimated regression coefficients. A higher sum of squares generally leads to more precise estimates and narrower confidence intervals for the predicted mean response.

Please give Ace some feedback