Based on a sample of n=20, the least-squares method was used...

Based on a sample of $n=20$, the least-squares method was used to develop the following prediction line: $\hat{Y}_i=-0.2977+$ $0.0466 X_i$. In addition, $$ \bar{X}=90 \quad S_{X Y}=0.6754 \quad S S X=178,500 $$ a. Calculate the value for $\hat{Y}_i$ when $X_i=85$. b. Construct a $95 \%$ confidence interval estimate of the population mean response for an individual response, $Y$. c. Interpret the confidence interval from part (b).

Based on a sample of $n=20$, the least-squares method was used to develop the following prediction line: $\hat{Y}_i=-0.2977+$ $0.0466 X_i$.
In addition,
$$
\bar{X}=90 \quad S_{X Y}=0.6754 \quad S S X=178,500
$$
a. Calculate the value for $\hat{Y}_i$ when $X_i=85$.
b. Construct a $95 \%$ confidence interval estimate of the population mean response for an individual response, $Y$.
c. Interpret the confidence interval from part (b).

Key Concepts

Least Squares Regression

This method estimates the parameters of the linear relationship between the independent and dependent variables by minimizing the sum of the squared differences between the observed values and the values predicted by the model. It provides a systematic way to derive the regression line characteristics, such as the intercept and the slope, which are essential for making predictions.

Prediction Using the Regression Equation

Once the regression line is established, it can be used to predict the response variable for a given value of the predictor. This involves substituting the specific predictor value into the regression equation, yielding an estimated response. This concept is fundamental in regression analysis to understand the relationship between variables.

Confidence Interval for the Mean Response

In regression, a confidence interval for the mean response quantifies the uncertainty around the estimated average value of the dependent variable at a specific value of the predictor. It relies on both the variability of the data and the distance of the predictor value from the mean of all predictor values. The interval provides a range that is likely to contain the true mean response with a specified level of confidence.

t-Distribution in Regression Inference

When constructing confidence intervals in regression analysis, especially with small sample sizes, the t-distribution is used instead of the normal distribution. This adjustment accounts for the additional uncertainty in estimating the standard deviation from the data and is based on degrees of freedom, which in simple linear regression is typically n-2.

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