Ideal proportions The students in Mr. Shenk's class measured...

Ideal proportions The students in Mr. Shenk's class measured the arm spans and heights (in inches) of a random sample of 18 students from their large high school. Some computer output from a least-squares regression analysis on these data is shown below. Construct and interpret a $90 \%$ confidence interval for the slope of the population regression line. Assume that the conditions for performing inference are met. $\begin{array}{lllrl}\text { Predictor } & \text { Coef } & \text { Stdev } & \text { t-ratio } & \text { p } \\ \text { Constant } & 11.547 & 5.600 & 2.06 & 0.056 \\ \text { Armspan } & 0.84042 & 0.08091 & 10.39 & 0.000 \\ \mathrm{~S}=1.613 & \mathrm{R}-\mathrm{Sq}=87.1 \% & \mathrm{R}-\mathrm{Sq}(\mathrm{adj}) & =86.3 \%\end{array}$

Ideal proportions The students in Mr. Shenk's class measured the arm spans and heights (in inches) of a random sample of 18 students from their large high school. Some computer output from a least-squares regression analysis on these data is shown below. Construct and interpret a $90 \%$ confidence interval for the slope of the population regression line. Assume that the conditions for performing inference are met.
$\begin{array}{lllrl}\text { Predictor } & \text { Coef } & \text { Stdev } & \text { t-ratio } & \text { p } \\ \text { Constant } & 11.547 & 5.600 & 2.06 & 0.056 \\ \text { Armspan } & 0.84042 & 0.08091 & 10.39 & 0.000 \\ \mathrm{~S}=1.613 & \mathrm{R}-\mathrm{Sq}=87.1 \% & \mathrm{R}-\mathrm{Sq}(\mathrm{adj}) & =86.3 \%\end{array}$

Key Concepts

Interpretation of the Slope Coefficient

The slope coefficient in a regression model represents the estimated change in the dependent variable for a one-unit increase in the independent variable. Interpreting this coefficient within the context of a confidence interval allows one to understand not just the point estimate but also the range of possible values for the true effect, providing insights into the reliability and practical significance of the relationship between the variables.

Least-Squares Regression

Least-squares regression is a method used to model the relationship between a dependent variable and one or more independent variables by minimizing the sum of the squared differences between observed and predicted values. This technique provides estimates for the parameters of the regression equation, most notably the intercept and the slope, which quantify the strength and direction of the association between variables.

Confidence Interval for the Slope

A confidence interval for the slope in a regression analysis provides a range of plausible values for the true population slope parameter. Constructing such an interval involves using the estimated slope, its standard error, and the appropriate t-distribution critical value based on the desired confidence level. This interval helps assess the precision of the slope estimate and whether there is a significant linear relationship between the predictor and the response variable.

t-Distribution

The t-distribution is used in regression analysis, especially when the sample size is small and the population standard deviation is unknown. It accounts for the additional uncertainty in the estimate of the standard error. The critical values from the t-distribution are used to construct confidence intervals and perform hypothesis tests for regression parameters, ensuring that the inference is valid under these conditions.

Inference Conditions in Regression

Performing inference in regression analysis requires that certain conditions be met, such as linearity of the relationship, independence of observations, constant variance (homoscedasticity) of the errors, and normally distributed errors. Ensuring these assumptions hold is crucial for the validity of the confidence intervals and hypothesis tests derived from the regression analysis.

Transcript

00:01 So we had a set of 18 pieces of data, and we have the arm span in inches versus the height.

00:09 And we want to find, and again, both are in inches.

00:14 And so we have a display of data, and we want to find a 90 % confidence interval for the slope.

00:26 And so we know that we'll end up taking our estimate.

00:30 For the b plus or minus, and then we will take our t star value that is 16 degrees of freedom because we take n minus two, and then we take the standard air of the slope.

00:45 And so when we look at our table, we see that there is a column for arm span, or i should say a row for arm span, and the arm span has the coefficient.

00:58 The coefficient is gonna end up being our slope and that is 0 .8402.

01:03 And then next to it it says standard deviation.

01:06 Well, that's going to be our standard error of the slope, which luckily for us we don't have to calculate longhand.

01:13 I usually make my students do that one time and that's about it.

01:18 And so we know we have 0 .8402 plus or minus.

01:23 And then when we look on our table for the 90 % confidence level for the t -stall.

01:30 Value with 16 degrees of freedom.

01:32 I actually used my calculator and did inverse t.

01:36 And if you do inverse t and you want 90 % confidence in here, that means you want a tail, upper tail of 5%, and a lower tail of 5%...