Beavers and beetles Do beavers benefit beetles? Researchers...

Beavers and beetles Do beavers benefit beetles? Researchers laid out 23 circular plots, each 4 meters in diameter, at random in an area where beavers were cutting down cottonwood trees. In each plot, they counted the number of stumps from trees cut by beavers and the number of clusters of beetle larvae. Ecologists think that the new sprouts from stumps are more tender than other cottonwood growth, so that beetles prefer them. If so, more stumps should produce more beetle larvae. ${ }^{8}$ Minitab output for a regression analysis on these data is shown below. Construct and interpret a $99 \%$ confidence interval for the slope of the population regression line. Assume that the conditions for performing inference are met. $$ \begin{aligned} &\text { Regression Analysis: Beetle larvae versus Stumps }\\ &\begin{array}{lllll} \text { Predictor } & \text { Coef } & \text { SE Coef } & \text { T } & \text { P } \\ \text { Constant } & -1.286 & 2.853 & -0.45 & 0.657 \\ \text { Stumps } & 11.894 & 1.136 & 10.47 & 0.000 \end{array}\\ &\begin{array}{ll} S=6.41939 & R-S q=83.9 \% & R-S q(a d j)=83.1 \% \end{array} \end{aligned} $$

Beavers and beetles Do beavers benefit beetles? Researchers laid out 23 circular plots, each 4 meters in diameter, at random in an area where beavers were cutting down cottonwood trees. In each plot, they counted the number of stumps from trees cut by beavers and the number of clusters of beetle larvae. Ecologists think that the new sprouts from stumps are more tender than other cottonwood growth, so that beetles prefer them. If so, more stumps should produce more beetle larvae. ${ }^{8}$ Minitab output for a regression analysis on these data is shown below. Construct and interpret a $99 \%$ confidence interval for the slope of the population regression line. Assume that the conditions for performing inference are met.
$$
\begin{aligned}
&\text { Regression Analysis: Beetle larvae versus Stumps }\\
&\begin{array}{lllll}
\text { Predictor } & \text { Coef } & \text { SE Coef } & \text { T } & \text { P } \\
\text { Constant } & -1.286 & 2.853 & -0.45 & 0.657 \\
\text { Stumps } & 11.894 & 1.136 & 10.47 & 0.000
\end{array}\\
&\begin{array}{ll}
S=6.41939 & R-S q=83.9 \% & R-S q(a d j)=83.1 \%
\end{array}
\end{aligned}
$$

Key Concepts

t-Distribution in Regression

The t-distribution is used in regression analysis when constructing confidence intervals or performing hypothesis tests, particularly when dealing with small sample sizes. It accounts for the additional uncertainty due to estimating the standard error from the sample data and relies on the degrees of freedom associated with the regression model.

Confidence Interval for the Slope

A confidence interval for the slope provides a range of values within which we are confident that the true population slope lies, based on sample data. Constructing this interval involves the estimated slope, its standard error, and a critical value from the t-distribution corresponding to the selected confidence level, allowing inference about the overall relationship between the predictor and response variables.

Simple Linear Regression

Simple linear regression is a statistical method used to model the relationship between a single predictor variable and a response variable by fitting a straight line through the observed data. It involves estimating an intercept and a slope, which describe how the expected value of the response changes with a one-unit change in the predictor.

Slope Coefficient

The slope coefficient represents the estimated change in the response variable for an additional unit of the predictor variable. It quantifies the strength and direction of the relationship between the two variables, indicating whether an increase in the predictor is associated with an increase or decrease in the response.

Transcript

00:01 So we're given this mini tab output for regression analysis on some data, and we're told that there were a total of n equals 23 circular plots, and so there's 23 data values, and within each plot they were counting the number of stumps from trees cut by beavers.

00:22 And we're looking at the number of clusters of bee larvae.

00:26 So we made a regression equation, y hat equals your constant, beta one or beta not, plus your slope coefficient, which is beta one x.

00:41 And x is the number of stumps, bee larvae versus stumps.

00:51 So y hat would be your bee larvae, x is your number of stumps, bee larvae versus stumps.

00:57 That's what it says here.

00:59 So we want to make a, and given this output, we want to make a 99 % confidence interval for the slope for beta one, the slope of the population regression line.

01:14 So this is what our form is.

01:17 Now, these are going to be estimators, so these will be hat values, beta not hat beta one hat.

01:26 And so the constant is beta not, the stumps coefficient is beta one.

01:30 And so the formula is as follows, beta hat one plus minus t alpha over two, noting the degrees of freedom, multiplied by the standard error of the coefficient.

01:47 Good.

01:47 So from the printout, we can see that the coefficient for the slope term is 11 .894 plus minus the t, we don't know yet, but the alpha is going to be 0 .01 because one minus the alpha gives us the confidence level.

02:06 So one minus 1%, it's 99%.

02:10 And then the degrees of freedom for this is given as n, big n minus one, number of, or excuse me, minus two.

02:19 And that's because we're taking an extra degree of freedom away for the predictor.

02:25 That's why it's minus two, because normally the degrees of freedom is n minus one, but we subtract another one for the extra parameter.

02:32 So the degrees of freedom in this case will be 23 minus two.

02:36 So it will be 21 degrees of freedom...

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