Question
Wildlife biologists are interested in the health of tule elk, a species of deer found in California. An important measurement of tule elk health is their weight. The weight of a tule elk is difficult to measure in the wild. However, chest circumference, which is believed to be related to the weight of a tule elk, can easily be measured from a safe distance using a harmless laser. A study was done to investigate whether chest circumference, in centimeters \((\mathrm{cm})\), could be used to accurately estimate the weight, in kilograms \((\mathrm{kg})\), of male tule elk. For the study, wildlife biologists captured 30 male tule elk, measured their chest circumference and weight, and then released the elk. The data for the 30 male tule elk are shown in the scatterplot.
(a) Describe the relationship between chest circumference and weight of male tule elk in context.
Following is the equation of the least-squares regression line relating chest circumference and weight for male tule elk.
$
\text { predicted weight }=-350.3+3.7455 \text { (chest circumference) }
$
(b) The weight of one male tule elk with a chest circumference of \(145.9 \mathrm{~cm}\) is \(204.3 \mathrm{~kg}\).
(i) Using the equation of the least-squares regression line, calculate the predicted weight for this male tule elk. Show your work.
(ii) Calculate the residual for this male tule elk. Show your work.
The equation of the least-squares regression line relating chest circumference and weight for male tule elk is repeated here.
$
\text { predicted weight }=-350.3+3.7455 \text { (chest circumference) }
$
(c) Interpret the slope of the least-squares regression line in context.
(d) The sambar, another species of deer, is similar in size to the tule elk. The slope of the population regression line relating chest circumference and weight for all male sambars is 4.5 kilograms per centimeter. A wildlife biologist wants to determine whether the slope of the population regression line for male tule elk is different than that for male sambars. Let \(\beta\) represent the slope of the population regression line for male tule elk. The wildlife biologist conducted a test of the following hypotheses using the sample of 30 tule elk.
$
\begin{aligned}
& \mathrm{H}_0: \beta=4.5 \\
& \mathrm{H}_{\mathrm{a}}: \beta \neq 4.5
\end{aligned}
$
The test statistic was calculated to be 3.408 . Assume all conditions for inference were met.
(i) Determine the \(p\)-value of the test.
(ii) At a significance level of \(\alpha=0.05\), what conclusion should the wildlife biologist make regarding the slope of the population regression line for male tule elk? Justify your response.
▶️Answer/Explanation
Ans:
(a) There is a strong, positive, linear relationship between chest circumference and weight of male tole elk.
(b) (i) weight \(=-350.3+3.7455(145.9)=196.17 \mathrm{~kg}\).
(ii) residual \(=204.3-196.17=8.13 \mathrm{~kg}\).
(c) For every increase of \(1 \mathrm{~cm}\) of chest circumference, the predicted weight for male tole elks increases by \(3.7455 \mathrm{~kg}\).
\(2 \times \operatorname{tcdf}(3.408,9999,28)\)
\(P\)-value \(=0.002\)
(ii) Assuming \(H_0\) is true \((B=4.5)\), there is a 0.002 probability of getting the obtained sample slope or more extreme in either direction purely by chance.
Because \(0.002<0.05\), we reject \(H_0\) and do have convincing evidence that the slope of the population regression line for male tole elk is different than \(4.5 \mathrm{~kg} / \mathrm{cm}\).
Question
A biologist gathered data on the length, in millimeters ( \(\mathrm{mm}\) ), and the mass, in grams ( \(\mathrm{g}\) ), for 11 bullfrogs. The data are shown in Plot 1.
(a) Based on the scatterplot, describe the relationship between mass and length, in context.
From the data, the biologist calculated the least-squares regression line for predicting mass from length. The least-squares regression line is shown in Plot 2.
(b) Identify and interpret the slope of the least-squares regression line in context.
(c) Interpret the coefficient of determination of the least-squares regression line, \(r^2 \approx 0.819\), in context.
(d) From Plot 2, consider the residuals of the 11 bullfrogs.
(i) Based on the plot, approximately what is the length and mass of the bullfrog with the largest absolute value residual?
(ii) Does the least-squares regression line overestimate or underestimate the mass of the bullfrog identified in part (d-i)? Explain your answer.
▶️Answer/Explanation
Ans:
(a) There appears to be a moderately strong linear relationship between mass and length for the 11 bullfrogs. There are no obvious outliers.
(b) The predicted mass in grams of a bullfrog is expected to increase by 6.086 for every \(1 \mathrm{~mm}\) increase in the length of a bullfrog.
(c)\(81.9 \%\) of the variation in the mass \((g)\) of a bull frog is accounted for by the least squares regression line on length \((\mathrm{mm})\) of a bullfrog.
(d)(i)The bullfrog with the largest absolute value residual has a length of approximately \(162 \mathrm{~mm}\) and a mass of approximately \(355 \mathrm{~g}\).
(ii) The least squares regression line would overestimate the mass of the bullfrog because the bullfrog has a negative residual with the observed value being less than the expected.