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The Practice of Statistics for AP Examination

Daren Starnes, Josh Tabor

$Math Studyset Vaia Explanations$ Math

6th Edition · 837 pages

ISBN: 9781319113339

Chapter 12: Q. 51 (page 817)

Suppose that the relationship between a response variable y and an explanatory variable x is modelled by $y = 2.7 {(0.316)}^{x}$ . Which of these scatterplots would approximately follow a straight line?
a. A plot of y against x
b. A plot of y against log x
c. A plot of log y against x
d. A plot of log y against log x
e. A plot of $y^{\sqrt{y}}$ against x

Short Answer

Expert verified

The scatterplots that approximately follow a straight line is option (c).

Step by step solution

01

Given information

The variable $x$ is modelled by $y = 2.7 {(0.316)}^{x}$

02

Explanation

The given data is

$y = 2.7 {(0.316)}^{x}$

Because this equation depicts a nonlinear exponential function, the scatter plot will be a straight line.

If both sides of the equation are logarithmized, the equation is a linear equation with variables $\log y$ and $x$ , and the scatter plot would be a straight line in this case.

As a result, the best solution is (c).

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Most popular questions from this chapter

Lamb’s quarters is a common weed that interferes with the growth of corn. An agriculture researcher planted corn at the same rate in $16$ small plots of ground and then weeded the plots by hand to allow a fixed number of lamb’s quarters plants to grow in each meter of cornrow. The decision on how many of these plants to leave in each plot was made at random. No other weeds were allowed to grow. Here are the yields of corn (bushels per acre) in each of the plots:

Here is some computer output from a least-squares regression analysis of these data. Do these data provide convincing evidence at the $α = 0.05$ level that more lamb’s quarters reduce corn yield?

$Predictor Coef SE Coef T P Constant 166.483 2.725 61.11 0.000 Weeds per - 1.0987 0.5712 - 1.92 0.075 meter S = 7.97665 R - Sq = 20.9 % R - Sq (adj) = 15.3 %$

Brawn versus brain How is the weight of an animal’s brain related to the weight of its body? Researchers collected data on the brain weight (in grams) and body weight (in Page Number: 813 Page Number: 814 kilograms) for $96$ species of mammals. The following figure is a scatterplot of the logarithm of brain weight against the logarithm of body weight for all $96$ species. The least-squares regression line for the transformed data is

${logy}^{\land} = 1.01 + 0.72 {logx}^{\hat{\log y}} = 1.01 + 0.72 \log x$

Based on footprints and some other sketchy evidence, some people believe that a large ape-like animal, called Sasquatch or Bigfoot, lives in the Pacific Northwest. Bigfoot’s weight is estimated to be about $127$ kilograms (kg). How big do you expect Bigfoot’s brain to be?

Which of the following statements about the t distribution with degrees of freedom $d f$ is (are) true?

I. It is symmetric.

II. It has more variability than the t distribution with $d f + 1$ degrees of freedom. III. III. As df increases, the t distribution approaches the standard Normal distribution.

a. I only

b. II only

c. III only

d. I and III

e. I, II, and III

Heart weights of mammals Here are some data on the hearts of various mammals:

a. Make an appropriate scatterplot for predicting heart weight from length. Describe what you see.

b. Use transformations to linearize the relationship. Does the relationship between heart weight and length seem to follow an exponential model or a power model? Justify your answer.

c. Perform least-squares regression on the transformed data. Give the equation of your regression line. Define any variables you use.

d. Use your model from part (c) to predict the heart weight of a human who has a left ventricle $6.8$ cm long.

Multiple Choice Select the best answer for Exercises 23-28. Exercises 23-28 refer to the following setting. To see if students with longer feet tend to be taller, a random sample of $25$ students was selected from a large high school. For each student, $x = f o o t l e n g t h & y = h e i g h t$ ere recorded. We checked that the conditions for inference about the slope of the population regression line are met. Here is a portion of the computer output from a least-squares regression analysis using these data:

26. Which of the following is the best interpretation of the value $0.4117$ in the computer output?

a. For each increase of $1 c m$ in foot length, the average height increases by about $0.4117 c m$

b. When using this model to predict height, the predictions will typically be off by about $0.4117 c m$ .

c. The linear relationship between foot length and height accounts for $41.17 %$ of the variation in height.

d. The linear relationship between foot length and height is moderate and positive.

e. In repeated samples of size $25$ the slope of the sample regression line for predicting height from foot length will typically vary from the population slope by about $0.4117$ .

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