Chapter 15, Understanding Relationships—Numerical Data Video Solutions, Statistics Learning From Data

Problem 1

Identify the following relationships as deterministic or probabilistic:
a. The relationship between the length of the sides of a square and its perimeter.
b. The relationship between the height and weight of an adult.
c. The relationship between SAT score and college freshman GPA.
d. The relationship between tree height in centimeters and tree height in inches.

Christopher Stanley

Numerade Educator

01:40

Problem 2

Let $x$ be the size of a house (in square feet) and $y$ be the amount of natural gas used (therms) during a specified period. Suppose that for a particular community, $x$ and $y$ are related according to the simple linear regression model with
$\beta=$ slope of population regression line $=.017$ $\alpha=y$ intercept of population regression line $=-5.0$
Houses in this community range in size from 1000 to 3000 square feet.
a. What is the equation of the population regression line?
b. Graph the population regression line by first finding the point on the line corresponding to $x=1000$ and then the point corresponding to $x=2000$, and drawing a line through these points.
c. What is the mean value of gas usage for houses with 2100 sq. ft. of space?
d. What is the average change in usage associated with a 1 sq. ft. increase in size?
e. What is the average change in usage associated with a 100 sq. ft. increase in size?
f. Would you use the model to predict mean usage for a 500 sq. ft. house? Why or why not?

Sheryl Ezze

Numerade Educator

04:40

Problem 3

Suppose that a simple linear regression model is appropriate for describing the relationship between $y=$ house price (in dollars) and $x=$ house size (in square feet) for houses in a large city. The population regression line is $y=23,000+47 x$ and $\sigma_{e}=5000$
a. What is the average change in price associated with one extra square foot of space? With an additional 100 sq. $f t$. of space?
b. Approximately what proportion of 1800 sq. ft. homes would be priced over $\$ 110,000 ?$ Under $\$ 100,000 ?$

Jon Southam

Numerade Educator

02:05

Problem 4

Identify the following relationships as deterministic or probabilistic:
a. The relationship between height at birth and height at one year of age.
b. The relationship between a positive number and its square root.
c. The relationship between temperature in degrees Fahrenheit and degrees centigrade.
d. The relationship between adult shoe size and shirt size.

Christopher Stanley

Numerade Educator

06:49

Problem 5

The flow rate in a device used for air quality measurement depends on the pressure drop $x$ (inches of water) across the device's filter. Suppose that for $x$ values between 5 and $20,$ these two variables are related according to the simple linear regression model with population regression line $y=-0.12+0.095 x$
a. What is the mean flow rate for a pressure drop of 10 inches? A drop of 15 inches?
b. What is the average change in flow rate associated with a 1 inch increase in pressure drop? Explain.

Cyra Jelle Calleja

Numerade Educator

00:39

Problem 6

The paper "Predicting Yolk Height, Yolk Width, Albumen Length, Eggshell Weight, Egg Shape Index, Eggshell Thickness, Egg Surface Area of Japanese Quails Using Various Egg Traits as Regressors" (International journal of Poultry Science [2008]: $85-88$ ) suggests that the simple linear regression model is reasonable for describing the relationship between $y=$ eggshell thickness (in micrometers) and $x=$ egg length (mm) for quail eggs. Suppose that the population regression line is $y=0.135+0.003 x$ and that $\sigma=0.005 .$ Then, for a fixed $x$ value, $y$ has a normal distribution with mean $0.135+0.003 x$ and standard deviation 0.005 .
a. What is the mean eggshell thickness for quail eggs that are $15 \mathrm{~mm}$ in length? For quail eggs that are $17 \mathrm{~mm}$ in length?
b. What is the probability that a quail egg with a length of $15 \mathrm{~mm}$ will have a shell thickness that is greater than $0.18 \mu \mathrm{m} ?$
c. Approximately what proportion of quail eggs of length $14 \mathrm{~mm}$ have a shell thickness of greater than $0.175 ?$ Less than $0.178 ?$

Sriparna Bhattacharjee

Numerade Educator

07:35

Problem 7

Tom and Ray are managers of electronics stores with slightly different pricing strategies for USB drives. In Tom's store, customers pay the same amount, $c,$ for each USB drive. In Ray's store, it is a little more exciting. The customer pays an up-front cost of $\$ 1.00$. Ray charges the same price per USB drive, $c$, but at the register the customer flips a coin. If the coin lands heads up, the customer gets his or her $\$ 1.00$ back, plus another dollar off the total cost of the USB drives purchased.
a. Which of these pricing strategies can be expressed as a deterministic model?
b. Using mathematical notation, specify a model using Tom's pricing strategy that relates $y=$ total cost to $x=$ number of USB drives purchased.
c. Using mathematical notation, specify a model using Ray's pricing strategy that relates $y=$ total cost to $x=$ number of USB drives purchased.
d. Describe the distribution of $e$ for the probabilistic model described above. What is the mean of the distribution of $e ?$ What is the standard deviation of $e ?$

Sophie Knight

Numerade Educator

View

Problem 8

Identify the following relationships as deterministic or probabilistic:
a. The relationship between the speed limit and a driver's speed.
b. The relationship between the price in dollars and the price in Euros of an object.
c. The relationship between the number of pages and the number of words in a text book.
d. The relationship between the possible numbers of pennies and the nickels in a pile if no other coins are in the pile and the amount of money in the pile is $\$ 3.00$.

Rashmi Sinha

Numerade Educator

02:10

Problem 9

Hormone replacement therapy (HRT) is thought to increase the risk of breast cancer. The accompanying data on $x=$ percent of women using HRT and $y=$ breast cancer incidence (cases per 100,000 women) for a region in Germany for 5 years appeared in the paper "Decline in Breast Cancer Incidence after Decrease in Utilisation of Hormone Replacement Therapy" (Epidemiology [2008]: $427-430$ ). The authors of the paper used a simple linear regression model to describe the relationship between HRT use and breast cancer incidence.
\begin{tabular}{|cc|}
\hline HRT Use & Breast Cancer Incidence \\
\hline 46.30 & 103.3 \\
40.60 & 105.0 \\
39.50 & 100.0 \\
36.60 & 93.8 \\
30.00 & 83.5 \\
\hline
\end{tabular}
a. What is the equation of the estimated regression line?
b. What is the estimated average change in breast cancer incidence associated with a 1 percentage point increase in HRT use?
c. What would you predict the breast cancer incidence to be in a year when HRT use was $40 \% ?$
d. Should you use this regression model to predict breast cancer incidence for a year when HRT use was $20 \%$ ? Explain.
e. Calculate and interpret the value of $r^{2}$.
f. Calculate and interpret the value of $s_{e}$.

Sheryl Ezze

Numerade Educator

16:41

Problem 10

Consider the accompanying data on $x=$ advertising share and $y=$ market share for a particular brand of soft drink during 10 randomly selected years. $\begin{array}{llllllllll}x & 0.103 & 0.072 & 0.071 & 0.077 & 0.086 & 0.047 & 0.060 & 0.050 & 0.070 & 0.052\end{array}$
$\begin{array}{llllllllll}\underline{y} & 0.135 & 0.125 & 0.120 & 0.086 & 0.079 & 0.076 & 0.065 & 0.059 & 0.051 & 0.039\end{array}$
a. Construct a scatterplot for these data. Do you think the simple linear regression model would be appropriate for describing the relationship between $x$ and $y ?$
b. Calculate the equation of the estimated regression line and use it to obtain the predicted market share when the advertising share is 0.090
c. Compute $r^{2}$. How would you interpret this value?
d. Calculate a point estimate of $\sigma_{e^{-}}$ How many degrees of freedom is associated with this estimate?

Jon Southam

Numerade Educator

05:40

Problem 11

The paper "Depression, Body Mass Index, and Chronic Obstructive Pulmonary Disease-A Holistic Approach" (International Journal of COPD [2016]:239-249) gives data on
$x=$ change in Body Mass Index (BMI, in kilograms/meter^{2} )
and
$y=$ change in a measure of depression
for patients suffering from depression who participated in a pulmonary rehabilitation program. The table below contains a subset of the data given in the paper and are approximate values read from a scatterplot in the paper.
a. What percentage of observed variation in depression score change can be explained by the simple linear regression model?
b. Give a point estimate of $\sigma_{e}$ and interpret this estimate.
c. Give an estimate of the average change in depression score change associated with a $1 \mathrm{~kg} / \mathrm{m}^{2}$ increase in BMI change.
d. Calculate a point estimate of the mean depression score change for a patient whose BMI change was $1.2 \mathrm{~kg} / \mathrm{m}^{2}$

Sheryl Ezze

Numerade Educator

03:23

Problem 12

The production of pups and their survival are the most significant factors contributing to gray wolf population growth. The causes of early pup mortality are unknown and difficult to observe. The pups are concealed within their dens for 3 weeks after birth, and after they emerge it is difficult to confirm their parentage. Researchers recently used portable ultrasound equipment to investigate some factors related to reproduction ("Diagnosing Pregnancy, in Utero Litter Size, and Fetal Growth with Ultrasound in Wild, Free-Ranging Wolves," Journal of Mammology [2006]: 85-92). A scatterplot of $y=$ length of an embryonic sac diameter (in $\mathrm{cm}$ ) and $x=$ gestational age (in days) is shown below. Computer output from a regression analysis is also given.
a. What is the equation of the estimated regression line?
b. What is the estimated embryonic sac diameter for a gestational age of 30 days?
c. What is the average change in sac diameter associated with a 1 -day increase in gestational age?
d. What is the average change in sac diameter associated with a 5 -day increase in gestational age?
e. Would you use this model to predict the mean embryonic sac diameter for all gestation ages from conception to birth? Why or why not?

Karl Schaefer

University of Chicago