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Chapter 12: Q. T12.10 (page 824)

T12.10We record data on the population of a particular country from 1960 to 2010. A
scatterplot reveals a clear curved relationship between population and year. However, a different scatterplot reveals a strong linear relationship between the logarithm (base 10) of the population and the year. The least-squares regression line for the transformed data is
log(population)=^鈭�13.5+0.01(year)
Based on this equation, which of the following is the best estimate for the population of the country in the year 2020?
a. 6.7
b. 812
c. 5,000,000
d. 6,700,000
e. 8,120,000

Short Answer

Expert verified

The correct answer is option (c) 5,000,000.

Step by step solution

01

Given information

To determine the best estimate for the population of the country in the year 2020.

02

Explanation

A scatterplot shows that the population and year have a clear curving relationship. A separate scatterplot, on the other hand, displays a significant linear link between the population logarithm and the year.
The following is the linear regression line:
ln(P~opulation)=-13.5+0.01(Year)
By 2020, the year will be replaced, and it will be:
ln(P~opulation)=-13.5+0.01(Year)
鈬�ln(Population)=-13.5+0.01(2020)
=6.7
Take each side's exponential:
Population=106.7
=5011872
鈮�5000000
As a result, option (c)5000000 is the correct option.

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Most popular questions from this chapter

T12.3 Inference about the slope 尾1 of a least-squares regression line is based on which of
the following distributions?
a. The tdistribution with n鈭�1 degrees of freedom
b. The standard Normal distribution
c. The chi-square distribution with n鈭�1 degrees of freedom
d. The t distribution with n-2 degrees of freedom
e. The Normal distribution with mean 渭 and standard deviation 蟽.

The town council wants to estimate the proportion of all adults in their medium-sized town who favor a tax increase to support the local school system. Which of the

following sampling plans is most appropriate for estimating this proportion?

a. A random sample of 250names from the local phone book

b. A random sample of 200parents whose children attend one of the local schools

c. A sample consisting of 500people from the city who take an online survey about the issue

d. A random sample of 300homeowners in the town

e. A random sample of 100people from an alphabetical list of all adults who live in the town

T12.2 Students in a statistics class drew circles of varying diameters and counted how many Cheerios could be placed in the circle. The scatterplot shows the results. The students want to determine an appropriate equation for the relationship between diameter and the number of Cheerios. The students decide to transform the data to make it appear more linear before computing a least-squares regression line. Which of the following transformations would be reasonable for them to try?

I. Plot the square root of the number of Cheerios against diameter.
II. Plot the cube of the number of Cheerios against diameter.
III. Plot the log of the number of Cheerios against the log of the diameter.
IV. Plot the number of Cheerios against the log of the diameter.

a. I and II
b. I and III
c. II and III
d. II and IV
e. I and IV

Exercises T12.4鈥揟12.8 refer to the following setting. An old saying in golf is 鈥淵ou drive for show and you putt for dough.鈥� The point is that good putting is more important than long driving for shooting low scores and hence winning money. To see if this is the case, data from a random sample of 69 of the nearly 1000 players on the PGA Tour鈥檚 world money list are examined. The average number of putts per hole (fewer is better) and the player鈥檚 total winnings for the previous season are recorded and a least-squares regression line was fitted to the data. Assume the conditions for
inference about the slope are met. Here is computer output from the regression analysis:

T12.8 Which of the following would make the calculation in Exercise T12.7 invalid?

a. If the scatterplot of the sample data wasn鈥檛 perfectly linear.

b. If the distribution of earnings has an outlier.

c. If the distribution of earnings wasn鈥檛 approximately Normal.

d. If the earnings for golfers with small putting averages was much more variable than the earnings for golfers with large putting averages.

e. If the standard deviation of earnings is much larger than the standard deviation of putting average.

Braking distance How is the braking distance for a motorcycle related to the speed at which the motorcycle was traveling when the brake was applied? Statistics teacher Aaron Waggoner gathered data to answer this question. The table shows the speed (in miles per hour) and the distance needed to come to a complete stop when the brake was applied (in feet).

Speed (mph)Distance (ft)Speed (mph)Distance (ft)61.423252.0894.924084191848110.333044.75

a. Transform both variables using logarithms. Then calculate and state the least-squares regression line using the transformed variables.

b. Use the model from part (a) to calculate and interpret the residual for the trial when the motorcycle was traveling at 48 mph.
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