Homework answers / question archive / 1) Assume that you have paired values consisting of heights (in inches) and weights (in Ib) from 40 randomly selected men

1) Assume that you have paired values consisting of heights (in inches) and weights (in Ib) from 40 randomly selected men

Statistics

Share With

---

1) Assume that you have paired values consisting of heights (in inches) and weights (in Ib) from 40 randomly selected men. The linear correlation coefficient ris 0.537. Find the value of the coefficient of determination. What practical information does the coefficient of determination provide?

Choose the correct answer below.

A. The coefficient of determination is 0.712. 28.8% of the variation is explained by the linear correlation, and 71.2% is explained by other factors.

B. The coefficient of determination is 0.712. 71.2% of the variation is explained by the linear correlation, and 28.8% is explained by other factors.

C. The coefficient of determination is 0.288. 28.8% of the variation is explained by the linear correlation, and 71.2% is explained by other factors.

D. The coefficient of determination is 0.288. 71.2% of the variation is explained by the linear correlation, and 28.8% is explained by other factors.

2. Use the value of the linear correlation coefficient r to find the coefficient of determination and the percentage of the total variation that can be explained by the linear relationship between the two variables.

r= 0.588

What is the value of the coefficient of determination?

r² = (Round to four decimal places as needed.)

What is the percentage of the total variation that can be explained by the linear relationship between the two variables?

Explained variation = _____ % (Round to two decimal places as needed.)

3. Use the value of the linear correlation coefficient r to find the coefficient of determination and the percentage of the total variation that can be explained by the linear relationship between the two variables.

r= - 0.453

What is the value of the coefficient of determination?

r² = (Round to four decimal places as needed.)

What is the percentage of the total variation that can be explained by the linear relationship between the two variables?

Explained variation = ____ % (Round to two decimal places as needed.)

4. The Minitab output shown below was obtained by using paired data consisting of weights (in Ib) of 32 cars and their highway fuel consumption amounts (in mi/gal). Along with the paired sample data, Minitab was also given a car weight of 5000 Ib to be used for predicting the highway fuel consumption amount. Use the information provided in the display to determine the value of the linear correlation coefficient. (Be careful to correctly identify the sign of the correlation coefficient.) Given that there are 32 pairs of data, is there sufficient evidence to support a claim of linear correlation between the weights of cars and their highway fuel consumption amounts?

¹Click the icon to view the Minitab display.

The linear correlation coefficient is _____.

(Round to three decimal places as needed. )

Is there sufficient evidence to support a claim of linear correlation?

(a) No

(b) Yes

1: Minitab output

The regression equation is

Highway = 50.8 — 0.00580 Weight

Predictor               Coef                     SE Coef                  T                 P

Constant             50.801                       2.844             17.46         0.000

Weight         -0.0057967              0.0007555            - 7.54         0.000

S = 2.29741               R – Sq = 63.8%                  R - Sq(adj) = 61.3%

Predicted Values for New Observations

New

Obs                         Fit             SE Fit                             95% Cl                             95% PI

1                       21.818             0.525                  (20.803, 22.833)           (17.227, 26.409)

Values of Predictors for New Observations

New

Obs          Weight

     1                5000

5. The accompanying technology output was obtained by using the paired data consisting of foot lengths (cm) and heights (cm) of a sample of 40 people. Along with the paired sample data, the technology was also given a foot length of 13.2 cm to be used for predicting height. The technology found that there is a linear correlation between height and foot length. If someone has a foot length of 13.2 cm, what is the single value that is the best predicted height for that person?

²Click the icon to view the technology output.

The single value that is the best predicted height is __________ cm.

(Round to the nearest whole number as needed. )

2: Technology Output

The regression equation is

Height = 54.7 + 4.15 Foot Length

Predictor	Coef	SE Coef	T	P
Constant	54.67	11.02	4.96	0.000
Foot Length	4.1547	0.4626	8.98	0.000

 

S = 5.50642           R-Sq = 70.6%               R-Sq(adj) = 69.8%

Predicted Values for New Observations

New Obs	Fit	SE Fit	95% CI	95% PI
1	109.512	1.783	(104.853, 114.171)	(98.497, 120.527)

 

Values of Predictors for New Observations

                             Foot

New Obs          Length

              1               13.2

6. Over the years, it was noticed that the cost of a slice of pizza and the cost of a subway fare in a certain city seemed to increase by the same amounts. Let x represent the cost of a slice of pizza and let y represent the corresponding subway fare. Use the following statistics that were obtained from a random sample of costs (in dollars) of pizza/subway fares to construct a prediction interval estimate of the subway fare with 99% confidence when the cost of a slice of pizza is $2.90.

n=6         b₀ = 0.03456          b₁ = 0.94502

X= 1.0833333        Sx =6.50          Sx² = 9.77      S_e = 0.123

The 99% prediction interval is _________ < y < ______.

(Round to two decimal places as needed.)

7. Over the years, it was noticed that the cost of a slice of pizza and the cost of a subway fare in a certain city seemed to increase by the same amounts. Let x represent the cost of a slice of pizza and let y represent the corresponding subway fare. Use the following statistics that were obtained from a random sample of costs (in dollars) of pizza/subway fares to construct a prediction interval estimate of the subway fare with 95% confidence when the cost of a slice of pizza is $1.20.

n=6          b₀ = 0.03456                           b₁ = 0.94502

X= 1.0833333                     Sx = 6.50          Sx² = 9.77          S_e = 0.123

The 95% prediction interval is ______ < y < __________.

(Round to two decimal places as needed.)

8. Listed below are altitudes (thousands of feet) and outside air temperatures (°F) recorded during a flight. Find the (a) explained variation, (b) unexplained variation, and (c) indicated prediction interval. There is sufficient evidence to support a claim of a linear correlation, so it is reasonable to use the regression equation when making predictions. For the prediction interval, use a 95% confidence level with the altitude of 6327 ft (or 6.327 thousand feet).

Altitude	2	9	16	24	29	31	34
Temperature	60	31	22	-4	-29	-41	-51

 

a. Find the explained variation.

(Round to two decimal places as needed.)

b. Find the unexplained variation.

(Round to five decimal places as needed.)

c. Find the indicated prediction interval.

___________________°F < y < ____________ °F

(Round to four decimal places as needed.)

9. Listed below are amounts of court income and salaries paid to the town justices for a certain town. All amounts are in thousands of dollars. Find the (a) explained variation, (b) unexplained variation, and (c) indicated prediction interval. There is sufficient evidence to support a claim of a linear correlation, so it is reasonable to use the regression equation when making predictions. For the prediction interval, use a 99% confidence level with a court income of $800,000.

Court Income	$70	$401	$1587	$1149	$292	$252	$113	$158	$27
Justice Salary	$27	$45	$99	$52	$40	$56	$25	$29	$23

 

a. Find the explained variation.

(Round to three decimal places as needed. )

b. Find the unexplained variation.

(Round to three decimal places as needed. )

c. Find the indicated prediction interval.

$______ < y < $______

(Round to four decimal places as needed. )

10. The table below lists measured amounts of redshift and the distances (billions of light-years) to randomly selected astronomical objects. Find the (a) explained variation, (b) unexplained variation, and (c) indicated prediction interval. There is sufficient evidence to support a claim of a linear correlation, so it is reasonable to use the regression equation when making predictions. For the prediction interval, use a 90% confidence level with a redshift of 0.0126.

Redshift	0.0236	0.0535	0.0717	0.0398	0.0441	0.0105
Distance	0.31	0.74	0.99	0.57	0.62	0.15

 

a. Find the explained variation.

(Round to six decimal places as needed.)

b. Find the unexplained variation.

(Round to six decimal places as needed.)

c. Find the indicated prediction interval.

__________________ billion light-years < y < _____________ billion light-years

(Round to three decimal places as needed. )

11. The table below lists weights (carats) and prices (dollars) of randomly selected diamonds. Find the (a) explained variation, (b) unexplained variation, and (c) indicated prediction interval. There is sufficient evidence to support a claim of a linear correlation, so it is reasonable to use the regression equation when making predictions. For the prediction interval, use a 95% confidence level with a diamond that weighs 0.8 carats.

Weight	0.3	0.4	0.5	0.5	1.0	0.7
Price	$514	$1149	$1335	$1421	$5664	$2277

 

a. Find the explained variation.

(Round to the nearest whole number as needed. )

b. Find the unexplained variation.

(Round to the nearest whole number as needed. )

c. Find the indicated prediction interval.

$___________ < y < $ __________

(Round to the nearest whole number as needed. )

12. Fill in the blank.

The _______ deviation of (x, y) is the vertical distance y- y, which is the distance between the point (x,y) and the horizontal line passing through the sample mean y.

The (1) ___ deviation of (x, y) is the vertical distance y - y, which is the distance between the point (x,y) and the horizontal line passing through the sample mean y.

(1)  explained

 partial

total

unexplained

13. Fill in the blank.

The coefficient of determination, r² is the amount of variation in ____________ that is explained by the regression line.

The coefficient of determination, r² is the amount of variation in (1) ___________ that is explained by the regression line.

(1) r

X

s

y

14. Fill in the blank.

A ___________ is an interval estimate of a predicted value of y.

A (1) ________ is an interval estimate of a predicted value of y.

(1) confidence interval

prediction interval

residual

standard error of estimate

15. Fill in the blank.

The ______ is a measure of the differences between the observed sample y-values and the predicted values y that are obtained using the regression equation.

The (1) ______ is a measure of the differences between the observed sample y-values and the predicted values y that are obtained using the regression equation.

(1) confidence interval

 prediction interval

standard error of estimate

residual

16. Fill in the blank.

The deviation is the vertical distance y — y, which is the distance between the predicted y-value and the horizontal line passing through the sample mean y.

The (1) ______  deviation is the vertical distance y - y, which is the distance between the predicted y-value and the horizontal line passing through the sample mean y.

(1)  unexplained

 explained

standard

total