stats 2 week 5

1) The following sample observations were randomly selected.

   X :    4        5         3         6      10    Y:    4        6          5         7      7

Click here for the Excel Data File

(1)

Fill in the blanks below: (Round your answers to 2 decimal places.)

  Sx

  Sy

  Coefficient of correlation

  Coefficient of determination

(2)

Choose the right option.

  The correlation coefficient obtained here indicates correlation between X and Y.

(3)

Fill in the blanks. (Round your answer to the nearest whole number.)

  The coefficient of determination obtained here indicates X accounts for approximately   percent of the variation in Y.

The following sample observations were randomly selected.

    X:      5     3     6      3    4      4     6     8     Y:     13   15     7    12   13    11    9     5

Click here for the Excel Data File

2) Determine the coefficient of correlation and the coefficient of determination. Interpret the association between X and Y. (Round your coefficient of determination to 4 decimal places. Round all other answers to 2 decimal places. Negative amounts should be indicated by a minus sign.)

X

Y

( )2

( )2

( )( )

5

13

2.38  

0.02  

5.64  

0.30  

3

15

4.38  

3.52  

19.14  

6

7

1.13  

1.27  

13.14  

-4.08  

3

12

-1.88  

3.52  

1.89  

4

13

-0.88  

2.38  

5.64  

-2.08  

4

11

-0.88  

0.38  

0.14  

-0.33  

6

9

1.13  

-1.63  

1.27  

8

5

3.13  

-5.63  

9.77  

 -17.58  

39

85

=

  ,

=

 ,

sx

=

sy

=

 ,

r

=

The correlation coefficient of  indicates a  relationship between X and Y. The coefficient of determination is  , found by squaring the value of . X accounts for   percent of the variation in Y.

3) Bi-lo Appliance Super-Store has outlets in several large metropolitan areas in New England. The general sales manager aired a commercial for a digital camera on selected local TV stations prior to a sale starting on Saturday and ending Sunday. She obtained the information for Saturday–Sunday digital camera sales at the various outlets and paired it with the number of times the advertisement was shown on the local TV stations. The purpose is to find whether there is any relationship between the number of times the advertisement was aired and digital camera sales. The pairings are:

  Location of

Number of

Saturday–Sunday Sales

  TV Station

Airings

($ thousands)

  Providence

4

15

  Springfield

2

8

  New Haven

5

21

  Boston

6

24

  Hartford

3

17

Click here for the Excel Data File

(a)

What is the dependent variable?

is the dependent variable.

(c)

Determine the correlation coefficient. (Round your answer to 2 decimal places.)

  Coefficient of correlation

(d)

Interpret these statistical measures.

  The statistical measures obtained here indicate correlation between the variables.

4) The production department of Celltronics International wants to explore the relationship between the number of employees who assemble a subassembly and the number produced. As an experiment, two employees were assigned to assemble the subassemblies. They produced 15 during a one-hour period. Then four employees assembled them. They produced 25 during a one-hour period. The complete set of paired observations follows.

Number of Assemblers

One-Hour Production (units)

2

15

4

25

1

10

5

40

3

30

The dependent variable is production; that is, it is assumed that different levels of production result from a different number of employees.

Click here for the Excel Data File

(b)

A scatter diagram is provided below. Based on it, does there appear to be any relationship between the number of assemblers and production?

   , as the number of assemblers , so does the production.

(c)

Compute the coefficient of correlation. (Negative amounts should be indicated by a minus sign. Round sx, sy and r  to 3 decimal places.)

X

Y

( )2

( )2

( )( )

2  

15  

−9  

81  

4  

25  

1  

1  

1  

1  

10  

−14  

196  

5  

40  

2  

4  

32  

3  

30  

6  

0  

0  

=

=

sx

=

sy

=

r

=

(d)

Evaluate the strength of the relationship by computing the coefficient of determination. (Round your answer to 4 decimal places, and the percentage to nearest whole percent.)

  The r2 is , so about percent of the variation in production is explained by   the variation in the number of assemblers.

5) The owner of Maumee Ford-Mercury-Volvo wants to study the relationship between the age of a car and its selling price. Listed below is a random sample of 12 used cars sold at the dealership during the last year.

Car

Age (years)

Selling Price ($000)

Car

Age (years)

Selling Price ($000)

1

9

8.1

7

8

7.6

2

7

6.0

8

11

8.0

3

11

3.6

9

10

8.0

4

12

4.0

10

12

6.0

5

8

5.0

11

6

8.6

6

7

10.0

12

6

8.0

(a)

If we want to estimate selling price on the basis of the age of the car, which variable is the dependent variable and which is the independent variable?

   is the independent variable and  is the dependent variable.

(b-1)

Determine the coefficient of correlation. (Negative amounts should be indicated by a minus sign. Round your answers to 3 decimal places.)

X

Y

( )2

( )2

( )( )

9.0  

8.1  

1.192  

0.007  

1.420  

0.099  

7.0  

6.0  

-0.908  

3.674  

0.825  

1.741  

11.0  

3.6  

2.083  

4.340  

10.945  

-6.892  

12.0  

4.0  

3.083  

9.507  

8.458  

-8.967  

8.0  

5.0  

-0.917  

-1.908  

3.642  

1.749  

7.0  

10.0  

-1.917  

3.092  

9.558  

-5.926  

8.0  

7.6  

-0.917  

0.692  

0.840  

-0.634  

11.0  

8.0  

2.083  

1.092  

4.340  

2.274  

10.0  

8.0  

1.083  

1.092  

1.174  

1.192  

12.0  

6.0  

3.083  

-0.908  

9.507  

0.825  

6.0  

8.6  

-2.917  

1.692  

8.507  

2.862  

-4.934  

6.0  

8.0  

-2.917  

1.092  

8.507  

1.192  

-3.184  

107.000  

82.900  

=

=

sx

=

sy

=

r

=

(b-2)

Determine the coefficient of determination. (Round your answer to 3 decimal places.)

(c)

Interpret these statistical measures. Does it surprise you that the relationship is inverse? (Round your answer to nearest whole number.)

   correlation between age of car and selling price. So,  percent of the variation in the selling price is explained by the variation in the age of the car.

6) The following hypotheses are given.

H0: ρ ≤ 0

H1: ρ > 0

A random sample of 12 paired observations indicated a correlation of .32.

(1)

State the decision rule for .05 significance level. (Round your answer to 3 decimal places.)

  Reject Ho if t >

(2)

Compute the value of the test statistic. (Round your answer to 2 decimal places.)

  Value of the test statistic

(3)

Can we conclude that the correlation in the population is greater than zero? Use the .05 significance level.

   Ho. We conclude that the correlation in the population is    greater than zero.

7) The following hypotheses are given.

A random sample of 15 paired observations have a correlation of −.46. Can we conclude that the correlation in the population is less than zero? Use the .05 significance level. (Negative amount should be indicated by a minus sign. Round your answer to 3 decimal places.)

  Reject H0 if t < −1.771

t

=

   Ho. 

8) The following sample observations were randomly selected.

Click here for the Excel Data File

X:

 5

 3

6

 3

 4

 4

6

8

Y:

13

15

7

12

13

11

9

5

(a)

Determine the regression equation. (Negative amounts should be indicated by a minus sign. Round your answers to 3 decimal places.)

X

Y

( )2

( )2

( )( )

5  

13   

2.375  

5.641  

3  

15   

−1.875  

3.516  

−8.203    

6  

7   

13.141  

−4.078    

3  

12   

−1.875  

1.375  

4  

13   

−0.875  

0.766  

−2.078    

4  

11   

0.375  

0.141  

6  

9   

1.125  

−1.625  

8  

5   

31.641  

−17.578    

=

=

  sx

=

sy

=

r

=

b

=

a

=

  Y' = + X

(b)

Determine the value of when X is 7. (Round your answer to 3 decimal places.)

9) Bradford Electric Illuminating Company is studying the relationship between kilowatt-hours (thousands) used and the number of rooms in a private single-family residence. A random sample of 10 homes yielded the following.

Number of Rooms

Kilowatt-Hours (thousands)

Number of Rooms

Kilowatt-Hours (thousands)

12

9

8

6

9

7

10

8

14

10

10

10

6

5

5

4

10

8

7

7

Click here for the Excel Data File

(a)

Determine the regression equation. (Round your answers to 3 decimal places.)

    =    +   X

(b)

Determine the number of kilowatt-hours, in thousands, for a six-room house. (Round your answer to 3 decimal places.)

 Number of kilowatt-hours

10) Mr. James McWhinney, president of Daniel-James Financial Services, believes there is a relationship between the number of client contacts and the dollar amount of sales. To document this assertion, Mr. McWhinney gathered the following sample information. The X column indicates the number of client contacts last month, and the Y column shows the value of sales ($ thousands) last month for each client sampled.

Number of Contacts, X

Sales ($ thousands), Y

Number of Contacts, X

Sales ($ thousands), Y

14  

24  

23  

30  

12  

14  

48  

90  

20  

28  

50  

85  

16  

30  

55  

120  

46  

80  

50  

110  

Click here for the Excel Data File

(a)

Determine the regression equation. (Negative amounts should be indicated by a minus sign.Do not round intermediate calculations. Round final answers to 2 decimal places.)

X

Y

( ) 2

( )2

( )( )

14  

376.36  

1376.41  

719.74  

12  

14  

−21.4  

−47.1  

20  

−13.4  

179.56  

443.54  

16  

30  

−31.1  

967.21  

46  

12.6  

357.21  

23  

−10.4  

967.21  

48  

90  

28.9  

213.16  

421.94  

50  

85  

23.9  

275.56  

396.74  

55  

466.56  

3469.21  

1,272.24  

50  

110.0  

16.6  

48.9  

=

=

  sx

=

sy

=

r

=

   b =

   a  =

  Y' = + X

(b)

Determine the estimated sales if 40 contacts are made.(Do not round intermediate calculations. Round final answers to 2 decimal places.)