# Patterned residuals

HWK 6

1) Analyze the residual plot below and identify which, if any of the conditions for an adequate linear model is not met.

2

0

-2 5 15 25

(I cannot make the plots but they are below and above the middle line. There is one dot in conjunction of -2 and 5; one dot on the line of 15 and one on the above 25 located high of both previous. All other plots were everywhere that is how I can explain it)

Which of the conditions for an adequate linear model is not met?

A-Patterned residuals

B-None

C-Constant error variance

D-Outlier

2) Consider the data set given in the accompanying table. Complete parts (a through g)

X -2 -1 0 1 2

Y -2 0 2 3 5

a) Draw a scatter diagram treating x as the explanatory variable and y as the response variable. Choose the correct graph below.(please just make the correct draw and I will figure it out)

b) Fine the equation of the line containing the points (-2,-2) and (2,5). The equation of the line is y = (…….)x + (……).

c) Graph the line found in part (b) on the scatter diagram. Choose the correct graph below (Please make the right graph then I will figure it out).

d) Determine the least-squares regression line. Choose the correct answer.

A-the least-squares regression line is y = 1.5x + 1.75.

B-the least-squares regression line is y =1.7x + 1.75.

C-the least-squares regression line is y =1.7x + 1.6.

D-the least-squares regression line is y =1.6x + 1.7.

e) Graph the least-squares regression line on the scatter diagram. Choose the correct graph below.(Please graph the least squares line and I will figure it out).

f) Compute the sum of the squared residuals for the line found in part (b). The sum of the squared residuals for the line founds in the part (b) is (…..)

g) Compute the sum of the squared residuals for the least-squares regression line found in part (d). The sum of the squared residuals for least squares regression line is (……).

3) For the data set shown below, do the following. (a) Compute the standard error, the point estimate for σ. (b) Assuming the residuals are normally distributed, determine Sb1. (c) Assuming the residuals are normally distributed, test H0: β1=0 verses H1: β1≠0 at he a=0.05 level of significance.

X 20 30 40 50 60

Y 100 93 91 85 72

a) Determine the point estimate for σ. Se = (…….) (do not round until the final answer then round to four decimal places as needed).

b) Find the sample standard error of b1. Sb1 = (……)(Use the answer from part (a) to find this answer. Round to four decimal places as needed.)

c) Which of the following conclusions is correct?

A-Do not reject H0 and conclude that a linear relation exists between x and y.

B-Do not reject H0 and conclude that a linear relation does not exist between x and y.

C-Reject H0 and conclude that a linear relation does not exist between x and y.

D-Reject H0 and conclude that a linear relation exists between x and y.

4) Match the linear correlation coefficient to the scatter diagram. The scales on the x- and y- axis are the same for each scatter diagram.

(a) r= -0.969, (b) r= -1, (c) r= -0.049.

R

R

Reponse

I Exp II Exp III Explanatory

few dots between the 2 lines; many dots over the place in 2 lines; dots on the straight line.

A-Scatter diagram ( Ii/ I/ III)

B-Scatter diagram (II/ I/ III)

C-Scatter diagram (I/III/II)

5) A scatter diagram is shown to the right with one of the points drawn in blue. In addition, two least squares regression lines are drawn. The line drawn in red is the least squares regression line with the point in blue excluded. The line drawn in blue is the least squares regression line with the point in blue included. On the basis of these graphs, do you think the point in the blue is influential?

20 red line

response

*15*

10 blue line

5 10 15 Explanatory

(There are 5 dots above the blue and red line and 5 below red and blue line and three on the red and blue)

Does the point in the blue seem to be influential?

A-No

B-Yes

6) Because colas tend to replace healthier beverages and colas contain caffeine and phosphoric acid, researcher wanted to know weather consumption of colas is associated with lower bone mineral density in women. The data shown in the accompanying table represent the typical number of cans of soda consumed in a week and the bone mineral density of the femoral neck for a sample of 15 women. The data were collected through a prospective cohort study. Complete parts (a) through (f).

Number of Bone mineral

Colas per week density (g/cm2)

0 0.897

0 0.884

1 0.891

1 0.877

2 0.888

2 0.871

3 0.868

3 0.876

4 0.873

5 0.875

5 0.871

6 0.867

7 0.862

7 0.872

8 0.865

a) Find the leas squares regression line treating cola consumption per week as the explanatory variable. Choose the correct answer below.

A-The least squares regression line is ŷ= -0.0030x + 0.8866.

B- The least squares regression line is ŷ= 0.8866x – 0.0030.

C- The least squares regression line is ŷ= 0.0030x – 0.8866.

D- The least squares regression line is ŷ= -0.8866x + 0.0030.

b) Interpret the slope.

For each additional cola consumed per week, bone mineral density will (increase/decrease) by (0.0030; 1.0053; 0.5423; 0.8866) g/cm2, on average.

c) Interpret the intercept. Choose the correct answer below.

A-For each additional cola consumed per week, bone mineral density will decrease by 0.0030g/cm2, on average.

B-For a woman who does not drink cola, bone mineral density will be 0.030 g/cm2.

C-it is not appropriate to interpret the y-intercept. It is outside the scope of the model.

D-For a woman who does not drink cola, bone mineral density will be 0.8866 g/cm2.

d) Predict the bone mineral density of the femoral neck of a woman who consumes four cola per week.

The Predict value of the bone mineral density of the femoral neck of this woman is (……) g/cm2. (Round to four decimal places as needed)

e) The researcher found a woman who consumes of four colas per week to have a bone mineral density of 0.873 g/cm2. Is this woman’s bone mineral density above or below average among all women who consume four cola per week?

A-Above average

B-Below average

f) Would you recommend using the model found in part (a) to predict the bone mineral density of a woman who consumes two cans of cola per day?

A- No

B- Yes

7) The time it takes for a planet to complete its orbit around a particular star is called a planet’s sidereal year. The sidereal year of the planet is related to the distance the planet is from the star. The companying data show the distances of the planets from a particular star and there sidereal years. Complete part (a) through (e).

Planet Distance from the star Sidereal year y,

X, millions of miles

Planet1 36 0.24

Planet2 67 0.62

Planet3 93 1.00

Planet4 142 1.86

Planet5 483 11.8

Planet6 887 29.3

Planet7 1785 82.0

Planet8 2797 163.0

Planet9 3675 248.0

a) Draw a scatter data of the diagram treating distance from the star as the explanatory variable.

b) Determine the correlation between distance and sidereal year. The correlation between distance and sidereal year is (…..). (Round to three decimal places as needed).

-Does this imply a linear relation between distance and sidereal year?

A-No

B-Yes

c) Compute the least squares regression line. Choose the correct answer below.

A- y= -0.0654x + 12.6256

B- y= 0.0654x + 12.6256

C- y= 0.0654x – 12.6256

D- y= -0.0654x + 132.1412

d) Plot the residuals against the distance from the star.

e) Do you think the least squares regression is a good model?

A-No

B-Yes

8) For the following data set (a) Draw a scatter diagram, (b) by hands compute the correlation coefficient, and (c) Comment on the type of relation that appears to exist between x and y.

X 1 4 8 8 9

Y 1.3 1.8 2.3 2.2 2.6

(b) r= (…..) (Round to four decimal place as needed)

(c) What type of relation appears to exist between x and y?

A- There appears to be a positive linear association

B- There is a little or no evidence of a linear association

C- There appears to be a negative linear association.

9) The data in the accompanying table represent the rate of return of a certain company stock for 11 months, compare with the rate of return of a certain index of 500 stocks. Both are in percent. Complete parts (a) through (b)

Month Rate of return of the index, x Rate of return of company stock, y

Apr 07 4.23 3.38

May07 3.25 5.09

Jun 07 -1.78 0.54

Jul 07 -3.20 2.88

Aug07 1.29 2.69

Sep 07 3.58 7.41

Oct 07 1.48 -4.83

Nov 07 -4.40 -2.38

Dec 07 -0.86 2.37

Jan 08 -6.12 -4.27

Feb 08 -3.48 -3.77

a) Treating the rate of return of the index as the explanatory variable, x, determine the estimates of β0 and β1.

The estimate of β0 is (…..)(Round to four decimal places as needed)

The estimate of β1 is (…..)(Round to four decimal places as needed)

b) Compute the standard error of the estimate. The standard error of the estimate is (….)( Round to four decimal places as needed).

c) Determine whether the residuals are normally distributed. Choose the correct answer below

A-The residuals are normally distributed

B-The residuals are not normally distributed

d) If the residuals are normally distributed, determine Sb1. Choose the correct answer below.

A-Sb1=0.2649

B-Sb1= 0.2884

C-Sb1= 0.4115

D-The residuals are not normally distributed.

e) If the residuals are normally distributed, test weather a linear relation exists between the rate of return of the index, x, and the rate of return for the company stock, y at the a= 0.1level of significant. State the appropriate conclusion. Choose the correct answer below.

A-There is a sufficient evidence to conclude that a linear relation exist between the rate of return of the index and the rate of return of the company.

B-The residuals are not normally distributed.

C-There is not sufficient evidence to conclude that a linear relation exist between the rate of the index and the rate of return of the company.

f) If the residuals are normally distributed, construct a 90% confidence interval for the slop of the true last squares regression line. Choose the correct answer below.

A-The residuals are not normally distributed

B-The 90% confidence interval is (0.2072, 1.2986)

C-The 90% confidence interval is (0.2072, 1.4546)

D-The 90% confidence interval is (0.2412, 1.2986)

g) What is the mean rate of return for the company stock if the rate of return of the index is 3.15%?

The mean rate of return for the company stock if the rate of return of the index is 3.15% is (…..) (Round to three decimal places as needed)

10) Supposed that the response variable y is related to the explanatory variable x1 and x2 by the regression equation parts (a) through (f)

Y= 2+0.5×1 – 1.4×2

(a) Construct a graph showing the relationship between the expected value of y and x1 for x2 = 10, 20, and 30.

(b) Construct a graph showing the relationship between the expected value of y and x2 for x1= 40, 50, and 60.

(c) How can we tell from the graphs alone that there is on interaction between x1 and x2?

A-The lines are not parallel.

B-The line are parallel

C-The slopes of the lines are positive

D-The slops of the line are negative

(d) Construct a graph showing the relationship between the expected value y and x1 for x2 = 10, 20, and 30 with the interaction term 0.05x1x2 added to the regression equation.

(e) Construct a graph showing a relationship between the expected value of y and x2 for x1 = 40, 50, and 60 with the interaction term 0.05x1x2 added to the regression equation.

(f) How do the graphs from parts (d) and (e) differ from the graphs in parts (a) and (b)?

A-The lines are now parallel in the graphs from parts (d) and (e)

B-The y intercepts of the lines are now equal in the graphs from parts (d) and (e)

C-The slopes of the lines are now negative in the graphs from parts (d) and (e)

D-The lines are no longer parallel in the graphs from parts (d) and (e)

11) Researchers initiated a long term study of population America black bears. One aspect of the study was to develop a model that could be used to predict a bear’s weight (since it is not practical to weight bears in the field). One variable thought to be related to weigh is the length of the bear. The accompanying data represent the lengths and weights of 12 American black bears. Complete part (a) through (d)

Table length cm Weight (kg)

138.0 110

138.0 60

130.0 90

120.5 60

149.0 95

141.0 105

141.0 110

150.0 85

166.0 155

151.5 140

129.5 105

150.0 110

(a) Which variable is explanatory variable based on the goals of the research?

A-The number of bears

B-The weight of the bear

C-The length of the bear

(b) Draw a scatter diagram of the data.

(c) Determine the linear correlation coefficient between weight and height

The linear correlation coefficient between weight and height is r= (….) (Round to three decimal places as needed)

(d) Does a linear relation exist between the weight of the bear and its height?

A- No, there is no linear relation between the weight of the bear and it height.

B- Yes, there is a negative linear relation between the weight of the bear and its height

C- Yes, there is a positive linear relation between the weight of the bear and its height

12) The multiple regression equation y = – 5 – x1 + 8×2 is obtained from a set of sample data. Complete parts (a) through (e).

(a) Interpret the slope coefficient for x1. Choose the correct answer below

A-The slope coefficient of x1 is -1. This indicates that y will decrease 1 unit for everyone unit increase in x1, x2 remain constant.

B- The slope coefficient of x1 is -5. This indicates that y will decrease 5 unit for everyone unit increase in x1, x2 remain constant.

C- The slope coefficient of x1 is -1. This indicates that y will increase 1 unit for everyone unit increase in x1, x2 remain constant.

Interpret the slope coefficient for x2. Choose the correct answer below.

A- The slope coefficient of x2 is 8. This indicates that y will increase 5 units, for everyone unit increase in x2, x1 remains constant.

B- The slope coefficient of x1 is -5. This indicates that y will increase 1/8 units, for everyone unit increase in x1, x2 remains constant.

C- The slope coefficient of x2 is 8. This indicates that y will increase 8 units, for everyone unit increase in x2, x1 remains constant.

(b) Determine the regression equation with x1 = 10. Choose the correct answer below.

A- y= -5 + 8×2

B- y= -15 + 8×2

C-y= -5 +8×1

D-y= 30 + 8×2

Graph the regression equation with x1 = 10

(c) Determine the regression equation with x1 = 15

A- y= -20 + 8×2

B- y= -5 + 8×1

C-y= 32 + 8×2

D-y= -5 + 8×2

Graph the regression equation with x1 = 15.

(d) Determine the regression equation with x1 = 20

A-y= -5 + 8×2

B-y= 45+ 8×2

C-y= -5 + 8×1

D-y= -25 + 8×2

Graph the regression equation with x1 = 20.

(e) What is the effect of changing the value x1 on the graph of the regression equation?

A- No changes

B- Changes in the slope

C-Changes in the y interception