MTH 305 Midterm Exam Complete Solution

Question Details: #1149
MTH 305 Midterm Exam Complete Solution

Assignment 3

Possible Total Points: 60 pts

Instructions:

  • Turn in your answers in paper-form (hand-written or typed) by the beginning of class.
  • Include complete sentences and show all calculations.
  • Each question is equally weighted.
  • Partial credit is possible even if an answer is incomplete.

 

EXERCISE #1.          Ch. 13 (section 13.2): Problem #13.1 (p. 523)

  1. Fitting a straight line to set a data yields the ff prediction line
    Yi=2+5Xi
  1. Interpret the meaning of the Y intercept b0.
  2. Interpret the meaning of the slope b1
  3. Predict the mean value of Y for X=3

 

EXERCISE #2.          Ch. 13 (section 13.2): Problem #13.9 all except part (a) (p. 525)

An  agent for residential real estate company in a large  city would like  to be able to predict the monthly rental cost for apartments based on the  size of the apartment as defined by square footage. A sample of 25 apartment RENT in a particular residential neighborhood was selected, and the information gathered revealed the following

  Apartment    Monthly Rent ($)    Size ( Square Feet)    Apartment       Monthly Rent          Size

         1                    950                             850                            14                1,800                  1,369

         2                   1,600                        1,450                            15               1,400                  1,175

         3                  1,200                         1,085                            16                1,450                 1,225

        4                    1,500                       1,232                             17                1,100                 1,245

        5                      950                         718                               18                 1,700              1,259

     6                 1,700                       1,485                                  19               1,200                1,150

     7                  1,650                      1,136                                  20               1,150                  896

     8                   935                        726                                     21               1,600                1,361

     9                   875                        700                                     22               1,650                1,040

     10                 1,150                     956                                     23               1,200                  755

     11                1,400                    1,100                                    24                 800                 1,000

     12                 1,650                  1,285                                     25                1,750               1,200

      13                2300                   1,985    

      14                1,800                   1,369

      15                1,400                   1,175

b)  Use the least –squares method to find the regression coefficients b0 and b1.

c) Interpret the meaning of b0 and b1 in this problem.

d) Predict the mean monthly rent for an apt. that has 1,000 square feet.

e)  Why would it not be appropriate to use the model to predict the monthly rent for apts. That have 500 square feet?

f) Your friend Jim and Jennifer are considering signing a lease for an apt. in this residential neighborhood. They are trying to decide between two apts, one with 1,000 square feet for a monthly rent of 1,275 and the other with 1,200 square feet for a monthly rent of 1,425. What would you recommend to them? Why?

 

 

For part (b): Below is the dataset you need to solve this problem in Excel. To get the intercept b0 and slope b1, you can use the “Regression Analysis” function in Excel or type these commands in any Excel cell:

=SLOPE(range of Y data, range of X data)

=INTERCEPT(range of Y data, range of X data)

Rent

Size

950

850

1600

1450

1200

1085

1500

1232

950

718

1700

1485

1650

1136

935

726

875

700

1150

956

1400

1100

1650

1285

2300

1985

1800

1369

1400

1175

1450

1225

1100

1245

1700

1259

1200

1150

1150

896

1600

1361

1650

1040

1200

755

800

1000

1750

1200

 

 

EXERCISE #3.          Ch. 13 (section 13.3): Problem #13.21 (p. 531)

In  problem 13.9 on page 255 an agent for real estate company wanted to predict the monthly rent for apts. Based on the size of the apt. Rent using of the apt, RENT using the results of that problem.

  1. Determine the coefficient of the determination r^2 and interpret its meaning.
  2. Determine the standard error of the estimate.
  3. How useful do you thing this regression model is  for predicting the monthly year?

 

For parts (a) and (b): To get the intercept r2 and the standard error you can use the output from the “Regression Analysis” function you got for Problem #13.19 above (see Exercise #2), or type these commands in any Excel cell:

=RSQ(range of Y data, range of X data)

=STEYX(range of Y data, range of X data)

 

EXERCISE #4.          Ch. 14 (section 14.6): Problem #14.38 (p. 600)

Suppose X1 is a numerical variable and X2 is a dummy variable and the following regression equation for a sample n= 20 is:   Y1 = 6+4 x 1i + 2X 2i

  1. Interpret the meaning of the slope for variable X1.
  2. Interpret the meaning of the slope for variable X2
  3. Suppose that the t statistic for testing the contribution of variable X2 is 3.27. At the 0.05 level of significance, is there evidence that variable X2 makes a significant contribution to the model?

 

EXERCISES  #5 & 6.            Ch 14: Problem #14.72 all part except (d) and (i) (p. 613)

The file AUTO2002 contains data on 121 automobile models from the year 2002. Among the variables included are the gasoline mileage ( in miles per gallon), the length ( in inches), and the weight (in pounds) of each automobile. Develop a model to predict the gasoline mileage based on the length and weight of each automobile.

  1. State the multiple regression equation.
  2. Interpret the meaning of the slopes in this equation.
  3. Predict the gasoline mileage for an automobile for an automobile that has the length of 195 inches and weight of 3,000 pounds.

e  )Is there a significant  relationship between gasoline mileage and the two independent variables (length and weight) at the 0.5 level of significance.

 f)Determine the p-value in (e) and interpret its meaning.

g) Interpret the meaning of the coefficient of multiple determination in this problem.

      h ) Determine the adjusted r2.

j)  Determine the p-value in (i) and interpret their meaning. ( (i)= At the 0.05 level of significance, determine whether each independent variable makes a significant contribution  to the regression model. Indicate the most appropriate regression model for this set of data).

k)  Construct a 95% confidence interval estimate of the population slope between gfasoline mileage and weight.

l) Compute and interpret the coefficients of partial determination.

The Excel output for this exercise is given below. Use this output to answer the questions.

 

 
   

 

SUMMARY OUTPUT

           
             

Regression Statistics

         

Multiple R

0.782187748

         

R Square

0.611817673

         

Adjusted R Square

0.60186428

         

Standard Error

2.952425134

         

Observations

121

         
             

ANOVA

           

 

                df

SS

MS

F

Significance F

 

Regression

3

1607.421998

535.8073326

61.46825227

6.20871E-24

 

Residual

117

1019.867258

8.716814173

     

Total

120

2627.289256

 

 

 

 
             

 

Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Intercept

42.43290086

8.218578926

5.163045977

1.00728E-06

26.15643651

58.70936521

Length

-0.00667189

0.036217633

-0.18421688

0.854162226

-0.07839902

0.065055222

Width

-0.03989444

0.182924039

-0.21809293

0.827736634

-0.40216590

0.322377022

Weight

-0.00487697

0.000600754

-8.11807648

5.3858E-13

-0.00606673

-0.00368720

 

 

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Solution Details: #1151
MTH 305 Midterm Exam Complete Solution

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