Question 1 the Irs

QUESTION 1 – THE IRS – CALCULATING AND INTREPERTING

The IRS would like to determine if individuals overstated charitable contributions on their tax returns
The IRS supplied data listing the AGI and charitable contributions of 11 audited taxpayers
1. Does there appear to be a linear relationship between the variables?
2. Develop a linear regression model used to predict charitable contributions from the AGI
3. What is the estimated regression equation?
4. Interpret the regression coefficient and t- test
5. How might the IRS use this model to identify returns with unusually high charitable contributions
AGI / Charitable
(in $1,000s) / Contributions
$55 / $4,200
$58 / $4,800
$63 / $6,329
$67 / $8,017
$74 / $7,400
$78 / $8,600
$83 / $12,290
$88 / $10,406
$92 / $11,820
$98 / $12,090
$105 / $14,675

QUESTION 2 – HOME HEATING – CALCULATING AND INTREPERTING

The attached data depicts the average heating cost of a home along with other variables. Often, prospective home owners request an estimate of the average heating cost .

1. Prepare scatter plots between average heating cost and each independent variable. What relationship does each plot suggest?

2. If the company wanted to develop a regression model using only one independent variable, which one would it choose, Why?

3. If the company wanted to develop a regression model using two independent variables which two would it choose, Why?

4. If the company wanted to develop a regression model using three independent variables which three would it choose, Why?

5. Calculate the regression equation using all four independent variables

6. At 95% confidence level, “What would be the average heating cost for a house with 4” insulation, a 5-year old furnace, 2,500 sq ft. and with an average outside winter temperature of 40 degrees?

Home / Avg Outside Temp (X1) / Attic Insulation (inches) (X2) / Age of Furnace (yrs) (X3) / Square Footage (X4) / Avg Heating Cost
1 / 29 / 5 / 4 / 1900 / 198
2 / 8 / 6 / 7 / 2800 / 355
3 / 6 / 10 / 9 / 2500 / 291
4 / 22 / 8 / 11 / 2000 / 230
5 / 55 / 2 / 4 / 1300 / 121
6 / 36 / 2 / 5 / 2100 / 250
7 / 28 / 4 / 9 / 2400 / 360
8 / 36 / 7 / 2 / 2300 / 164
9 / 59 / 5 / 9 / 1300 / 42
10 / 64 / 5 / 6 / 1500 / 90
11 / 19 / 4 / 8 / 2300 / 271
12 / 57 / 5 / 3 / 1400 / 96
13 / 39 / 7 / 11 / 1900 / 187
14 / 25 / 9 / 8 / 2100 / 235
15 / 28 / 6 / 4 / 1800 / 138
16 / 53 / 11 / 2 / 1200 / 71
17 / 47 / 5 / 2 / 2000 / 206
18 / 20 / 4 / 14 / 2900 / 398
19 / 39 / 4 / 6 / 2600 / 319
20 / 60 / 8 / 6 / 1500 / 72

QUESTION 3 INTREPETING A REGRESSION MODEL

A company has built a regression model to predict the number of labor hours (Yi) required to process a batch of parts (Xi). It has developed the following Excel spreadsheet of the results.

A / B / C / D / E / F / G
1 / Regression Statistics
2 / Multiple R / 0.9970
3 / R Square / 0.9941
4 / Adjusted R Square / 0.9933
5 / Standard Error / 0.3679
6 / Observations / 10
7
8 / ANOVA
9 / df / SS / MS / F / Significance F
10 / Regression / 1 / 181.5971 / 181.5971 / 1341.5500 / 0.0000
11 / Residual / 8 / 1.0829 / 0.1354
12 / Total / 9 / 182.6800
13
14 / Coefficients / Standard Error / t Stat / P-value / Lower 95% / Upper 95%
15 / Intercept / 4.8400 / 0.2513 / 19.2571 / 0.0000 / 4.2604 / 5.4196
16 / X Variable 1 / 1.4836 / 0.0405 / 36.6272 / 0.0000 / 1.3902 / 1.5770

What is the estimated regression function for this problem? Explain what the terms in your equation mean.

Predict the mean number of labor hours for a batch of 5 parts.

Provide a rough 95% confidence interval on the number of labor hours for a bath of 5 parts.

Test the significance of the model and explain which values you used to reach your conclusions.

Interpret the meaning of the "Lower 95%" and "Upper 95%" terms in cells F16:G16 of the spreadsheet.

Interpret the meaning of R Square in cell B3 of the spreadsheet.

QUESTION 4 CUSTOMER SERVICE QUEUING

A company has recorded the following customer interarrival times and service times for 10 customers at one of its single teller service lines. Assume the data are exponentially distributed and the 10 data points represent a reasonable sample.

All time in minutes
Customer / Interarrival / Service
1 / 11.08 / 2.20
2 / 2.50 / 2.50
3 / 6.00 / 1.10
4 / 5.75 / 14.50
5 / 8.50 / 2.00
6 / 4.15 / 2.70
7 / 15.50 / 5.00
8 / 13.00 / 8.50
9 / 10.50 / 5.00
10 / 6.00 / 1.50

What is the mean arrival rate per hour?

What is the mean service rate per hour?

What is the average time a customer spends in the service line?

What is the average number of customers in the service line?

QUESTION 5 SIMULATION

What is the expected number of phone calls per hour based on the following distribution on the number of phone calls per hour?

A / B / C / D
1 / # of phone calls / P(# of phone calls) / # of phone calls / P(# of phone calls)
2 / 1 / 0.10 / 1 / 0.10
3 / 2 / 0.40 / 2 / 0.50
4 / 3 / 0.30 / 3 / 0.80
5 / 4 / 0.15 / 4 / 0.95
6 / 5 / 0.05 / 5 / 1.00

What is the probability that 3 or more phone calls are received in any hour of operation?