Statistics

Statistics

RealEstate

720 179,000

937

1050

1057

274,900

,900

1230

799,900

,900

,900

425,000

299,900

1600

1600

,900

499,900

,000

,900

464,900

,000

,000

995,000

,000

5200

Total Area (sq.ft.) List Price ($)
500 184,900
515 164,900
518 167,510
550 289,900
609 239,000
612 192,779
713 244,900
720 179,000
742 237,500
749 241,745
780 335,000
781 425,000
800 339,900
831 269,000
840 189,000
880 299,000
937 332,000
380,900
1000 449,900
1010 220,000
1014 227,000
1032 429,900
1038 469,900
1050 214,900
229,000
1057 292,000
585,000
1085 499,900
1092 450,500
1102 387,900
1104 274,900
1105
1117 368,500
1130 219,500
1150 249,999
1154 430,000
1155 354,200
1190 349,900
1200 295,000
1230 509
575,000
1293 249,000
1296 699,900
1304 451,900
1310 404,900
1313 459,000
1316 440,500
1318 799,900
1342 449,000
1346 1,300,000
1350
1352 639
1370 539
1379 299,900
1380 479,900
1390 448,900
1400
1

428 222,400
1524 495,000
1560
1600 799,000
515,000
324,900
1627 599
1670
1686 392,900
1692 379,200
1

726 489,900
1840 699,000
1911 1,575,000
1952 535,000
2100 685
2180 649
2181 464,900
2188 477,900
2191
2261 484
2273 859,900
2274 495,500
2347 995,000
2440 759,000
2472 1,425,000
2520 379,900
2800 795,000
3085 499,000
3218 1,399,000
3220 709
3307 999,999
3

580
3

617 1,579,

750
4000 999,900
4357 1,180,000
4724 1,395,000
4860 975,000
5025 2,

490
5200 1,599,000
1,895,000

Cereal

509

539

428

617

568

607

754

566 539
639

607

649

580

679

593
685
800
663
596
637
679
599
566
484
739

663

462
646
547
693
726
490
580

624
566

709
518
573

596

750
582
Consumers Nonconsumers
568 705
498 819
589 706
681
540 613
646 582
636 601
739 608
787
596 573
607
529 754
637 741
628
633 537
555 748
565 663
526
584 541
462
530 719
566
687 740
694 688
714 725
711
693
556 816
473 426
593 773
551 480
683 632
667 569
547
647 710
679
532 674
467 505
622 527
650
629 830
651 602
765
723
730
701
672
369
758
553
620
717
642
563
733
664
625
655
466
603
588
476
421
812
643
624
549
645
794
514
554
623
747
583
536
833
644
594
788

Instructions and Advice:

· This assignment consists of six questions. They each have lots of parts but most of them are very short!

· Data for Questions 3 and 6 are in the companion Excel spreadsheet .

· Present the parts of your answers in the same order as the questions are asked.

· Do not include any original data in your printed submission.

· Maintain all precision in your calculator or in Excel as you do your multi-step computations. Round off to fewer decimal places only when you write your work and the final answer down to hand in.

· When formatting numbers in Excel, display only as many decimal places as provide decision-making value to the reader.

Question 1 – Interpreting or Misinterpreting Correlation

a) Various factors are associated with the gross domestic product (GDP) of nations. State whether each of the following statements is reasonable or not. If not, explain the blunder.

(i) A correlation of –0.722 shows that there is almost no association between GDP and Infant Mortality Rate.

(ii) There is a correlation of 0.44 between GDP and Continent.

(iii) There is a very strong correlation of 1.22 between Life Expectancy and GDP.

(iv) The correlation between Literacy Rate and GDP was 0.83. This shows that countries wanting to increase their standard of living should invest heavily in education.

b) An article in a business magazine reported that Internet E-commerce has doubled nearly every three years. It then stated that there was a high correlation between sales made on the Internet and year. Do you think this is an appropriate summary? Explain in one sentence.

c) Simpson’s Paradox can occur in regression, when a relationship between variables within groups of observations is reversed if all the data are combined. Here is an example.

Group

X

Y

Group

X

Y

1

1

10.1

2

6

18.3

1

2

8.9

2

7

17.1

1

3

8.9

2

8

16.2

1

4

6.9

2

9

15.1

1

5

6.1

2

10

14.3

(i) Make a scatterplot of the data for Group 1 and add the least squares line. Describe the relationship between Y and X for Group 1. Find the correlation (using Excel).

(ii) Do the same for Group 2.

(iii) Make a scatterplot using all 10 observations and add the least squares line. Find the correlation (using Excel).

(iv) Summarize your findings in one or two sentences.

d) Since 1980, average mortgage interest rates in the U.S. have fluctuated from a low of under 6% to a high of over 14%. Is there a relationship between the amount of money people borrow and the interest rate that’s offered? Here is a scatterplot of Total Mortgages in the U.S. (in millions of 2005 dollars) vs. Interest Rates at various times over the past 26 years. The correlation is -0.84.

(i) Describe the relationship between Total Mortgages and Interest Rate.

(ii) If we standardized both variables, what would the correlation coefficient between the standardized variables be?

(iii) If we were to measure Total Mortgages in thousands of dollars instead of millions of dollars, how would the correlation coefficient change?

(iv) Suppose in another year, interest rates were 11%, and mortgages totalled $250 million. How would including that year with these data affect the correlation coefficient?

(v) Do these data provide proof that if mortgage rates are lowered, people will take out more mortgages? Explain.

Question 2 – Regression and the Market Model (Calculations from Summary Statistics)

It is usual in finance to describe the returns from investing in a single stock by regressing the stock’s returns of the returns from the stock market as a whole. This helps us see how closely the stock follows the market. We analyzed the monthly percent total return y on Research in Motion (RIM), now called BlackBerry, stock and the monthly return x on the NASDAQ index, which represents the market, for the period between January 2005 and December 2009. Here are the results.

A scatterplot shows no very influential observations.

a) Find the equation of the least-squares line. What percent of the variation in RIM stock is explained by the linear relationship with the market as a whole?

b) Interpret what the slope and the y-intercept of the regression line indicate. The slope is called “beta” in investment theory.

c) Based on the statistics above, how effective do you think the monthly return on the NASDAQ index would be in predicting the monthly percent total return on RIM stock? Explain.

d) Returns on most individual stocks have a positive correlation with returns on the entire market. Explain why an investor should prefer stocks with beta > 1 when the market is rising and stocks with beta < 1 when the market is falling.

Question 3 – Residual Plots – Halifax Real Estate Listings (yes, again)

In Assignment 2, Question 3, you provided scatterplots and regression calculations. One more step in the analysis is a residual plot. For your convenience, I have extracted the data needed from the original spreadsheet. The worksheet RealEstate has the List Price and Total Area data for all 98 listings of properties, sorted from lowest to highest total area. And, to make your life a little easier, here is the regression equation that you should have computed in Assignment 2: Price = 42424.75 + 307.06Area.

Please use this equation here.

a) Compute the residuals, and construct a residual plot. Your solution should only show the plot; do not include the listing of the 98 residuals!

b) Does the plot show that the linear regression model is appropriate here? Explain in one sentence.

c) Compute the standard deviation of the residuals (se).

Question 4 – Project Management and Random Variables

PERT (Project Evaluation and Review Technique) and CPM (Critical Path Method) are related management science techniques that help operations managers control the activities and amount of time it takes to complete a project. The longest path from starting point to completion is called the critical path because any delay along this path will result in a project delay.

The operations manager of a large plant wishes to overhaul a machine. His critical path has five activities. The mean (i.e. expected value) and the variance of completion time for each activity is listed below. Assume the activities are independent of one another.

Activity

Mean (mins.)

Variance

1. Disassemble machine

35

10

2. Determine parts that need replacing

20

6

3. Find needed parts in inventory

20

4

4. Reassemble machine

50

13

5. Test machine

20

3

a) What are the mean, variance and standard deviation of the project total completion time?

b) Assuming that the total completion time is approximately normally distributed:

i) Find the probability that the project will take more than 165 minutes to complete.

ii) Find the probability that the project will take less than 141 minutes to complete.

iii) Find the probability that the project will take between 141 and 165 minutes to complete.

iv) If this project were repeated many times, what total completion time would be exceeded by at most 5% of such projects. Report your answer rounded to the nearest whole minute.

Question 5 – Sampling Distribution of Proportions

A university bookstore claims that 50% of its customers are satisfied with the service and prices.

a) Suppose a simple random sample of 600 customers is taken, and we assume that the bookstore’s claim is true (i.e. that the true proportion is indeed 0.50). What is general shape of the sampling distribution of the sample proportion of customers who are satisfied? Why can assume that? Then give the mean and standard deviation of the sample proportion .

b) Again assuming that the bookstore’s claim is true (i.e. that the true proportion is indeed 0.50), what is the probability that in simple random sample of 600 customers less than 45% are satisfied? Use the mean and standard deviation you computed in part a).

c) Suppose that in a random sample of 600 customers, 270 express satisfaction with the bookstore. What does this tell you about the bookstore’s claim? Hint: Refer to part b).

d) Repeat parts a) and b) but now with a simple random sample of 1200.

Question 6 – Sampling Distribution of Means

A scientist claims that people who eat high-fibre cereal for breakfast will consume, on average, fewer calories for lunch than people who don’t eat high-fibre cereal for breakfast. If this is true, high-fibre cereal manufacturers will be able to claim potential weight reduction for dieters as an advantage of eating their product. As a preliminary test of the claim, 150 people were randomly selected and asked what they regularly eat for breakfast and lunch. Each person was identified as either as a Consumer or a Non-consumer of high-fibre cereal, and the number of calories eaten at lunch was measured and recorded. The data are provided in the accompanying spreadsheet under the tab Cereal.

a) For the 43 Consumers, compute the mean and standard deviation of the number of lunchtime calories.

b) Assume that the mean and standard deviation computed in (a) are true for the general population of Consumers. What is the general shape of the sampling distribution of the sample mean? What are the mean and standard deviation of the sample mean?

c) What is the probability that the sample mean number of lunchtime calories of Consumers will be less than 600?

d) Repeat parts a) and b) for the 107 Non-consumers.

e) This part is harder: What is the probability that the mean number of lunchtime calories of Non-consumers exceeds the mean number of lunchtime calories of Consumers? Assume that the means and standard deviations computed for the samples are true of the general population.

Hints: Let X = Non-consumer calories and Y = Consumer calories. Then find Pr( > 0). You will need the sampling distribution of the difference of means and the mean and standard deviation of the difference; then you’re on your way.

*** END ***

Total Mortgages 12.5 14.4 14.7 12.3 12 11.2 9.8000000000000007 8.9 9 9.8000000000000007 9.7000000000000011 9.1 7.8 6.9 7.3 7.7 7.6 7.5 7 7.1 7.9 6.9 6.4 5.7 5.7 5.9 122.7 116.4 112.4 117.5 121.2 127.4 141.30000000000001 153.19999999999999 160.80000000000001 164.6 155.4 152.4 151.30000000000001 144.6 144.80000000000001 141.5 147.80000000000001 154.1 157.9 163.30000000000001 168.2 171.7 177.4 178.2 191.8 210.8

Interest Rate (%)

Total Mortgage (millions of $)

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