Problem 4: House Prices

Use the “Fairfax City Home Sales” dataset for parts of this problem.

a) Use StatCrunch to construct an appropriately titled and labeled relative frequency histogram of Fairfax home closing prices stored in the “Price” variable. Copy your histogram into your document.

b) What is the shape of this distribution? Answer this question in one complete sentence.

c) Assuming the population has a similar shape as the sample with population mean $510,000 and population standard deviation $145,000; calculate the probability that in a random sample of size 10, the mean of the sample will be greater than $600,000. You may assume a random sample was taken and the sample came from a big population. However, be sure to check the central limit theorem condition of a large sample size before completing this problem using one complete sentence. If this condition is not met, you cannot complete the problem.

d) Assuming the population has a similar shape as the sample with population mean $510,000 and population standard deviation $145,000; calculate the probability that in a random sample of size 36, the mean of the sample will be greater than $600,000. You may assume a random sample was taken and the sample came from a big population. However, be sure to check the central limit theorem condition of a large sample size before completing this problem using one complete sentence. If this condition is not met, you cannot complete the problem.

Data:

Price Year, Days, TLArea, Acres

369900   1922   44   1870   0.39

373000   1952   0   1242   0.27

375000   1952   8   932   0.15

375000   1950   2   768   0.19

379000   1952   31   816   0.21

380000   1941   53   1092   0.19

385000   1951   5   984   0.27

387700   1953   5   975   0.36

395000   1954   18   957   0.29

395000   1951   12   1105   0.22

399900   1954   29   1206   0.28

399900   1951   6   1226   0.18

400000   1954   31   957   0.27

410000   1949   6   1440   0.2

410000   1954   17   1344   0.23

412500   1954   4   1008   0.25

415000   1953   17   1371   0.28

420000   1954   2   957   0.25

426000   1952   3   1694   0.25

430000   1953   19   975   0.23

434900   1950   5   1128   0.18

435000   1954   32   1252   0.24

440000   1960   3   1161   0.26

440000   1954   2   1036   0.28

440000   1955   12   1645   0.28

440000   1960   5   1746   0.31

441000   1952   133   1062   0.23

442000   1961   4   1414   0.32

443000   1951   26   962   0.2

444900   1955   4   1122   0.19

446500   1953   3   962   0.26

450000   1952   2   1488   0.15

450000   1955   49   1122   0.23

450000   1979   0   1092   0.28

450000   1951   70   962   0.2

450000   1957   23   1300   0.51

451000   1947   12   1325   0.34

455000   1952   7   2267   0.81

455000   1962   4   1050   0.31

460000   1955   5   997   0.3

460000   1954   10   1125   0.17

465000   1954   77   1288   0.46

465900   1947   21   1309   0.19

469000   1963   153   1149   0.27

474000   1959   5   1319   0.32

475000   1955   4   1530   0.28

475000   1953   29   1008   0.2

475000   1955   6   1530   0.28

475000   1956   116   1345   0.5

475000   1956   1   1530   0.28

480000   1960   27   1236   0.27

480000   1959   133   1527   0.24

485000   1955   4   1008   0.24

485000   1956   74   977   0.24

488000   1960   11   1972   0.33

500000   1963   0   2145   0.25

500000   1953   14   1758   0.54

500500   1955   6   1630   0.28

510000   1959   5   1680   0.34

512000   1963   0   1968   0.22

519000   1961   1   1312   0.29

520000   1954   15   1492   0.25

520000   1958   80   1443   0.33

520000   1963   122   1822   0.32

530000   1962   6   1393   0.29

540000   1962   12   1414   0.25

543600   1962   4   1414   0.24

560000   1967   5   1530   0.28

560000   1961   16   1438   0.53

565000   1947   6   1510   0.25

565500   1967   5   1217   0.26

589000   1954   32   2368   0.3

593000   1954   9   2044   0.25

610000   1978   140   2091   0.09

655000   1976   180   2728   0.24

660000   1947   10   2635   0.22

665000   1950   37   2645   0.57

685000   1982   120   2752   0.09

795000   2002   259   3402   0.12

852000   2000   4   3215   0.11

895000   2000   63   3230   0.11

930000   2015   135   3175   0.15

940000   1860   42   3038   0.57

968500   1850   74   3630   0.34

1100000   2004   161   3640   0.19

Can someone please answer these 3 by Friday?

Problem 2 Let D, E, F, G, and H be events such that P(D) = 0.7, P(E) = 0.6, P(F) = 0.8, P(G) = 0.9, and P(H) = 0.5. Suppose that Dc, E, Fc, Gc, and H are independent.
(a) Find the probability that all of the events D, E, F, G, and H occur.
(b) Find the probability that at least one of the events D, E, F, G, and H occurs.

Problem 3 Five men and five women are ranked according to their scores on an examination. Assume that no two scores are alike, and all 10! possible rankings are equally likely. Let X denote the highest ranking achieved by a woman (for example, X = 2 if the top-ranked person was a man, and the next-ranked person was a woman). Find the probability mass function (pmf) of the random variable X, and plot the cumulative distribution function (cdf) of X.

Problem 4 Suppose there are three cards numbered 2, 7, 10, respectively. Suppose you are to be offered these cards in random order. When you are offered a card, you must immediately either accept it or reject it. If you accept a card, the process ends. If you reject a card, then the next card (if there is one) is offered. If you reject the first two cards, you have to accept the final card. You plan to reject the first card offered, and then to accept the next card if and only if its value is greater than the value of the first card. Let X be a number on the card you have accepted in the end. Find the pmf of X and plot the cdf of X.

compare and contrast the county data to your state as a whole. Present the county and state data in the same table to make comparisons easier. Finally, discuss the overall patterns you find in social inequalities and any suggestions for social change to address these issues in your community.   

Garcia Real Estate is involved in commercial real estate ventures throughout the United States. Some of these ventures are much riskier than other ventures because of market conditions in different regions of the country.

1) If Garcia does not risk-adjust its discount rate for specific ventures properly, which of the following is likely to occur over time? Check all that apply.

A. The firm will increase in value.

B. The firm’s overall risk level will increase.

C. The firm could potentially reject projects that provide a higher rate of return than the company should require.

2) How do managers typically deal with within-firm risk and beta risk when they are evaluating a potential project?

A. Subjectively

B. Quantitatively

Consider the case of another company. Turnkey Printing is evaluating two mutually exclusive projects. They both require a $5 million investment today and have expected NPVs of $1,000,000. Management conducted a full risk analysis of these two projects, and the results are shown below.

3) Which of the following statements about these projects’ risk is correct? Check all that apply.

A. Project B has more market risk than Project A.

B. Project B has more stand-alone risk than Project A.

C. Project A has more stand-alone risk than Project B.

D. Project A has more market risk than Project B.

Let z denote a random variable having a normal distribution with μ = 0 and σ = 1. Determine each of the probabilities below. (Round all answers to four decimal places.)

(a) P(z < 0.2) =  

(b) P(z < -0.2) =  

(c) P(0.40 < z < 0.86) =  

(d) P(-0.86 < z < -0.40) =  

(e) P(-0.40 < z < 0.86) =  

(f) P(z > -1.24) =  

(g) P(z < -1.5 or z > 2.50) =

Calculate the expected return and standard deviation of the following. T.Bills, HT, Coll, USR, MP

Object E is dependent on Objects A and B.

P(A works) = 0.90

P(A fails) = 0.10

P(B works) = 0.90

P(B fails) = 0.10

If Object A works, then Probability of Object E working is 0.6

If Object A fails, then Probability of Object E working is 0.2

If Object B works, then Probability of Object E working is 0.6

If Object B fails, then Probability of Object E working is 0.2

What is the Probability of Object E working?

Is the national crime rate really going down? Some sociologists say yes! They say that the reason for the decline in crime rates in the 1980s and 1990s is demographics. It seems that the population is aging, and older people commit fewer crimes. According to the FBI and the Justice Department, 70% of all arrests are of males aged 15 to 34 years†. Suppose you are a sociologist in Rock Springs, Wyoming, and a random sample of police files showed that of 38 arrests last month, 24 were of males aged 15 to 34 years. Use a 10% level of significance to test the claim that the population proportion of such arrests in Rock Springs is different from 70%.

(a) What is the level of significance?


State the null and alternate hypotheses.

H₀: p = 0 .7; H₁: p < 0.7H₀: p = 0.7; H₁: p > 0.7    H₀: p < 0 .7; H₁: p = 0.7H₀: p = 0.7; H₁: p ≠ 0.7H₀: p ≠ 0.7; H₁: p = 0.7

(b) What sampling distribution will you use?

The standard normal, since np > 5 and nq > 5.The Student's t, since np < 5 and nq < 5.    The Student's t, since np > 5 and nq > 5.The standard normal, since np < 5 and nq < 5.

What is the value of the sample test statistic? (Round your answer to two decimal places.)


(c) Find the P-value of the test statistic. (Round your answer to four decimal places.)


Sketch the sampling distribution and show the area corresponding to the P-value.

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?

At the α = 0.10 level, we reject the null hypothesis and conclude the data are statistically significant.At the α = 0.10 level, we reject the null hypothesis and conclude the data are not statistically significant.    At the α = 0.10 level, we fail to reject the null hypothesis and conclude the data are statistically significant.At the α = 0.10 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

(e) Interpret your conclusion in the context of the application.

There is sufficient evidence at the 0.10 level to conclude that the true proportion of arrests of males aged 15 to 34 in Rock Springs differs from 70%.There is insufficient evidence at the 0.10 level to conclude that the true proportion of arrests of males aged 15 to 34 in Rock Springs differs from 70%.    

Is the national crime rate really going down? Some sociologists say yes! They say that the reason for the decline in crime rates in the 1980s and 1990s is demographics. It seems that the population is aging, and older people commit fewer crimes. According to the FBI and the Justice Department, 70% of all arrests are of males aged 15 to 34 years†. Suppose you are a sociologist in Rock Springs, Wyoming, and a random sample of police files showed that of 36 arrests last month, 25 were of males aged 15 to 34 years. Use a 10% level of significance to test the claim that the population proportion of such arrests in Rock Springs is different from 70%.

(a) What is the level of significance?


State the null and alternate hypotheses.

H₀: p = 0.7; H₁: p ≠ 0.7H₀: p < 0 .7; H₁: p = 0.7    H₀: p = 0 .7; H₁: p < 0.7H₀: p = 0.7; H₁: p > 0.7H₀: p ≠ 0.7; H₁: p = 0.7

(b) What sampling distribution will you use?

The Student's t, since np > 5 and nq > 5.The standard normal, since np < 5 and nq < 5.    The standard normal, since np > 5 and nq > 5.The Student's t, since np < 5 and nq < 5.

What is the value of the sample test statistic? (Round your answer to two decimal places.)


(c) Find the P-value of the test statistic. (Round your answer to four decimal places.)


Sketch the sampling distribution and show the area corresponding to the P-value.

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?

At the α = 0.10 level, we reject the null hypothesis and conclude the data are statistically significant.At the α = 0.10 level, we reject the null hypothesis and conclude the data are not statistically significant.    At the α = 0.10 level, we fail to reject the null hypothesis and conclude the data are statistically significant.At the α = 0.10 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

(e) Interpret your conclusion in the context of the application.

There is sufficient evidence at the 0.10 level to conclude that the true proportion of arrests of males aged 15 to 34 in Rock Springs differs from 70%.There is insufficient evidence at the 0.10 level to conclude that the true proportion of arrests of males aged 15 to 34 in Rock Springs differs from 70%.    

Is the national crime rate really going down? Some sociologists say yes! They say that the reason for the decline in crime rates in the 1980s and 1990s is demographics. It seems that the population is aging, and older people commit fewer crimes. According to the FBI and the Justice Department, 70% of all arrests are of males aged 15 to 34 years†. Suppose you are a sociologist in Rock Springs, Wyoming, and a random sample of police files showed that of 39 arrests last month, 29 were of males aged 15 to 34 years. Use a 5% level of significance to test the claim that the population proportion of such arrests in Rock Springs is different from 70%.

(b) What sampling distribution will you use?

The Student's t, since np > 5 and nq > 5.The standard normal, since np > 5 and nq > 5.    The Student's t, since np < 5 and nq < 5.The standard normal, since np < 5 and nq < 5.

What is the value of the sample test statistic? (Round your answer to two decimal places.)

(c) Find the P-value of the test statistic. (Round your answer to four decimal places.)

Sketch the sampling distribution and show the area corresponding to the P-value.

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?

(e) Interpret your conclusion in the context of the application.