GAM470 Name_____Answers________

March 29, 2005 Homework

 

  1. You roll a die 1000 times to test if it is fair. Fill in the last column to determine the chi-squared statistic, which will be in the lower right cell.
    2.

    Number

    Actual

    Expected

    Measure of Deviation

    1

    161

    166.6667

    0.1927

    2

    158

    166.6667

    0.4507

    3

    156

    166.6667

    0.6827

    4

    171

    166.6667

    0.1127

    5

    161

    166.6667

    0.1927

    6

    193

    166.6667

    4.1607

    Total

    1000

    1000.0000

    5.7920
  2. How many degrees of freedom are there?

    3. There are 6 groupings so there are 5 degrees of freedom.

     

     

  3. What is the probability that a fair die would produce a distribution more skewed than this?

    4. Using Excel: =chidist(5.792,5) = 0.326987.

    Here is a piece of the chi-squared table:

    Degrees of Freedom

    10%

    20%

    30%

    40%

    50%

    60%

    70%

    80%

    90%

    2

    4.61

    3.22

    2.41

    1.83

    1.39

    1.02

    0.71

    0.45

    0.21

    3

    6.25

    4.64

    3.66

    2.95

    2.37

    1.87

    1.42

    1.01

    0.58

    4

    7.78

    5.99

    4.88

    4.04

    3.36

    2.75

    2.19

    1.65

    1.06

    5

    9.24

    7.29

    6.06

    5.13

    4.35

    3.66

    3.00

    2.34

    1.61

    6

    10.64

    8.56

    7.23

    6.21

    5.35

    4.57

    3.83

    3.07

    2.20

    7

    12.02

    9.80

    8.38

    7.28

    6.35

    5.49

    4.67

    3.82

    2.83

    Using the table for 5 degrees of freedom we see the chi-squared statistic of 5.792 falls between 5.13 and 6.06. The p value for 5.13 is 40% and for 6.06 is 30%. So the probability must fall between 30% and 40%.

     

  4. You have a video poker machine you suspect is not dealing the ranks fairly on the deal. You test 1000 cards with the following results. Fill in the Measure of Deviation column to find the chi-squared statistic in the lower right cell.
    5.

    Number

    Actual

    Expected

    Measure of Deviation

    2

    79

    76.9231

    0.0561

    3

    77

    76.9231

    0.0001

    4

    73

    76.9231

    0.2001

    5

    92

    76.9231

    2.9551

    6

    69

    76.9231

    0.8161

    7

    73

    76.9231

    0.2001

    8

    70

    76.9231

    0.6231

    9

    80

    76.9231

    0.1231

    10

    75

    76.9231

    0.0481

    J

    70

    76.9231

    0.6231

    Q

    69

    76.9231

    0.8161

    K

    94

    76.9231

    3.7911

    A

    79

    76.9231

    0.0561

    Total

    1000

    1000

    10.308

     

  5. How many degrees of freedom are there?

    6. There are 13 groupings so there are 12 degrees of freedom.

     

  6. What is the probability a fair game would produce results more skewed than this?

    7.  

    Chidist(10.308,12) = 0.5890

    Or using a chi-squared table:

    Degrees of Freedom

    10%

    20%

    30%

    40%

    50%

    60%

    70%

    80%

    90%

    10

    15.99

    13.44

    11.78

    10.47

    9.34

    8.30

    7.27

    6.18

    4.87

    11

    17.28

    14.63

    12.90

    11.53

    10.34

    9.24

    8.15

    6.99

    5.58

    12

    18.55

    15.81

    14.01

    12.58

    11.34

    10.18

    9.03

    7.81

    6.30

    13

    19.81

    16.98

    15.12

    13.64

    12.34

    11.13

    9.93

    8.63

    7.04

    14

    21.06

    18.15

    16.22

    14.69

    13.34

    12.08

    10.82

    9.47

    7.79

    For 12 degrees of freedom we see 10.308 falls between 10.18 and 11.34. The p value of 10.18 is 60%, and for 11.34 is 50%, so the p value for 10.308 must fall between 50% and 60%.

  7. You suspect your dice in sic-bo may not be fair. You roll the dice 500 times and record the total each time. You get the following results. Fill in the rest of the table to determine the chi-squared statistic.
    8.

    Number

    Actual

    Expected

    Measure of Deviation

    3

    3

    2.3148

    0.2028

    4

    4

    6.9444

    1.2484

    5

    15

    13.8889

    0.0889

    6

    23

    23.1481

    0.0009

    7

    36

    34.7222

    0.0470

    8

    42

    48.6111

    0.8991

    9

    62

    57.8704

    0.2947

    10

    60

    62.5000

    0.1000

    11

    65

    62.5000

    0.1000

    12

    57

    57.8704

    0.0131

    13

    34

    48.6111

    4.3917

    14

    48

    34.7222

    5.0774

    15

    26

    23.1481

    0.3513

    16

    15

    13.8889

    0.0889

    17

    7

    6.9444

    0.0004

    18

    3

    2.3148

    0.2028

    Total

    500

    500

    13.10761

    To get the expected number per total you can either refer to the sic-bo notes or calculate manually:

    Total

    Dice

    Combinations

    3

    1,1,1

    1

    4

    1,1,2

    3

    5

    1,1,3

    3

    5

    1,2,2

    3

    6

    1,1,4

    3

    6

    1,2,3

    6

    6

    2,2,2

    1

    7

    1,1,5

    3

    7

    1,2,4

    6

    7

    1,3,3

    3

    7

    2,2,3

    3

    8

    1,1,6

    3

    8

    1,2,5

    6

    8

    1,3,4

    6

    8

    2,2,4

    3

    8

    2,3,3

    3

    9

    1,2,6

    6

    9

    1,3,5

    6

    9

    1,4,4

    3

    9

    2,2,5

    3

    9

    2,3,4

    6

    9

    3,3,3

    1

    10

    1,3,6

    6

    10

    1,4,5

    6

    10

    2,2,6

    3

    10

    2,3,5

    6

    10

    2,4,4

    3

    10

    3,3,4

    3

    Totals of 11 to 18 are the mirror image of 3 to 10, resulting in the following combinations.

    Total

    Total

    Probability

    3

    1

    0.0046

    4

    3

    0.0139

    5

    6

    0.0278

    6

    10

    0.0463

    7

    15

    0.0694

    8

    21

    0.0972

    9

    25

    0.1157

    10

    27

    0.1250

    11

    27

    0.1250

    12

    25

    0.1157

    13

    21

    0.0972

    14

    15

    0.0694

    15

    10

    0.0463

    16

    6

    0.0278

    17

    3

    0.0139

    18

    1

    0.0046

    Total

    216

    1.0000
  8. How many degrees of freedom are there?

There are 16 groupings so there are 15 degrees of freedom.

9. What is the probability fair dice would produce results this skewed or more?

Chidist(13.10761,15) = 0.5940

Or using a chi-squared table:

Degrees of Freedom

10%

20%

30%

40%

50%

60%

70%

80%

90%

13

19.81

16.98

15.12

13.64

12.34

11.13

9.93

8.63

7.04

14

21.06

18.15

16.22

14.69

13.34

12.08

10.82

9.47

7.79

15

22.31

19.31

17.32

15.73

14.34

13.03

11.72

10.31

8.55

16

23.54

20.47

18.42

16.78

15.34

13.98

12.62

11.15

9.31

17

24.77

21.61

19.51

17.82

16.34

14.94

13.53

12.00

10.09

For 15 degrees of freedom we see 13.10761 falls between 13.03 and 14.34. The p value of 13.03 is 60%, and for 14.34 is 50%, so the p value for 10.308 must fall between 50% and 60%.

10. Your sock drawer has the following distribution of socks:

Red

1

Green

2

Blue

3

Yellow

4

Orange

5

You enjoy drawing socks at random but wonder if you are drawing fairly. So you draw socks one at a time, 1500 times, with replacement, and observe the following the following distribution. Fill in the rest of the table to determine the chi-squared statistic.

 

Color

Actual

Expected

Measure of Deviation

Red

109

100

0.8100

Green

202

200

0.0200

Blue

321

300

1.4700

Yellow

396

400

0.0400

Orange

472

500

1.5680

Total

1500

1500

3.908

11. How many degrees of freedom are there?

There are 5 groupings so there are 4 degrees of freedom.

12. What is the probability fair dice would produce results this skewed or more?

Chidist(3.908,4) = 0.4186

Or using a chi-squared table:

Degrees of Freedom

10%

20%

30%

40%

50%

60%

70%

80%

90%

2

4.61

3.22

2.41

1.83

1.39

1.02

0.71

0.45

0.21

3

6.25

4.64

3.66

2.95

2.37

1.87

1.42

1.01

0.58

4

7.78

5.99

4.88

4.04

3.36

2.75

2.19

1.65

1.06

5

9.24

7.29

6.06

5.13

4.35

3.66

3.00

2.34

1.61

6

10.64

8.56

7.23

6.21

5.35

4.57

3.83

3.07

2.20

For 4 degrees of freedom we see 3.908 falls between 3.36 and 4.04. The p value of 3.36 is 50% and of 4.04 is 40%. So the p value must fall between 40% and 50%.

  1. Your sock drawer has 8 black socks, 6 white socks, and 1 red sock. You draw 3 socks at random without replacement. What is the probability you draw 2 black socks and 1 white sock?

    2. There are combin(8,2)=8*7/2 = 28 ways to draw 2 black socks out of 8. There are 6 ways to draw the white sock. There are combin(15,3) = (15*14*13)/(1*2*3) = 455 ways to draw any 3 socks out of 15. So the probability is 28*6/455 = 36.92%.

  2. You are playing blackjack in a 2-deck game. You have a jack and queen and the dealer has an ace up. You have no idea what cards are previously been played. What is the probability the dealer has a blackjack?

    3. There are 16 10-point cards per deck, so 32 in two decks. You have two of them so there are 30 left. There are a total of 52*2=104 cards in 2 decks, but three of them are known, so there are 101 left. The probability the dealer’s whole card is a 10, giving the dealer a blackjack, is 30/101 = 29.70%.

  3. The dealer offers insurance, which pays 2 to 1, that his hole card is a 10-point card. What is the expected value of this bet?

Probability of win = 29.70%

Probability of loss = 70.30%.

Expected value = .297*2 - .703 = -.2575 = -10.90%.