GAM470 Name_____________

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

    .

    3

    156

    166.6667

    .

    4

    171

    166.6667

    .

    5

    161

    166.6667

    .

    6

    193

    166.6667

    .

    Total

    1000

    1000.0000

    .
  2. How many degrees of freedom are there?

    3.  

     

     

     

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

    4.  

     

     

     

    You may use Excel or the following 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

     

     

  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

    .

    4

    73

    76.9231

    .

    5

    92

    76.9231

    .

    6

    69

    76.9231

    .

    7

    73

    76.9231

    .

    8

    70

    76.9231

    .

    9

    80

    76.9231

    .

    10

    75

    76.9231

    .

    J

    70

    76.9231

    .

    Q

    69

    76.9231

    .

    K

    94

    76.9231

    .

    A

    79

    76.9231

    .

    Total

    1000

    1000

    .

     

  5. How many degrees of freedom are there?

    6.  

     

     

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

    7.  

     

     

    You may use Excel or the following 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

     

     

     

  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

    .

    .

    5

    15

    .

    .

    6

    23

    .

    .

    7

    36

    .

    .

    8

    42

    .

    .

    9

    62

    .

    .

    10

    60

    .

    .

    11

    65

    .

    .

    12

    57

    .

    .

    13

    34

    .

    .

    14

    48

    .

    .

    15

    26

    .

    .

    16

    15

    .

    .

    17

    7

    .

    .

    18

    3

    .

    .

    Total

    500

    .

    .

     

  8. How many degrees of freedom are there?

 

 

 

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

 

 

 

 

You may use Excel or the following 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

 

 

 

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

.

.

Green

202

.

.

Blue

321

.

.

Yellow

396

.

.

Orange

472

.

.

Total

1500

.

.

11. How many degrees of freedom are there?

 

 

 

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

 

 

 

 

 

 

You may use Excel or the following 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

 

 

 

 

 

 

 

  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.  

     

     

     

     

     

     

     

     

  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.  

     

     

     

     

     

     

     

     

     

  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?