1. At the State Fair the rifle range offers 4 different hats as prizes for perfe
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Question
1. At the State Fair the rifle range offers 4 different hats as prizes for perfect scores, one hat for each of the State University campuses. The hats are in boxes, and a box is chosen at random and given to the winners; no substitutions are allowed. A local football fan (an ace shot who never misses) wishes to collect all 4 hat designs. There are a very large number of hats on hand and there are equal numbers of each hat design, so each design has a probability of 0.25 of appearing as the prize. You are to design a simulation that could be used to estimate the average number of perfect scores needed to complete the set of hats. a) To simulate this strategy, assign digits to the hat designs that will result in the probability of selection of each design to be 0.25. Hat 1 Digits: Hat 2 Digits: Hat 3 Digits: Hat 4 Digits: b) Describe how you would use a random digit table to conduct one run of your simulation, where one run continues until a complete set of the 4 hats is acquired. Chapter 6, Quiz 3, Form A Page 2 of 3 c) Using the following lines from a random number table, demonstrate how your assignment of digits in part (b) would work for simulating the winning of hats. (You may mark on the digits to help explain your procedure.) 61790 55300 05756 72765 96409 12531 35013 82853 73676 57890 99400 37754 42648 82425 36290 45467 71709 77558 00095 32863 29485 82226 Chapter 6, Quiz 3, Form A Page 3 of 3 d) Suppose that the rifle range manager decides to order different numbers of hats, consistent with the popularity of each campus’s football team. Hat 1 will be put in 50% of the boxes, Hats 2 and 3 will be put in 20% of the boxes, and Hat 4 will be put in 10% of the boxes. Assign digits to the hats that will be consistent with these probabilities. Hat 1 Digits: Hat 2 Digits: Hat 3 Digits: Hat 4 Digits: e) Perform 5 runs, and use your results to estimate the probability that it would take more than 10 boxes to complete the set of 4 hats. (You may mark on the digits to help explain your procedure.) 19223 95734 05756 28713 96409 12531 42544 82853 73676 47150 99400 37754 42648 82425 36290 45467 71709 77558 00095 32863 29485 82226 68417 35013 15529 72765 85089 57067 50211 47487 82739 57890 20807 81676 55300 94383 14893 60940 72024 17868 24943 61790 90656 87964
Explanation / Answer
Solution
Part (a)
Assignment of random digits to Hat Designs
Random Digits
01 to 25
26 to 50
51 to 75
76 to 99 and 00
Hat Design
1
2
3
4
As can be seen from the above table, each Hat Design is assigned 25 random digits. Since random digits are based on Uniform distribution, each design has 25 out of 100 i.e., 0.25 chance of being selected.
Part (b)
Starting from any point at random of the random digits line, keep picking 2-digit numbers sequentially and following the rule in the Table under Part (a), keep assigning the Hat Design. Continue the process till one of each Hat Design is obtained. Keep entering the results of the simulation in a Tabular form as shown below:
Simulation #
Random Digits
Hat Design
1
2
3
.
.
.
.
Part (c)
61790 55300 05756 72765 96409 12531 35013 82853 73676 57890 99400 37754 42648 82425 36290 45467 71709 77558 00095 32863 29485 82226
The run using the given lines from a random number table:
Simulation #
Random Digits
Hat Design
Remarks, if any
1
61
3
2
79
4
3
05
1
4
53
3
5
00
4
6
05
1
7
75
3
8
67
3
9
27
2
Since all the four designs have been covered, the simulation is stopped. Summary of results:
Hat Design
Frequency
Relative Frequency
1
||
0.22
2
|
0.11
3
||||
0.44
4
||
0.22
Total
9
0.99
Comments:
Clearly, the run size is too small for a simulation resulting in skewed probability distribution as represented by the relative frequency.
Part (d)
Assignment of random digits to Hat Designs
Random Digits
1 to 5
6 and 7
8 and 9
0
Hat Design
1
2
3
4
As can be seen from the above table, Hat Design 1 is assigned 5 random digits, Hat Design 2 and 3 are assigned 2 random digits each and Hat Design 4 is assigned 1 random digits. Since random digits are based on Uniform distribution, each design has stipulated chance of being selected.
Part (e)
19223 95734 05756 28713 96409 12531 42544 82853 73676 47150 99400 37754 42648 82425 36290 45467 71709 77558 00095 32863 29485 82226 68417 35013 15529 72765 85089 57067 50211 47487 82739 57890 20807 81676 55300 94383 14893 60940 72024 17868 24943 61790 90656 87964
The run using the given lines from a random number table:
Simulation #
Random Digits
Hat Design
Remarks, if any
1
1
1
2
9
3
3
2
1
4
2
1
5
3
1
6
9
3
7
5
1
8
7
2
9
3
1
10
4
1
11
0
4
Since all the four designs have been covered, the simulation is stopped. Summary of results:
Hat Design
Frequency
Relative Frequency
1
|||| |||
0.64
2
|
0.09
3
||
0.18
4
|
0.09
Total
11
1.00
Comments:
Clearly, the run size is too small for a simulation resulting in skewed probability distribution as represented by the relative frequency.
Random Digits
01 to 25
26 to 50
51 to 75
76 to 99 and 00
Hat Design
1
2
3
4
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