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2014
WORLD
SERIES
CBS
Celebrating America’s
Pastimes
Khari Bell
Mrs. Wheeler, Discrete Mathematics
27 November 2013
Bell 1
Data Analysis of the World Series
Major League Baseball has played a total of 104 World Series games. CBS wants to buy
the rights to air next year’s World Series games. They want to make revenue but they also want
to make a competitive bid for the air time. As the CBS President, a full analysis of past lengths
of World Series’ must be completed before a final bid is offered.
1.
FREQUENCY CHART
7
6
5
4
36
23
Category 1
24
21
0
5
10
15
20
25
30
HISTOGRAM
1
2.
2
3
4
5
6
7
40
36
35
30
24
25
20
23
21
15
10
5
0
0
0
0
1
2
3
4
5
# of Games Played
6
7
35
40
Bell 2
3. Mean:
4 ∗ 21 = 84
5 ∗ 24 = 120
6 ∗ 23 = 138
7 ∗ 36 = 252
84 + 120 + 138 + 252 = 594
594
104
= 5.71 = 𝑥̅
4. Standard Deviation
xi
4
5
6
7
(xi - 𝑥̅ )
-1.71
-0.71
0.29
1.29
(xi - 𝑥̅ )2
2.9241
0.5041
0.0841
1.6641
(xi - 𝑥̅ )2 *Frequency
61.4061
12.0984
1.9343
59.9076
𝑠2 =
135.3464
= 1.314
103
2
√𝑠 2 = 2√1.314
𝑠 = 1.1463
5.
S1= 5.71 − 1.1463 = 4.5637
Mean
=5.71
5.71 + 1.1463 = 6.8563
𝑠2 = 4.5637 − 1.1463 = 3.4174
6.8563 + 1.1463 = 8.0026
𝑠3 = 3.417 − 1.1463 = 2.2711
8.0026 + 1.1463 = 9.1489
The mean is an effective measure of center. The data is close together without any outliers. The
mean accurately measures the center of the majority of the data.
The standard deviation is not an efficient measure of the data’s spread because 100 percent of
the data lies between two standard deviations. The data is not a symmetrical curve, so standard
deviation does not work in this graph.
Bell 3
6. Measures of Center: Mean, Median, Mode
a. Mean = 5.71
b. Median = 6
i.
(𝑛+1)
2
=
104+1
2
= 52.5
𝑇ℎ𝑒 52.5𝑡ℎ 𝑛𝑢𝑚𝑏𝑒𝑟 𝑖𝑛 𝑑𝑎𝑡𝑎 𝑠𝑒𝑡 𝑖𝑠 𝑡ℎ𝑒 𝑚𝑒𝑑𝑖𝑎𝑛.
21 + 24 + 7.5= 52.5. -> 6 games played
c. Mode = 7
Measures of Spread
d. Standard Deviation = 1.1463
e. Range = 7 - 4 = 3
f. Quartiles =
i. Quartile 1 = 5.71 − 0.67(1.1463)
5.71 − 0.768021 = 4.941979
ii. Quartile 3 = 5.71 + 0.67(1.1463) = 0.768021
5.71 + 0.768021 = 6.478021
Quartiles are a much better measure of spread because it shows that 50 percent of the series’
played were between 4.9 and 6.5 games.
7.
Outcomes
Probability
4 games
21
104
5 games
24
3
=
104 13
6 games
23
104
7 games
36 9
=
104 26
# 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑜𝑢𝑡𝑐𝑜𝑚𝑒 𝑜𝑐𝑐𝑢𝑟𝑒𝑑
𝑇𝑜𝑡𝑎𝑙 # 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠
Probability =
The probability model for the World Series number of games played suggests that it is more likely to
play a seven game series as opposed to a four, five, or six game series. This is probably because the best
two teams in Major League Baseball are paired together for the series, battling out until the final game.
8. If each game aired on television earns 2 million dollars then:
a. The least amount of revenue = 2 million * 4 games = 8 million dollars in revenue.
b. The most amount of revenue = 2 million * 7 games = 14 million dollars in revenue.
c. I will be willing to bid 5.5 million dollars to the major league for exclusive rights. This bid
will keep CBS competitive while also maintaining a profit. The average amount of games
played in a series over the past 104 years is 5.71 games. An estimation can be made that
on average, 6 games will be played in the World Series. On average 2 million dollars
times 6 games will produce 12 million dollars in revenue. Twelve-million dollars in
revenue minus 5.5 million dollars of cost to the major league is 6.5 million dollars profit.
Based on these averages it is almost guaranteed over years of broadcasting the World
Series the profits will always be more than the cost.
Bell 4
Conclusion:
As the president of CBS, the task at hand was to analyze the number of games played in the
World Series over a span of 104 years. The data was first put into a frequency chart to organize the data
by games played. Then a histogram showed the comparison of the games played graphically. By totaling
all of the data observations and dividing by the number of years a World Series was played, a mean was
found. The mean was used to find the standard deviation by subtracting the mean from each
observation and squaring it. All of the deviations squared were totaled and divided by one less than the
number of years a World Series was played. To save time, when a certain number of games were played,
the squared deviations were multiplied by the amount of years that number of games were played in a
World Series. Mean was then compared to median and mode as a measure of center. Mean was
concluded to be the best measure of center for this data set. Quartiles and range were compared to
standard deviation and quartiles were deemed the best way to measure this data set’s spread. A
probability model was made to identify the most likely number of games to be played in the World
Series. To decide CBS’s bid to the MLB, the mean was used to measure on average, the length of a World
Series. It was concluded that because on average about six games will be played, creating a revenue of
12 million, the bid should always be less than the profit made. Therefore, the bid would have to be less
than 6 million and the profit would be more than 6 million. CBS’s bid was 5.5 million dollars to remain
competitive while also generating a profit of 6.5 million, greater than the amount bid.