Stats project

MAT 137: Business Statistics
Project 2 – Binomial Distribution
Due by 11:59 pm Monday October 17, 2022
******* Please submit the Excel worksheet and the written report.
A high school has a population of 200 students. The school is considering opening a new
on-campus cafe. The opinions of the 200 students [in Column B] are stored in the Excel file
StudentOpinion.xlsx. If a student is in favor of a new on-campus cafe, his/her/their opinion is
coded as 1. Otherwise, his/her/their opinion is coded as 0. We denote the coded opinion [in
Column C] of a student as xi, i = 1, …, 200. The true proportion of students who are in
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 1′𝑠
∑200 𝑥
𝑖
= 𝑖=1
. S upp os e an analyst
favor of a new on-campus café equals
200
200
who doesn’t have access to this spreadsheet, would like to conduct a study about the students’
opinions. She randomly selected five students [with replacement, i.e., one student can be
repeatedly selected.] and asked their opinions about having a new on-campus cafe. Define a
random variable x: the number of students who support the proposal of a new on-campus
cafe in the sample of five students.
Complete the following tasks.
1. Compute the true proportion of students who are in favor of a new on-campus cafe.
We will see in Chapters 6 and 7 that this proportion is called the population/binomial
proportion. [Excel: = sum(data)/200, where data is the range of the data set, C2:C201
in this question.]
2. Now, we will simulate the value of x 300 times. This means we repeat the experiment
(randomly select five students) 300 times and record the value of x each time. [Excel:
To perform the experiment once (i.e., take one sample of five students),
SUM(INDEX(data,RANDBETWEEN(1,200))+INDEX(data,RANDBETWEEN(1,200))
+INDEX(data,RANDBETWEEN(1,200))+INDEX(data,RANDBETWEEN(1,200))
+INDEX(data,RANDBETWEEN(1,200)))
Explanation of the Excel functions:
(a) RANDBETWEEN(1,200)): randomly select one student from 200 students.
(b) INDEX(data,RANDBETWEEN(1,200)): locate the opinion of the randomly selected student.
(c) SUM(INDEX(data,RANDBETWEEN(1,200))+INDEX(data,RANDBETWEEN(1,200))
+INDEX(data,RANDBETWEEN(1,200))+INDEX(data,RANDBETWEEN(1,200))
+INDEX(data,RANDBETWEEN(1,200))): sum up the coded opinions of the five
randomly selected students].
3. Now, we have 300 observations of x. Compute the relative frequencies that x = 0,
x = 1, x = 2, x = 3, x = 4, and x = 5. These relative frequencies are estimated
probabilities for each possible value of x. If we repeat the sampling procedure in
Question 2 for a very large number of times (say, thousands or millions of times), the
estimated probabilities will be very close to the true probabilities, P (x = 0), P (x = 1),
…, P (x = 5).
4. Use “bar chart” function in Excel to plot the relative frequency histogram of the data
(300 observations) you generated for x.
5. Compute the sample mean and sample standard deviation of the 300 data values you
generated for x. [Excel: average(data), stdev(data)]
6. The probability that a randomly selected student is in favor of a new cafe is 0.57
(the proportion you calculated in Question 1). The selections of the five students are
identical and independent five trials. Thus, from what you have learned in class, this
experiment is a binomial experiment and x follows a binomial distribution with n = 5
trials and p = 0.57. Now, let’s use Excel to compute the exact probabilities of x, i.e.,
5
𝑝(𝑘) = ( ) (0.57)𝑘 (1 − 𝑝)𝑠−𝑘 , 𝑘 = 0, 1, 2, 3, 4, 5. (If done by hand.)
𝑘
[Excel: binom.dist(k,5,0.57,FALSE), when “FALSE”, Excel computes p(k); when “TRUE”,
Excel computes Pr(x ≤ k).]
7. Use “bar chart” function in Excel to plot the exact probability distribution of x. Compare
it to the relative frequency histogram in Question 4. Are they close? Explain.
8. Use Excel to compute the exact probability that fewer than two students in the sample
are in favor of a new cafe. [Excel: binom.dist(1,5,0.57,TRUE). Can you tell why the
first argument is 1 not 2?]
9. If the analyst randomly selected ten students instead of five, would you be surprised to
observe that fewer than three students support having a new cafe? Explain.
Student
Opinion
1 not favor
2 favor
3 favor
4 favor
5 favor
6 not favor
7 favor
8 not favor
9 favor
10 favor
11 not favor
12 favor
13 not favor
14 favor
15 not favor
16 not favor
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20 favor
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22 favor
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28 not favor
29 favor
30 not favor
31 favor
32 not favor
33 not favor
34 favor
35 favor
36 favor
37 favor
38 favor
39 not favor
40 favor
41 not favor
42 not favor
Coded Opinion
0
1
1
1
1
0
1
0
1
1
0
1
0
1
0
0
1
0
0
1
0
1
0
0
0
1
1
0
1
0
1
0
0
1
1
1
1
1
0
1
0
0
43 not favor
44 favor
45 favor
46 not favor
47 favor
48 not favor
49 not favor
50 favor
51 favor
52 not favor
53 favor
54 not favor
55 not favor
56 not favor
57 not favor
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59 favor
60 favor
61 favor
62 favor
63 favor
64 favor
65 favor
66 not favor
67 favor
68 not favor
69 not favor
70 not favor
71 favor
72 favor
73 not favor
74 favor
75 favor
76 favor
77 not favor
78 favor
79 not favor
80 favor
81 not favor
82 favor
83 favor
84 favor
85 favor
0
1
1
0
1
0
0
1
1
0
1
0
0
0
0
0
1
1
1
1
1
1
1
0
1
0
0
0
1
1
0
1
1
1
0
1
0
1
0
1
1
1
1
86 not favor
87 favor
88 not favor
89 favor
90 favor
91 favor
92 not favor
93 not favor
94 favor
95 not favor
96 favor
97 favor
98 not favor
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100 not favor
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103 favor
104 favor
105 favor
106 favor
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108 favor
109 favor
110 favor
111 favor
112 favor
113 favor
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116 favor
117 favor
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119 favor
120 favor
121 not favor
122 not favor
123 favor
124 favor
125 favor
126 not favor
127 favor
128 not favor
0
1
0
1
1
1
0
0
1
0
1
1
0
1
0
0
0
1
1
1
1
0
1
1
1
1
1
1
0
0
1
1
0
1
1
0
0
1
1
1
0
1
0
129 not favor
130 favor
131 favor
132 not favor
133 not favor
134 not favor
135 not favor
136 favor
137 favor
138 favor
139 favor
140 favor
141 not favor
142 favor
143 favor
144 favor
145 favor
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150 favor
151 favor
152 favor
153 favor
154 favor
155 favor
156 favor
157 favor
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159 favor
160 favor
161 favor
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163 favor
164 favor
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167 favor
168 favor
169 not favor
170 favor
171 favor
0
1
1
0
0
0
0
1
1
1
1
1
0
1
1
1
1
0
1
0
0
1
1
1
1
1
1
1
1
0
1
1
1
0
1
1
0
0
1
1
0
1
1
172 not favor
173 favor
174 not favor
175 favor
176 favor
177 favor
178 favor
179 not favor
180 not favor
181 favor
182 not favor
183 not favor
184 not favor
185 favor
186 not favor
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188 not favor
189 favor
190 favor
191 not favor
192 favor
193 favor
194 favor
195 favor
196 favor
197 not favor
198 not favor
199 favor
200 favor
0
1
0
1
1
1
1
0
0
1
0
0
0
1
0
0
0
1
1
0
1
1
1
1
1
0
0
1
1