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

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7 favor

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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

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85 favor

0

1

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1

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0

1

1

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1

0

0

0

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0

1

1

1

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1

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1

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1

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0

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1

1

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1

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1

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1

1

86 not favor

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0

1

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1

1

1

0

0

1

0

1

1

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1

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0

0

1

1

1

1

0

1

1

1

1

1

1

0

0

1

1

0

1

1

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1

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1

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129 not favor

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171 favor

0

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1

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1

172 not favor

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200 favor

0

1

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1

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