Quantitative Methods Assignment 3: Statistics and Probability Analysis

This assignment solution addresses a quantitative methods problem set. Question 1 focuses on a fruit stall scenario, analyzing watermelon sales using the Poisson distribution to calculate probabilities of profit and loss, expected profit, and optimal stocking levels. Question 2 explores website sales, using binomial and normal distributions to model sales data, and includes a hypothesis test to evaluate the impact of a new website design. Question 3 involves a tree diagram to illustrate marginal, conditional, and joint probabilities related to sales calls. The solution utilizes statistical concepts, probability distributions, and data analysis to answer the questions and offers references to relevant literature.

Question 1
a) Filling the table where some entries have been filled.
Watermelons sold Probability Profit/Loss
($)
0 0.006738 -30
1 0.03369 -20
2 0.08422 -10
3 0.1404 0
4 0.1755 10
5 0.1755 20
6 0.1462 30
b) The probability that Wally will make a loss is given as;
This is given as;
P(X<=2) = P (0) + P (1) + P (2)
= 0.006738+0.03369+0.08422
=0.1247
c) Wally’s expected profit is given as;
Profit = 10+20+30
= $ 60
The variance of the profit is $ 60. This is because both the expected value and the
variance of X are equal.
d) The number of watermelons that Wally should stock to maximize the expected profit.
Choose k so that
P (X=K) = 0.4972
Using R
qpois (0.4972, 5)
K = 5

e) The number of watermelons that Wally should stock to get a probability of making loss
less than 10% (Inouye et al., 2017).
Choose K so that
P(X>K) < 0.1
Using R
qpois (0.1, 5)
K = 2
X> 2
Question 2
a) How the 100 people should be selected.
Divide the 1000 visitors into groups or strata of 10. Then use the random sampling
technique to choose one visitors from each strata ,then 100 visitors shall have been
selected .The use of strata prevent duplicate selection and also make the selections
easy and simple. Random sampling technique helps in selecting all the people with
different feature and characteristics based on how they access the site and place
orders.
b)
Assuming that the new website design has had no effect, the distribution that best
describes the number of sales out of 100 .The binomial distribution best describe the
number of sales out of 100 .That is X~B(n,p), that is X is binomially distributed
random variable ,n being the total number of visitors and p the probability of each
visitors buying from the website (Butler and Stephens, 2017).
c)

It should be reasonable to approximate the distribution in (b) with a normal
distribution. This only happen when n in is large enough. The mean would be zero
and standard deviation would be one. This is because X would have a standard
normal distribution.
d)
The trial runs and of the 100 visitors indicated , the new website makes 35
sales .Assuming that the underlying rate of sale is still at 28.3% ,the probability of
getting 35 or more sale out of 100 visits.
P(x) = n!/(n-x)x! *px*qn-x
N=100
X=35
P=28.3%
P (X >= 35) = 0.08626
e)
Assuming that there would be some cost involved with swapping to the new design, I
think it is worthwhile for the store to change its website. This is because the p. value
(0.08626) is greater than α=0.05 and therefore the null hypothesis is not rejected.
Question 3
a) Information to be presented in a tree diagram which shows marginal, conditional and
joint probabilities.
3/20
A
calls
No calls

1/2 17/20
2/5 1/5
4/5
1/10
3/10
7/10
b) The proportion of calls resulting in sales
15/100+20/100+30/100=65/100
Ans.65%
c)
Given that a sale in made, the probability that call was made by employer A, B or C
P (A) +P(B)+P(C)= 3/40+2/25+3/100
= 0.185
d)
It is not possible to tell the best sales person. This is because this business has its peak
during weekends and therefore any person working during weekends would make the highest
sales.
References
C
B
No calls
Calls
No calls
calls

Butler, K. and Stephens, M.A., 2017. The distribution of a sum of independent binomial random
variables. Methodology and Computing in Applied Probability, 19(2), pp.557-571.
Inouye, D.I., Yang, E., Allen, G.I. and Ravikumar, P., 2017. A review of multivariate
distributions for count data derived from the Poisson distribution. Wiley Interdisciplinary
Reviews: Computational Statistics, 9(3), p.e1398.