#### Mid-Semester Examination – 2020 Semester 1  (i) [1 mark] What proportion of purchases was paid by credit card and was completed

online?

(ii) [1 mark] What proportion of purchases was paid by a non-cash method and was

completed in store?

– 2 –QBUS5001 (2020 S1): Mid-semester Exam

(iii) [1 mark] What proportion of purchases was paid by debit card?

(iv) [1 mark] What proportion of purchases was completed online?

(v) [2 marks] Find the probability that a purchase was completed online given that this

purchase was paid by debit card.

(vi) [2 marks] Find the probability that a purchase was paid by cash given that this purchase

was completed online.

(vii) [2 marks] Are the events “Mode of Shopping” and “Method of Payment” independent?

(Yes/No)

Question 3 (20 marks)

Let X and p be the number of heads and the proportion of heads, respectively, in a binomial

experience of tossing eighty identical and unfair coins. Obviously, X ~ Bin(n, p) where n = 80

and p is unknown. This binomial experiment is repeated independently for 1000 times and

the data are given in the Spreadsheet “Q3”. For each experiment, the sample proportion of 𝑝𝑝 is

calculated.

Instructions: Go to Spreadsheet “Q3”. Perform statistical analysis on the data and

(i) [2 marks] Find the 90th percentile of the proportion of heads.

(ii) [2 marks] Find the third quartile of the proportion of heads.

(iii) [2 marks] Find the IQR of the proportion of heads.

(iv) [2 marks] Find the CV of the proportion of heads.

(v) [4 marks] How many outliers are there in these 1000 values of 𝑝𝑝 ? Give your answer as

an integer.

(vi) [1 mark] Is the distribution of 𝑝𝑝 more likely to be symmetric? (Yes/No)

(vii) [1 mark] Can the Central Limit Theorem be applied to approximate the sampling

distribution for 𝑝𝑝 ? (Yes/No)

Answer the following questions based on the outcome from the first binomial experiment.

(viii) [2 marks] What is the standard error of 𝑝𝑝?

– 3 –QBUS5001 (2020 S1): Mid-semester Exam

(ix) [2 marks] What is the lower confidence limit of a 99% confidence interval for p?

(x) [2 marks] What is the upper confidence limit of a 99% confidence interval for p?

Question 4 (20 marks)

Find the following probabilities using Excel. Give your answer to 4 decimal places.

(i) [3 marks] Find P(15 < X ≤ 25) where X ~ Poi(20).

(ii) [3 marks] Find P(3 ≤ X < 6) where X ~ Exp(0.25).

(iii) [4 marks] A random sample of size 25 is taken from the normal N(70, 100) distribution.

Find the probability that the average of these 25 values is greater than 71.

(iv) [5 marks] For a binomial experiment with n = 150 and p = 0.60, use the normal

approximation to find the probability that the number of successes, X, is between 84 and

99, inclusive.

(v) [4 marks] For a binomial experiment with n = 150 and p = 0.60, find the probability that

the sample proportion of success 𝑝𝑝 is between 0.56 and 0.66, inclusive.

(vi) [1 mark] Is the answer in (v) exact? (Yes/No)

Question 5 (15 marks)

A training course claims that it can increase an adult’s intelligence quotient (IQ) score by µ

points. A business analyst is interested in studying this claim. She takes a random sample of

200 adults who participated in this training course and obtains 200 increases in the IQ score.

Data are saved in Spreadsheet “Q5”. She wants to construct a confidence interval for µ. It is

assumed that the increase in IQ score follows a normal distribution with unknown variance.

Instruction: Go to Spreadsheet “Q5”. Perform statistical analysis on the data and

specified.

(i) [2 marks] What is the point estimate of µ?

(ii) [2 marks] What is the z-value or t-value required for the calculation of a 95% confidence

interval for µ?

– 4 –QBUS5001 (2020 S1): Mid-semester Exam

– 5 –

(iii) [2 marks] What is the lower confidence limit of a 95% confidence interval for µ?

(iv) [2 marks] What is the upper confidence limit of a 95% confidence interval for µ?

(v) [1 mark] Is the upper confidence limit of a 95% confidence interval for µ smaller than

that of a 90% confidence interval? (Yes/No)

(vi) [1 mark] Is the lower confidence limit of a 99% confidence interval for µ smaller than

that of a 95% confidence interval? (Yes/No)

(vii) [5 marks] If the business analyst requires that the difference between the population

value and the sample value of the increase in IQ score to be at most 0.5 points with a 95%

confidence. What is the minimum sample size that she should take? From a very similar

training course, the range of increase is 14 points. Your answer must be an integer.

Question 6 (10 marks)

A business analyst wants to study the impact of the Coronavirus outbreak on people’s online

shopping habits in Sydney. She collects a random sample of 81 households from an online

survey and obtains the weekly expenditures on online shopping before and after the

outbreak. After analyzing the data, she finds that the average increase in weekly expenditure

on online shopping is \$250 with a standard deviation of \$75. The business analyst is

interested in estimating the mean, \$µ, of the increase in weekly expenditure on online

(i) [1 mark] Is this a matched pair experiment? (Yes/No)

(ii) [1 mark] Can the Central Limit Theorem be applied to this situation? (Yes/No)

(iii) [1 mark] Shall we assume that the population standard deviation of the increase in

weekly expenditure is unknown? (Yes/No)

(iv) [5 marks] Find the width of a 99% confidence interval for µ.

(v) [1 mark] Is a 99% confidence interval for µ wider than a 90% confidence interval?

(Yes/No)

(vi) [1 mark1] Suppose that the population standard deviation is known to be \$75. Will a

99% confidence interval for µ be wider than a 99% confidence interval in (iv)? (Yes/No)QBUS5001 (2020 S1): Mid-semester Exam

Question 7 (15 marks)

Many Australians who buy property need a mortgage from a leader which is a financial

institution. A borrower can obtain a home loan directly from a lender or indirectly via a

mortgage broker. A business analyst wants to study whether borrowers are satisfied with the

service provided by the lenders and mortgage brokers during their home loan application

process. Let p1 be the proportion of borrowers who obtain a home loan directly from a lender

and are satisfied with the service provided by the lenders, and p2 be the proportion of

borrowers who obtain a home loan via a mortgage broker and are satisfied with the service

provided by the brokers.

The business analyst takes a random sample of 80 borrowers who directly obtain their home

loan from lenders and 64 of them are satisfied with the service provided. In another random

sample of 60 borrowers who obtain their home loan via mortgage brokers, 45 of them are

satisfied with the service provided. For the following questions, give your answers to 4

decimal places.

(i) [1 mark] Are the two samples likely to be independent? (Yes/No)

(ii) [1 mark] What is the best-unbiased estimate of p1 – p2?

(iii) [3 marks] What is the standard error of the estimate of p1 – p2?

(iv) [3 marks] What is the lower confidence limit of a 99% confidence interval for p1 – p2?

(v) [3 marks] What is the upper confidence limit of a 95% confidence interval for p1 – p2?

(vi) [4 marks] Now, suppose that the business analyst wants to estimate p1 – p2 within a

margin of error of 0.05 from the true difference with 90% confidence. Suppose that he

has no prior knowledge about the values of p1 and p2. Find the minimum sample size, n1

= n2 = n (say) that he needs to take from each population. Your answer must be an

integer.

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