Question 1 (10 marks)

A business analyst is interested in studying the weekly range data in stock prices of two Australian banks: ANZ bank and NAB bank. The weekly range is defined as the highest stock price minus the lowest stock price in a week. He collects weekly range data from 2015 to 2019 and the data are given in Spreadsheet Q1. Summary statistics are given below.

Instruction: Go to Spreadsheet Q1. Perform statistical analysis on the data and answer the following questions. Give your answers to 2 decimal places unless specified.

(i) [1 mark] How many weekly data that the business analyst has collected? Give your answer as an integer.

(ii) [2 marks] Is it possible that ANZ range data and NAB range data have a linear relationship? You may draw a scatterplot to help you answer this question. (Yes/No)

(iii) [2 marks] What is the correlation coefficient between the range data of these two stocks?

(iv) [1 mark] What is the mean weekly range of ANZ stock price?

(v) [1 mark] What is the standard deviation of the weekly range of NAB stock price?

(vi) [1 mark] Is the distribution of the weekly range of the two stocks skewed to the left? (Yes/No)

(vii) [2 marks] Does NAB have a smaller coefficient of variation compared to ANZ? (Yes/No)

Question 2 (10 marks)

A department store analysed its most recent sales and determined the relationship between the way the customers paid for the item and the way they shop. The joint probabilities and the marginal probabilities are given in the table below.

Answer the following questions and give your answers to 2 decimal places.

(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?

(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 Spreadsheet “Q3”. For each experiment, the sample proportion 𝑝𝑝̂ is calculated.

Instructions: Go to Spreadsheet “Q3”. Perform statistical analysis on the data and answer the following questions. Give your answers to 4 decimal places unless specified.

(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 CV of the proportion of heads.

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

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

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

distribution for 𝑝𝑝̂ ? (Yes/No)

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

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

(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 𝑝𝑝̂ 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 answer the following questions. Give your answers to 2 decimal places unless 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?

(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 point 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 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 analysing 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 shopping. For numerical questions, give your answers to 2 decimal places.

(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)

Question 7 (15 marks)

Many Australians who buy a 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.