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HA1011 Applied Quantitative Methods

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Questions:

1.

Data were collected on the number of passengers at each train station in Melbourne. The numbers for the weekday peak time, 7am to 9:29am, are given below.

456 1189 410 318 648 399 382 248 379 1240 2268 272

267 1113 733 262 682 906 338 1750 530 1584 7729 323

1311 1632 1606 982 878 169 583 548 429 658 344 2630

538 494 1946 268 435 862 866 579 1359 1022 1618 1021

401 1181 1178 637 2830 1000 2958 962 697 401 1442 1115

Tasks:

  1. Construct a frequency distribution using 10 classes, stating the Frequency, Relative Frequency, Cumulative Relative Frequency and Class Midpoint
  2. Using (a), construct a histogram. (You can draw it neatly by hand or use Excel)
  3. Based upon the raw data (NOT the Frequency Distribution), what is the mean, median and mode? (Hint – first sort your data. This is usually much easier using Excel.)

2.

You are the manager of the supermarket on the ground floor below Holmes. You are wondering if there is a relation between the number of students attending class at Holmes each day, and the amount of chocolate bars sold. That is, do you sell more chocolate bars when there are a lot of Holmes students around, and less when Holmes is quiet? If there is a relationship, you might want to keep less chocolate bars in stock when Holmes is closed over the upcoming holiday. With the help of the campus manager, you have compiled the following list covering 7 weeks:

Weekly attendance Number of chocolate bars sold

472 6 916

413 5 884

503 7 223

612 8 158

399 6 014

538 7 209

455 6 214

Tasks:

  1. Is above a population or a sample? Explain the difference.
  2. Calculate the standard deviation of the weekly attendance. Show your workings. (Hint – remember to use the correct formula based upon your answer in (a).)
  3. Calculate the Inter Quartile Range (IQR) of the chocolate bars sold. When is the IQR more useful than the standard deviation? (Give an example based upon number of chocolate bars sold.)
  4. Calculate the correlation coefficient. Using the problem we started with, interpret the correlation coefficient. (Hint – you are the supermarket manager. What does the correlation coefficient tell you? What would you do based upon this information?)

3.

(We are using the same data set we used in Question 2)

You are the manager of the supermarket on the ground floor below Holmes. You are wondering if there is a relation between the number of students attending class at Holmes each day, and the amount of chocolate bars sold. That is, do you sell more chocolate bars when there are a lot of Holmes students around, and less when Holmes is quiet? If there is a relationship, you might want to keep less chocolate bars in stock when Holmes is closed over the upcoming holiday. With the help of the campus manager, you have compiled the following list covering 7 weeks:

Weekly attendance Number of chocolate bars sold

472 6 916

413 5 884

503 7 223

612 8 158

399 6 014

538 7 209

455 6 214

Tasks:

  1. Calculate AND interpret the Regression Equation. You are welcome to use Excel to check your calculations, but you must first do them by hand. Show your workings.
  1. Calculate AND interpret the Coefficient of Determination.

4.

You are the manager of the Holmes Hounds Big Bash League cricket team. Some of your players are recruited in-house (that is, from the Holmes students) and some are bribed to come over from other teams. You have 2 coaches. One believes in scientific training in computerised gyms, and the other in “grassroots” training such as practising at the local park with the neighbourhood kids or swimming and surfing at Main Beach for 2 hours in the mornings for fitness. The table below was compiled:

 

Scientific training

Grassroots training

Recruited from Holmes students

35

 

92

External recruitment

54

 

12


Tasks :

  1. What is the probability that a randomly chosen player will be from Holmes OR receiving Grassroots training?
  2. What is the probability that a randomly selected player will be External AND be in scientific training?
  3. Given that a player is from Holmes, what is the probability that he is in scientific training?
  4. Is training independent from recruitment? Show your calculations and then explain in your own words what it means.

5.

A company is considering launching one of 3 new products: product X, Product Y or Product Z, for its existing market. Prior market research suggest that this market is made up of 4 consumer segments: segment A, representing 55% of consumers, is primarily interested in the functionality of products; segment B, representing 30% of consumers, is extremely price sensitive; and segment C representing 10% of consumers is primarily interested in the appearance and style of products. The final 5% of the customers (segment D) are fashion conscious and only buy products endorsed by celebrities.

To be more certain about which product to launch and how it will be received by each segment, market research is conducted. It reveals the following new information.

  • The probability that a person from segment A prefers Product X is 20%
  • The probability that a person from segment B prefers product X is 35%
  • The probability that a person from segment C prefers Product X is 60%
  • The probability that a person from segment C prefers Product X is 90%

Tasks:

  1. The company would like to know the probably that a consumer comes from segment A if it is known that this consumer prefers Product X over Product Y and Product Z.
  2. Overall, what is the probability that a random consumer’s first preference is product X?

6.

You manage a luxury department store in a busy shopping centre. You have extremely high foot traffic (people coming through your doors), but you are worried about the low rate of conversion into sales. That is, most people only seem to look, and few actually buy anything.

You determine that only 1 in 10 customers make a purchase. (Hint: The probability that the customer will buy is 1/10.)

Tasks

  1. During a 1 minute period you counted 8 people entering the store. What is the probability that only 2 or less of those 8 people will buy anything? (Hint: You have to do this by hand, showing your workings. Use the formula on slide 11 of lecture 6. But you can always check your calculations with Excel to make sure they are correct.)
  2. (Task A is worth the full 2 marks. But you can earn a bonus point for doing Task B.)

On average you have 4 people entering your store every minute during the quiet 10-11am slot. You need at least 6 staff members to help that many customers but usually have 7 staff on roster during that time slot. The 7th staff member rang to let you know he will be 2 minutes late. What is the probability 9 people will enter the store in the next 2 minutes?

7.

You are an investment manager for a hedge fund. There are currently a lot of rumours going around about the “hot” property market on the Gold Coast, and some of your investors want you to set up a fund specialising in Surfers Paradise apartments.

You do some research and discover that the average Surfers Paradise apartment currently sells for $1.1 million. But there are huge price differences between newer apartments and the older ones left over from the 1980’s boom. This means prices can vary a lot from apartment to apartment. Based on sales over the last 12 months, you calculate the standard deviation to be $385 000.

There is an apartment up for auction this Saturday, and you decide to attend the auction.

Tasks :

  1. Assuming a normal distribution, what is the probability that apartment will sell for over $2 million?
  2. What is the probability that the apartment will sell for over $1 million but less than $1.1 million?

8.

You are an investment manager for a hedge fund. There are currently a lot of rumours going around about the “hot” property market on the Gold Coast, and some of your investors want you to set up a fund specialising in Surfers Paradise apartments.

Last Saturday you attended an auction to get “a feel” for the local real estate market. You decide it might be worth further investigating. You ask one of your interns to take a quick sample of 50 properties that have been sold during the last few months. Your previous research indicated an average price of $1.1 million but the average price of your assistant’s sample was only $950 000.

However, the standard deviation for her research was the same as yours at $385 000.

Tasks :

  1. Since the apartments on Surfers Paradise are a mix of cheap older and more expensive new apartments, you know the distribution is NOT normal. Can you still use a Z-distribution to test your assistant’s research findings against yours? Why, or why not?
  2. You have over 2 000 investors in your fund. You and your assistant phone 45 of them to ask if they are willing to invest more than $1 million (each) to the proposed new fund. Only 11 say that they would, but you need at least 30% of your investors to participate to make the fund profitable. Based on your sample of 45 investors, what is the probability that 30% of the investors would be willing to commit $1 million or more to the fund?
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