Statistics And Probability Exercises

Question 1 of 8

Question 1 of 8

HINT: We cover this in Lecture 1 (Summary Statistics and Graphs)

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.)

Question 2 of 8

HINT: We cover this in Lecture 2 (Measures of Variability and Association)

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?)

Question 3 of 8

HINT: We cover this in Lecture 3 (Linear Regression)

(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

Question 2 of 8

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.

(Hint 1 - As manager, which variable do you think is the one that affects the other variable? In other words, which one is independent, and which variable’s value is dependent on the other variable? The independent variable is always x.

Hint 2 – When you interpret the equation, give specific examples. What happens when Holmes are closed? What happens when 10 extra students show up?)

1. Calculate AND interpret the Coefficient of Determination.

Question 4 of 8

HINT: We cover this in Lecture 4 (Probability)

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 (show all your workings):

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.

Question 5 of 8

HINT: We cover this in Lecture 5 (Bayes’ Rule)

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 (show your workings):

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?

Question 6 of 8

HINT: We cover this in Lecture 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 (show your workings):

6. 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.)
7. (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 7^th 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? (Hint 1: It is a Poisson distribution. Hint 2: What is the average number of customers entering every 2 minutes? Remember to show all your workings.)

Question 7 of 8

HINT: We cover this in Lecture 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 (show your workings):

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?

Question 8 of 8

HINT: We cover this in Lecture 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 (show your workings):

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?

Question 1 of 8

HINT: We cover this in Lecture 1 (Summary Statistics and Graphs)

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

	Mid-point	Frequency	Relative frequency	Cumulative relative frequency
100-299	199.5	6	0.098	0.098
300-499	399.5	14	0.230	0.328
500-999	749.5	17	0.279	0.607
1000-1999	1499.5	18	0.295	0.902
2000-2999	2499.5	4	0.066	0.967
3000-3999	3499.5	0	0.000	0.967
4000-4999	4499.5	0	0.000	0.967
5000-5999	5499.5	0	0.000	0.967
6000-6999	6499.5	1	0.016	0.984
7000-7999	7499.5	1	0.016	1.000

1. Using (a), construct a histogram. (You can draw it neatly by hand or use Excel)
2. 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.)

number of passengers
Mean	1033.433
Median	715
Mode	401

Question 2 of 8

HINT: We cover this in Lecture 2 (Measures of Variability and Association)

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.

This is a sample since it is covering a portion of the period. A population covers the entire period.

1. Calculate the standard deviation of the weekly attendance. Show your workings. (Hint – remember to use the correct formula based upon your answer in (a).)

x
472	-13	158.041
413	-72	5122.469
503	18	339.612
612	127	16238.041
399	-86	7322.469
538	53	2854.612
455	-30	874.469
3392	0	32910

1. 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.)
2. 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?)

The correlation coefficient is 0.968; this means that a very strong positive linear relationship exists between weekly attendance and number of chocolates sold. Based on this information, as a manager I would try to see on ways of increasing the weekly attendance through attractive events in order to increase on sales.

Question 3 of 8

HINT: We cover this in Lecture 3 (Linear Regression)

(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:

Question 2 of 8

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.

x	y	x^2	xy
472	6916.00	222784	3264352
413	5884.00	170569	2430092
503	7223.00	253009	3633169
612	8158.00	374544	4992696
399	6014.00	159201	2399586
538	7209.00	289444	3878442
455	6214.00	207025	2827370
=3392	=47618	=1676576	=23425707

Weekly attendance affects number of chocolate bar sold. Weekly attendance is the independent variable while number of chocolate bar sold is the dependent variable. From the above calculations we observe that the regression model is;

The coefficient of the weekly attendance is 10.6772; this means that a unit increase in the weekly attendance would result to an increase in the number of Holmes sold by 10.6772. If the number of weekly attendance increases by 10 then number of Holmes sold would increase by 106.772. It also means that a unit decrease in the weekly attendance would result to a decrease in the number of Holmes sold by 10.6772. If the number of weekly attendance decreases by 10 then number of Holmes sold would also decrease by 106.772.

Using Excel We obtained;

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.967993
R Square	0.93701
Adjusted R Square	0.924412
Standard Error	224.5952
Observations	7
ANOVA
	df	SS	MS	F	Significance F
Regression	1	3751817	3751817	74.37736	0.000346
Residual	5	252215	50442.99
Total	6	4004032
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	1628.689	605.9	2.688049	0.0434	71.1734	3186.205
x	10.67723	1.238051	8.624231	0.000346	7.494724	13.85974

1. Calculate AND interpret the Coefficient of Determination.

Thus the coefficient of determination r² is 0.937; this means that 93.7% of the variation in the dependent variable (number of Holmes sold) is explained by the number of weekly attendances.

Question 4 of

HINT: We cover this in Lecture 4 (Probability)

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 (show all your workings):

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.

If A and B are independent events, the probability of this event happening can be calculated as shown below:

If the two events are independent then we would expect;

Hence from the above, we conclude that training is not independent from recruitment

Question 5 of 8

HINT: We cover this in Lecture 5 (Bayes’ Rule)

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.

Question 3 of 8

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 (show your workings):

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?

Question 6 of 8

HINT: We cover this in Lecture

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 (show your workings):

6. 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.)
7. (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 7^th 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? (Hint 1: It is a Poisson distribution. Hint 2: What is the average number of customers entering every 2 minutes? Remember to show all your workings.)

Question 7 of 8

HINT: We cover this in Lecture 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 (show your workings):

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?

Question 8 of 8

HINT: We cover this in Lecture 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 (show your workings):

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?

Yes you can still use normal distribution since the sample size used is relatively large and using the Central limit Theorem the samples will be considered to be approximately normally distributed hence z-distribution can be used to test.

1. 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?