DB/BB108: BUSINESS STATISTICS

Group Assignment (3-4 students per group)

SEMESTER 2, 2017

_________________________________________________________________________

The assignment has a total of 7 (7) tasks for students to complete.The total marks available for the entire assignment is 75. The assignment accounts for 25% of the overall assessment in the unit. So the total mark you receive for your assignment will be converted to a mark (out of 25).

The assignment consists of two parts. Part I is to collect a set of sample data which will be used to answer Part II.

• Your answers to Part 1, sample data, have to be submitted for marking by week 8. Late submission will attract a penalty of 1 mark per day.

• Your answers to Part 2 have to be submitted by week 10. Late submission will attract a penalty of 1 mark per day.

Presentation

• Your answers must be presented in task number order and be clearly labelled with the appropriate task number.

• Your assignment must be presented in Microsoft (MS) Word. Copy and paste any relevant Excel outputs to this document immediately before (above) any relevant written answers to each task.

• If you are unfamiliar with the use of the MS Word Equations Editor, you may write algebraic/mathematical/statistical symbols and notation in neat handwritten form.

• Your answers must be clear. You must highlight relevant items on any required Excel outputs and make reference to them in your written answers.

• When asked to perform a manual calculation (i.e. the use of MS Excel is not specified) you must show all working. This must include intermediate steps where relevant. Failure to do so will result in a loss of marks.

• Do not include the assignment questions nor the population property data with your submitted assignment.

Introduction

The Assignment Data (PopulationPropertyData.xls) file contains, in the range A1:I401, real estate sales data for a population of 400 properties around Melbourne in a particular week. You are required to select a random sample of 50 properties from this population.

The variables in the data set are as follows:

V1 = Region where property is located (1 = North, 2 = West, 3 = East, 4 = Central)

V2 = Property type (0 = Unit, 1 = House)

V3 = Sale result (1 = Sold at auction, 2 = Passed-in, 3 = Private sale, 4 = Sold before auction). Note that a blank cell for this variable indicates that the property did not sell. V4 = Building type (1 = Brick, 2 = Brick veneer, 3 = Weatherboard, 4 = Vacant land)

V5 = Number of rooms

V6 = Land size (Square metres)

V7 = Sold Price ($000s)

V8 = Advertised Price ($000s)

Column A (PN), contains the property identification numbers from 001 to 400 properties.

Selecting your Random Sample and Creating your Sample Data File

To select your random sample, you need:

• A printed copy of the Random Number Table handy.

• Open the PopulationPropertyData.xls file on computer screen.

• Create a SamplePropertyData Excel file and keep it open on computer screen.

In order to select the sample data that will form the basis of your assignment you will need to make use of the random number table provided as a pdf file (RandomNumbers.pdf) The provided table of random numbers is, as the title suggests, a sequence of randomly generated numerical digits (0 to 9). These digits are arranged in a table with ten columns (numbered 0 to 9) and one hundred rows (numbered 01 to 00) spread over two pages. The entries in each column of each row consist of six single digits.

Your first task is to select 50 three-digit random (property) numbers ranging from 001 to 400 from the table of random numbers. The type of simple random sampling that we will be engaged in here is termed “without replacement” because we specifically do not want to allow a property number to be selected more than once. If we allowed this to occur we would run the risk of the sample being biased and so not representative of the population. In the population, a particular property only occurs once and so it would not do to allow a particular property to occur more than once in your sample. In this way we can be more assured that the sample is typical of the population and so perform inferential statistical analyses about the population with some confidence.

In order to select your 50 random property numbers you will need to first go to a starting position row and column in the random number table (Note ~ not the population property data) defined by the last three digits of your MIT student identification number (the assignment marker will check your student ID number against the three digits number you use to collect the random sample). SINCE YOU ARE IN A GROUP OF STUDENTS ONLY ONE OF THE MEMBERS OF THE GROUP NEEDS TO DO THIS. AS SUCH ONLY THE LAST 3 DIGITS OF ONE OF THE GROUP MEMBER WILL BE REQUIRED.

The last two digits of your MIT ID number identify the row and the third last digit identifies the column of your (relatively) “unique” starting position.

For demonstration purposes, if the last three digits of your student identification were 7, 4 and 9 (i.e. 749), you would commence your property number selection at the starting position - row 49 and column 7 of the random number table. You are required to colour/highlight the starting row number 49 and the starting column number 7. You should be able to see that the six digit number occupying that position is 217035.

Then, moving across the row, from left to right from the starting position, examine the first three digits of each six digit number and then the second three digits in each of the columns of the table. If any of these three digit numbers are between 001 to 400 inclusive, they are “good” numbers (the population data numbered from 001 to 400). Ignore any number greater than 400 or equal to 000. They are “not-good” numbers.

Continue reading across row 49 from left to right starting at column 7 as instructed, you would encounter the following three digit good numbers:

217, 035, 306, 150, …

You need to record the first good property numbers, i.e. 217, and open the PopulationPropertyData.xls Excel file. On the spreadsheet, scroll down the PN column to locate 217 (note: do not select the Excel spreadsheet row number 217. Select the row with 217 in the PN column). At this row, highlight from 217 under the PN column across to the right up to the V8 column, use Cut and Paste procedure to cut the row of data and paste the data into a new Excel file (name it and save it as SamplePropertyData.xlsx). Next is to repeat the Cut and Paste process for PN 035, and for PN 306 and the subsequent three digit good numbers selected from Random Number Table up to the point when the row of the spreadsheet in the SamplePropertyData file grown up to 50 rows of data. Make sure you copy the column headings, PN, V1, ... V8 into your sample data file as the heading for the columns.

Each time a number is selected from the Random Number Table, insert a strikethrough mark over the selected number on the Random Number Table to mark it off. It is possible that you may come across some three digit good numbers more than once (we call them “repeated” number). The use of the Cut and Paste procedure is the “without replacement” sampling procedure to ensure that no repeated PN number and the corresponding data can be select more than once in this sample selection process. When a repeated number is found, colour/highlight/cross-out it in the Random Number Table to indicate that this good number has not been used to select the sample data.

Note that if you reach to the end of Row 50 on the first page of the Random Number Table but still not yet to collect 50 good numbers, continue the process on to Row 51 on the top of the second page of the Random Number Table (as the same practice in the Assignment Part I Model Answer). Similarly if you reach to the end of Row 00 on the second page, proceed on to row 01 on the top of the first page. Once 50 good numbers are selected and the 50 rows of data have been copied from the PopulationPropertyData file into the SamplePropertyData file, this will form a completed sample data set occupying spreadsheet columns A to I and spreadsheet rows 1 to 51.

Assignment Part I

Part I of the assignment simply requires the submission of a SOFT copy of your sample property data. This sample data set will form the basis of the statistical presentation and analysis tasks contained in Part II of the assignment.

Task 1 (10 marks)

(a) Make a hard copy of your Random Number Table containing the following:

(i) The highlight of the starting row and starting column of the sample selection process. (Refer to the Assignment Part I Model Answer). (1 mark)

(ii) The strikethrough/mark on the three digits good numbers and the cross-out of the repeated number(s). (4 marks)

(b) Submit a SOFT copy (see note below) of your sample property data (9 columns x 51 rows of data plus the column headings row) from the Excel file (SamplePropertyData) obtained per the above instructions. (5 marks)

END OF PART I OF THE ASSIGNMENT

Assignment Part II

Answers to the nine (9) assignment tasks in Part II must be based on the sample data file that you have created in Part I. All tasks in this assignment require you to obtain an Excel output prior to performing some analysis. Copy and Paste these outputs to your assignment MS- Word document immediately preceding any subsequent analysis. Explanations must be precise and to the point. Charts and tables must have appropriate titles and numerical values must be rounded to an appropriate number of decimal places and accompanied by the correct units of measure.

Task 2 (10 marks)

Use Excel to produce a Frequency Column Chart (4 marks) and a Relative Frequency PieChart (3 marks) for your sample to show the number and proportion, respectively, of each building type.

Use these graphical summaries to answer the following questions:

(a) How many properties in your sample consist of brick buildings? (1 mark)

(b) Which building type occurs most frequently in your sample? (1 mark) (c) What proportion of properties in your sample consists of weatherboard buildings?

(1 mark)

Task 3 (10 marks)

(a) Use Excel to sort your sample “Sold Price” data and paste into your MS Word assignment document. (1 mark)

(b) Use the percentile location formula;

P

LP =(n+1)100 , and the three associated rules (1 mark) to determine:

(i) The 70th percentile. (1 mark) Remember to show all working!

(ii) The first and third quartiles. (2 mark)

(c) Briefly explain what the 70th percentile that you have determined informs you about your sample “Sold Price” data. (2 mark)

(d) Determine the Inter-Quartile Range of your sample “Sold Price” data and provide a brief explanation of what information this statistic provides about your sample data. (3 marks)

Task 4 (15 marks)

(a) Use Excel to produce a Descriptive Statistics table for your sample “Sold Price” data and paste into your MS Word assignment document. (4 marks)

(b) Use results from Task 3 to determine manually for this data, the upper and lower inner fence limits;

IFUL = Q3 + 1.5 x IQR (1.5 marks)

Remember to show all working! and IFLL = Q1 – 1.5 x IQR (1.5 marks)

(c) Based on the limits calculated in (b), choose from the numerical summary measures provided in the Descriptive Statistics table, and/or measures calculated previously in

Task 3;

(i) an appropriate measure of central tendency, and, (1 mark)

(ii) an appropriate measure of dispersion for your sample “Sold Price” data.

(1mark)

Provide a brief explanation of the reasoning behind your choice in both cases.

(1mark)

(d) Write a brief report on the “Sold Price” data paying particular attention, on the mean, median, quartiles and measures of variation. (5 marks)

Task 5 (10 marks)

Remember to show all working! Failure to do so will result in the loss of marks.

(a) From the Descriptive Statistics table obtained in Task 5, identity three pieces of evidence that indicate whether your sample “Sold Price” data has been obtained from a normally distributed population or not. What is your conclusion? Note: Make sure only one piece of evidence relates to the shape of the sample data. ( 2 marks)

(b) Regardless of your conclusion in above, assume the “Sold Price” population data is normally distributed. Applying the Standard Normal tables, calculate how many “Sold Price” observations in your sample would expect to lie within 1.5 standard deviations of the mean (i.e. between z = –1.5 and z = +1.5). (4 marks)

(c) Use the mean and standard deviation from the Descriptive Statistics table of Task 5 to calculate the bound for 1.5 standard deviation spread from the mean. Using the “Sold Price” sample data, manually count the number of observations fall within the bound. State whether this count matches, approximately, your answer to (b) and hence whether this result confirms (or not) your conclusion in (a). (4 marks)

Task 6 (10 marks)

Remember to show all working! Failure to do so will result in the loss of marks.

(a) Use Excel to produce a Descriptive Statistics table for the “Sold Price” variable in your sample suitable for constructing an interval estimate of the population mean “Sold Price”. (2 marks) Hence determine:

(i) A point estimate of the mean “Sold Price” of the population of properties. (1 mark)

(ii) A 90% confidence interval estimate of the mean “Sold Price” of the population of properties. (2 marks)

(iii) Make a brief verbal statement explaining the meaning of the confidence interval estimate obtained in (ii) in the context of the variable in this task. (3 marks)

(b) If the population mean “Sold Price” is actually 650 ($000s), would you consider the interval estimate obtained in (a), to be satisfactory? Explain why or why not. (2 marks)

Task 7 (10 marks)

Remember to show all working! Failure to do so will result in the loss of marks.

(a) Use Excel to produce a Descriptive Statistics table for the brick veneer properties in your sample suitable for constructing an interval estimate of the population proportion of brick veneer properties. Hence determine: (2 marks)

(i) A point estimate of the proportion of brick veneer properties in the population. (1 mark)

(ii) A 99% confidence interval estimate of the proportion of brick veneer properties in the population. (1 mark)

(b) Using the following formula:

(sample statistic) ? (critical z or t) ? (standard error of the sample statistic)

Use the Empirical Rule for a Normal distribution to determine a 95% confidence interval estimate of the proportion of brick veneer properties in the population. (4 marks)

(c) Compare, in terms of the precision, the interval manually calculated in (b) with the interval obtained from the Descriptive Statistics table in (a). Explain why the direction of the change in precision is expected. (2 marks)

END OF PART II OF THE ASSIGNMENT

Group Assignment (3-4 students per group)

SEMESTER 2, 2017

_________________________________________________________________________

The assignment has a total of 7 (7) tasks for students to complete.The total marks available for the entire assignment is 75. The assignment accounts for 25% of the overall assessment in the unit. So the total mark you receive for your assignment will be converted to a mark (out of 25).

The assignment consists of two parts. Part I is to collect a set of sample data which will be used to answer Part II.

• Your answers to Part 1, sample data, have to be submitted for marking by week 8. Late submission will attract a penalty of 1 mark per day.

• Your answers to Part 2 have to be submitted by week 10. Late submission will attract a penalty of 1 mark per day.

Presentation

• Your answers must be presented in task number order and be clearly labelled with the appropriate task number.

• Your assignment must be presented in Microsoft (MS) Word. Copy and paste any relevant Excel outputs to this document immediately before (above) any relevant written answers to each task.

• If you are unfamiliar with the use of the MS Word Equations Editor, you may write algebraic/mathematical/statistical symbols and notation in neat handwritten form.

• Your answers must be clear. You must highlight relevant items on any required Excel outputs and make reference to them in your written answers.

• When asked to perform a manual calculation (i.e. the use of MS Excel is not specified) you must show all working. This must include intermediate steps where relevant. Failure to do so will result in a loss of marks.

• Do not include the assignment questions nor the population property data with your submitted assignment.

Introduction

The Assignment Data (PopulationPropertyData.xls) file contains, in the range A1:I401, real estate sales data for a population of 400 properties around Melbourne in a particular week. You are required to select a random sample of 50 properties from this population.

The variables in the data set are as follows:

V1 = Region where property is located (1 = North, 2 = West, 3 = East, 4 = Central)

V2 = Property type (0 = Unit, 1 = House)

V3 = Sale result (1 = Sold at auction, 2 = Passed-in, 3 = Private sale, 4 = Sold before auction). Note that a blank cell for this variable indicates that the property did not sell. V4 = Building type (1 = Brick, 2 = Brick veneer, 3 = Weatherboard, 4 = Vacant land)

V5 = Number of rooms

V6 = Land size (Square metres)

V7 = Sold Price ($000s)

V8 = Advertised Price ($000s)

Column A (PN), contains the property identification numbers from 001 to 400 properties.

Selecting your Random Sample and Creating your Sample Data File

To select your random sample, you need:

• A printed copy of the Random Number Table handy.

• Open the PopulationPropertyData.xls file on computer screen.

• Create a SamplePropertyData Excel file and keep it open on computer screen.

In order to select the sample data that will form the basis of your assignment you will need to make use of the random number table provided as a pdf file (RandomNumbers.pdf) The provided table of random numbers is, as the title suggests, a sequence of randomly generated numerical digits (0 to 9). These digits are arranged in a table with ten columns (numbered 0 to 9) and one hundred rows (numbered 01 to 00) spread over two pages. The entries in each column of each row consist of six single digits.

Your first task is to select 50 three-digit random (property) numbers ranging from 001 to 400 from the table of random numbers. The type of simple random sampling that we will be engaged in here is termed “without replacement” because we specifically do not want to allow a property number to be selected more than once. If we allowed this to occur we would run the risk of the sample being biased and so not representative of the population. In the population, a particular property only occurs once and so it would not do to allow a particular property to occur more than once in your sample. In this way we can be more assured that the sample is typical of the population and so perform inferential statistical analyses about the population with some confidence.

In order to select your 50 random property numbers you will need to first go to a starting position row and column in the random number table (Note ~ not the population property data) defined by the last three digits of your MIT student identification number (the assignment marker will check your student ID number against the three digits number you use to collect the random sample). SINCE YOU ARE IN A GROUP OF STUDENTS ONLY ONE OF THE MEMBERS OF THE GROUP NEEDS TO DO THIS. AS SUCH ONLY THE LAST 3 DIGITS OF ONE OF THE GROUP MEMBER WILL BE REQUIRED.

The last two digits of your MIT ID number identify the row and the third last digit identifies the column of your (relatively) “unique” starting position.

For demonstration purposes, if the last three digits of your student identification were 7, 4 and 9 (i.e. 749), you would commence your property number selection at the starting position - row 49 and column 7 of the random number table. You are required to colour/highlight the starting row number 49 and the starting column number 7. You should be able to see that the six digit number occupying that position is 217035.

Then, moving across the row, from left to right from the starting position, examine the first three digits of each six digit number and then the second three digits in each of the columns of the table. If any of these three digit numbers are between 001 to 400 inclusive, they are “good” numbers (the population data numbered from 001 to 400). Ignore any number greater than 400 or equal to 000. They are “not-good” numbers.

Continue reading across row 49 from left to right starting at column 7 as instructed, you would encounter the following three digit good numbers:

217, 035, 306, 150, …

You need to record the first good property numbers, i.e. 217, and open the PopulationPropertyData.xls Excel file. On the spreadsheet, scroll down the PN column to locate 217 (note: do not select the Excel spreadsheet row number 217. Select the row with 217 in the PN column). At this row, highlight from 217 under the PN column across to the right up to the V8 column, use Cut and Paste procedure to cut the row of data and paste the data into a new Excel file (name it and save it as SamplePropertyData.xlsx). Next is to repeat the Cut and Paste process for PN 035, and for PN 306 and the subsequent three digit good numbers selected from Random Number Table up to the point when the row of the spreadsheet in the SamplePropertyData file grown up to 50 rows of data. Make sure you copy the column headings, PN, V1, ... V8 into your sample data file as the heading for the columns.

Each time a number is selected from the Random Number Table, insert a strikethrough mark over the selected number on the Random Number Table to mark it off. It is possible that you may come across some three digit good numbers more than once (we call them “repeated” number). The use of the Cut and Paste procedure is the “without replacement” sampling procedure to ensure that no repeated PN number and the corresponding data can be select more than once in this sample selection process. When a repeated number is found, colour/highlight/cross-out it in the Random Number Table to indicate that this good number has not been used to select the sample data.

Note that if you reach to the end of Row 50 on the first page of the Random Number Table but still not yet to collect 50 good numbers, continue the process on to Row 51 on the top of the second page of the Random Number Table (as the same practice in the Assignment Part I Model Answer). Similarly if you reach to the end of Row 00 on the second page, proceed on to row 01 on the top of the first page. Once 50 good numbers are selected and the 50 rows of data have been copied from the PopulationPropertyData file into the SamplePropertyData file, this will form a completed sample data set occupying spreadsheet columns A to I and spreadsheet rows 1 to 51.

Assignment Part I

Part I of the assignment simply requires the submission of a SOFT copy of your sample property data. This sample data set will form the basis of the statistical presentation and analysis tasks contained in Part II of the assignment.

Task 1 (10 marks)

(a) Make a hard copy of your Random Number Table containing the following:

(i) The highlight of the starting row and starting column of the sample selection process. (Refer to the Assignment Part I Model Answer). (1 mark)

(ii) The strikethrough/mark on the three digits good numbers and the cross-out of the repeated number(s). (4 marks)

(b) Submit a SOFT copy (see note below) of your sample property data (9 columns x 51 rows of data plus the column headings row) from the Excel file (SamplePropertyData) obtained per the above instructions. (5 marks)

END OF PART I OF THE ASSIGNMENT

Assignment Part II

Answers to the nine (9) assignment tasks in Part II must be based on the sample data file that you have created in Part I. All tasks in this assignment require you to obtain an Excel output prior to performing some analysis. Copy and Paste these outputs to your assignment MS- Word document immediately preceding any subsequent analysis. Explanations must be precise and to the point. Charts and tables must have appropriate titles and numerical values must be rounded to an appropriate number of decimal places and accompanied by the correct units of measure.

Task 2 (10 marks)

Use Excel to produce a Frequency Column Chart (4 marks) and a Relative Frequency PieChart (3 marks) for your sample to show the number and proportion, respectively, of each building type.

Use these graphical summaries to answer the following questions:

(a) How many properties in your sample consist of brick buildings? (1 mark)

(b) Which building type occurs most frequently in your sample? (1 mark) (c) What proportion of properties in your sample consists of weatherboard buildings?

(1 mark)

Task 3 (10 marks)

(a) Use Excel to sort your sample “Sold Price” data and paste into your MS Word assignment document. (1 mark)

(b) Use the percentile location formula;

P

LP =(n+1)100 , and the three associated rules (1 mark) to determine:

(i) The 70th percentile. (1 mark) Remember to show all working!

(ii) The first and third quartiles. (2 mark)

(c) Briefly explain what the 70th percentile that you have determined informs you about your sample “Sold Price” data. (2 mark)

(d) Determine the Inter-Quartile Range of your sample “Sold Price” data and provide a brief explanation of what information this statistic provides about your sample data. (3 marks)

Task 4 (15 marks)

(a) Use Excel to produce a Descriptive Statistics table for your sample “Sold Price” data and paste into your MS Word assignment document. (4 marks)

(b) Use results from Task 3 to determine manually for this data, the upper and lower inner fence limits;

IFUL = Q3 + 1.5 x IQR (1.5 marks)

Remember to show all working! and IFLL = Q1 – 1.5 x IQR (1.5 marks)

(c) Based on the limits calculated in (b), choose from the numerical summary measures provided in the Descriptive Statistics table, and/or measures calculated previously in

Task 3;

(i) an appropriate measure of central tendency, and, (1 mark)

(ii) an appropriate measure of dispersion for your sample “Sold Price” data.

(1mark)

Provide a brief explanation of the reasoning behind your choice in both cases.

(1mark)

(d) Write a brief report on the “Sold Price” data paying particular attention, on the mean, median, quartiles and measures of variation. (5 marks)

Task 5 (10 marks)

Remember to show all working! Failure to do so will result in the loss of marks.

(a) From the Descriptive Statistics table obtained in Task 5, identity three pieces of evidence that indicate whether your sample “Sold Price” data has been obtained from a normally distributed population or not. What is your conclusion? Note: Make sure only one piece of evidence relates to the shape of the sample data. ( 2 marks)

(b) Regardless of your conclusion in above, assume the “Sold Price” population data is normally distributed. Applying the Standard Normal tables, calculate how many “Sold Price” observations in your sample would expect to lie within 1.5 standard deviations of the mean (i.e. between z = –1.5 and z = +1.5). (4 marks)

(c) Use the mean and standard deviation from the Descriptive Statistics table of Task 5 to calculate the bound for 1.5 standard deviation spread from the mean. Using the “Sold Price” sample data, manually count the number of observations fall within the bound. State whether this count matches, approximately, your answer to (b) and hence whether this result confirms (or not) your conclusion in (a). (4 marks)

Task 6 (10 marks)

Remember to show all working! Failure to do so will result in the loss of marks.

(a) Use Excel to produce a Descriptive Statistics table for the “Sold Price” variable in your sample suitable for constructing an interval estimate of the population mean “Sold Price”. (2 marks) Hence determine:

(i) A point estimate of the mean “Sold Price” of the population of properties. (1 mark)

(ii) A 90% confidence interval estimate of the mean “Sold Price” of the population of properties. (2 marks)

(iii) Make a brief verbal statement explaining the meaning of the confidence interval estimate obtained in (ii) in the context of the variable in this task. (3 marks)

(b) If the population mean “Sold Price” is actually 650 ($000s), would you consider the interval estimate obtained in (a), to be satisfactory? Explain why or why not. (2 marks)

Task 7 (10 marks)

Remember to show all working! Failure to do so will result in the loss of marks.

(a) Use Excel to produce a Descriptive Statistics table for the brick veneer properties in your sample suitable for constructing an interval estimate of the population proportion of brick veneer properties. Hence determine: (2 marks)

(i) A point estimate of the proportion of brick veneer properties in the population. (1 mark)

(ii) A 99% confidence interval estimate of the proportion of brick veneer properties in the population. (1 mark)

(b) Using the following formula:

(sample statistic) ? (critical z or t) ? (standard error of the sample statistic)

Use the Empirical Rule for a Normal distribution to determine a 95% confidence interval estimate of the proportion of brick veneer properties in the population. (4 marks)

(c) Compare, in terms of the precision, the interval manually calculated in (b) with the interval obtained from the Descriptive Statistics table in (a). Explain why the direction of the change in precision is expected. (2 marks)

END OF PART II OF THE ASSIGNMENT

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