Download Introduction - Pearson Higher Education

Instructor’s Manual
Quantitative Methods for
Business and Economics
2nd edition
Modular texts in business and economics
Glyn Burton
George Carrol
and
Stuart Wall
ISBN 0 273 65524 8
 Pearson Education 2002
Lecturers adopting the main text are permitted to photocopy the pack as required.
© Pearson Education Limited 2002
Pearson Education Limited
Edinburgh Gate
Harlow
Essex CM20 2JE
England
and Associated Companies around the world
Visit us on the World Wide Web at:
www.pearsoneduc.com
____________________________
First published 2002
© Pearson Education Limited 2002
The rights of Glyn Burton, George Carrol and Stuart Wall
to be identified as authors of this Work have been asserted by them
in accordance with the Copyright, Designs and Patents Act 1988.
ISBN 0 273 65524 8
All rights reserved. Permission is hereby given for the material in this
publication to be reproduced for OHP transparencies and student handouts,
without express permission of the Publishers, for educational purposes only.
In all other cases, no part of this publication may be reproduced, stored in
a retrieval system, or transmitted in any form or by any means, electronic,
mechanical, photocopying, recording, or otherwise without either the prior
written permission of the Publishers or a licence permitting restricted copying
in the United Kingdom issued by the Copyright Licensing Agency Ltd,
90 Tottenham Court Road, London W1P 0LP.
© Pearson Education Limited 2002
Contents
A variety of spreadsheet applications and exercises using Excel are presented for the
following topic areas (not all chapter headings are covered). Many of these are
integrated with materials in the existing textbook, others are freestanding. A separate
data file contains the raw data to be used in many of the spreadsheet exercises. The
materials conclude with two computerised practice tests.
1
2
3
4
5
6
7
8
9
10
Introduction: Excel
Central location and dispersion
Regression and correlation
Probability distributions
Sampling and tests of hypotheses 1
Sampling and tests of hypotheses 2
Index numbers
Time value of money
Basic mathematics
Exam Practice
Test 1
Test 2
Data files
iii
© Pearson Education Limited 2002
Burton, Carrol and Wall 2nd edition – Instructor’s Manual
Introducing Excel
By the end of this chapter you should be able to:
 understand the Excel screen;

enter data;

edit data;

save your workbook;

work with ranges of cells;

open a saved workbook;

create graphs and charts.
1.1 The use of spreadsheets
For many large collections of real data, computer spreadsheets are used to organise and
analyse information. Although the spreadsheet was designed for the calculation and
organisation of financial data, it is an extremely powerful tool for most forms of
business and economic data.
The value of the spreadsheet is that once the data is entered it is a simple and quick
process to analyse and manipulate it in a variety of sophisticated ways. This means that
the analysis (and the arithmetic) is automatic. In order to do this, of course, you need a
level of computer literacy and you need to learn the basics of spreadsheets. Microsoft’s
Excel is the largest-selling spreadsheet package in the business world. Consequently,
we will use the package to support the textbook and to provide you with a very
marketable skill when you graduate.
1.2 To start work in Excel
Once you have logged into the network you will find the Excel icon:
Move the mouse over the icon and select it (double-click on it with the left mouse
button).
The Excel application should load, presenting you with the screen as in Figure 1.1.
1
© Pearson Education Limited 2002
Burton, Carrol and Wall 2nd edition – Instructor’s Manual
figure 1.1 The Excel window
1.3 Understanding the worksheet screen
Overview
A spreadsheet or worksheet is the equivalent of a large sheet of paper, divided into
columns and rows. A workbook consists of one or more worksheets – each worksheet
can be regarded as a ‘page’ in the book.
When you start Excel it opens a workbook window containing a group of related
worksheets named, initially, Sheet1, Sheet2 etc. An examination of the screen shows
that Excel consists of two windows:
 The application window: found at the top of the screen, it contains all the Excel
commands – menus, tool bars etc.
 The document window: the main part of the screen, which consists of the worksheet
itself – divided into columns and rows, plus other features such as scroll bars and
sheet numbers.
2
© Pearson Education Limited 2002
Burton, Carrol and Wall 2nd edition – Instructor’s Manual
The application window
The menu bar
Provides a list of menu options. The Excel commands are grouped under these
menus
The tool bars
There are two tool bars immediately below the menu bar; each consists of a row of
buttons that you ‘click’ to carry out Excel tasks. Often a button is a shortcut
alternative to a menu command; occasionally there is no menu alternative to using a
button.
The standard toolbar
Contains the buttons used for formatting, editing, file handling and printing.
The formatting toolbar
Contains buttons used for controlling the appearance of the contents of each cell or
range of cells. This includes, for example, fonts, borders, shading, alignment and
numeric formats.
The formula bar
Shows whatever is in entered in the active cell. This is a new workbook so all the cells
are blank.
The Reference Area – shows the row and column number of the active cell.
The title bar
Whenever a new workbook is opened Excel gives it a temporary or default name,
Book1, Book2 etc. You should change the name when you save the workbook.
3
© Pearson Education Limited 2002
Burton, Carrol and Wall 2nd edition – Instructor’s Manual
Maximise/restore buttons
At the top right of the screen are two groups of three buttons; these control
the size of the worksheet screen. One set is for the whole application
window and one for the inner cell area, the document window.
The minimise button
The maximise button
The close button
The restore button
Reduces the size of the screen to a small icon.
Increases the window to full screen size.
Closes the workbook.
Restores the window to its original size.
The document window
A worksheet window can only show a small fraction (see Figure 1.2) of the total
worksheet size; potentially each sheet in a workbook can be 256 columns across and
16,384 rows down!
figure 1.2 The document window
The scroll bars
Horizontal and vertical bars enable you to move around a worksheet.
A scroll box in the bar shows the current display’s position relative to the entire
workbook contained in that window.
In the workbook illustrated in Figure 1.1, the scroll boxes show that we are
positioned at the top left portion of the worksheet (as in Figure 1.2). The length of the
scroll boxes indicates the amount of the worksheet in use.
4
© Pearson Education Limited 2002
Burton, Carrol and Wall 2nd edition – Instructor’s Manual
Cell references
Column headings contain the column references.
Row headings
contain the row references.
Jointly they give the cell
reference, for example, A1, D5.
Spreadsheets use cell references, based on column and row headings, to:

identify cells or groups of cells on a worksheet – a group of cells is known as a
cell range;

point to cells containing the value to be used in a formula.
A
B
C
D
E
1
2
3
4
5
6
7
8
9
10
11
12
13
14
The reference of this cell is D4
The reference of this range is B7:E11
The active cell
A heavy border surrounding
the cell indicates the active cell.
Sheet names
A new workbook consists of a number of blank worksheets, having the default
names Sheet1, Sheet2 etc. Collectively these form a workbook. Each sheet is
marked with a name tab – the name highlighted indicates which sheet is currently
selected or ‘active’. In the example above, Sheet1 is currently selected. To open a
new sheet click on the name tab.
On the left of the sheet names are a number of arrow buttons to move quickly
through a group of worksheets.
5
© Pearson Education Limited 2002
Burton, Carrol and Wall 2nd edition – Instructor’s Manual
The status bar
Is positioned below the sheet names and displays information about the current
command; no command has been issued yet so it reads ‘Ready’. When a cell is
made active it will read ‘Enter’.
1.4 Entering data
At the data entry stage, you should ignore the appearance of the worksheet and
concentrate on entering your data accurately. Therefore, avoid all the intricacies of
formatting.
Types of data
Before data can be converted into information, you need to enter the raw data into the
cells that make up the worksheet. There are many types of data that you can enter,
including:
 text;
 numbers;
 dates;
 times;
 formulas and functions.
Entering text
You can enter any combination of letters and numbers as text. Text is automatically leftaligned in a cell.
To enter text into a cell:
1
Select the cell into which you wish to enter text.
2
Type the text. As you type, your text appears in the cell and in the formula
bar, as shown in Figure 3.
3
Click the ACCEPT button
or use the key marked  on the keyboard.
6
© Pearson Education Limited 2002
Burton, Carrol and Wall 2nd edition – Instructor’s Manual
Whatever you type also appears on the formula bar.
figure 3 The formula bar
If you make a mistake and wish to cancel an entry before it is complete, press ESC or
click on the Clear button
on the formula bar.
Number text
You may want to enter a number as text (for example, a telephone number). Precede
your entry with a single quotation mark (‘), as in ‘946220. The single quotation mark is
an alignment prefix that tells Excel to treat the following characters as text and left-align
them in the cell.
Entering numbers
Numbers include all the numeric characters zero to nine (0–9) as well as the
mathematical operators and, on some occasions, ( ) , $ %. Numbers are automatically
right-aligned.
To enter a number:
1
Select the cell into which the number is to be entered.
2
Type the number. To enter a negative number, precede it with a minus sign,
or surround it with parentheses.
3
However, it is not necessary to use a comma to separate thousands.
4
Click on the Enter button
on the keyboard.
on the formula bar, or use the Enter key ()
It is better to format a cell to a currency format rather than type a column of pound
sterling amounts including the pound signs and decimal points. For example, you can
type numbers such as 241 and 25.17, and then format the column with currency
formatting. Excel would then change your entries to £241.00 and £25.17, respectively.
There is, therefore, no need to enter:

a pound sign (£), or

the dollar sign ($) before the number.
Where possible, calculate percentages rather than use the % icon.
7
© Pearson Education Limited 2002
Burton, Carrol and Wall 2nd edition – Instructor’s Manual
If you enter a number, and it appears in the cell as hatch signs,
#######, it means that the cell is not wide enough to display the
number. To remedy this, position the cursor on the extreme right
boundary of the column heading – when you are in the correct position
the cursor changes to a double-pointed arrow
– then adjust the width using the
mouse.
An important point to consider is that the accuracy of your results depends upon the
accuracy of your data. You should, therefore, always enter the data as accurately as
possible. This is true not only when using a spreadsheet but whenever you use
quantitative methods in the decision-making process.
8
© Pearson Education Limited 2002
Burton, Carrol and Wall 2nd edition – Instructor’s Manual
Central location and dispersion
By the end of this chapter you should be able to:
 construct a frequency table from data collected in a group format;
calculate the descriptive statistics of grouped data:

measures of central tendency;
measure of dispersion.
2.1 Analysing grouped data
On many occasions, data are grouped into a frequency table with various class intervals.
To deal with such data the simplifying assumption must be made that within any given
class interval the items of data fall on the class mid-point. This is equivalent to
assuming that the items of data are evenly spread within any given class interval.
Exercise 2.1
The descriptive statistics for Worked Example 2.2 on page 32 of your textbook show the
results from a survey of the prices of 60 items sold in a shop (Table 2.1).
Table 2.1 The frequency of items within a given price range
1
Price of item (£)
Number of items sold
1.5–2.5
15
2.5–3.5
2
3.5–4.5
19
4.5–5.5
10
5.5–6.5
14
Construct a table of 7 columns.
(a) Label the 7 columns in exactly the same cells as below:
2
3
B
C
Price of item (£)
LCB
UCB
D
Nº of items sold
Fi
E
F
G
H
Xi
FiXi
FiXi^2
Cu Freq
9
© Pearson Education Limited 2002
Burton, Carrol and Wall 2nd edition – Instructor’s Manual
(b) In the column headed LCB enter the values for the lower class price boundary.
(c) In the column headed UCB enter the values for the upper class price.
(d) In the column headed Fi enter the data for the number of items sold.
(e) Calculate the class mid-point (Xi) – sum the lower and upper class boundaries
then divide by 2 which is equivalent to the average of the upper and lower class
boundaries.
In cell E4: ........................................... =(B4+C4)/2
(f) Calculate F iX i – (e.g. multiply the first frequency value (F1) by the first class
mid-point (X1).
In cell F4: ........................................... =D4*E4
(g) Calculate F iX i2 – remember that only the X value is squared, so multiply the first
class mid-point (F1) by the first value of (F1 X1).
In cell G4: ........................................... =E4*F4
(h) Highlight cells E4:G4.
X2…X8;
(i) Calculate the remaining values for:
F2X2…F8X8;
F2X22…F8X82.
Copy the formulae to the bottom row of your table (row 8).
2 Cumulative frequency.
(a) Start the cumulative frequency
In cell H4: ........................................... =D4
(b) Add each new frequency to the previous frequencies.
In cell H5: ........................................... =H4+D5
Copy the formula in cell H5 to the bottom row of your data.
3 Determine the sums.
You should recall from the textbook that we have constructed the table to find the
sum of three columns (F i), (FiXi) and (FiXi2).
(a) Calculate the sum of the frequency data (F i).
(i)
Make cell D9 the active cell.
(ii)
Click on the AutoSum icon on the Standard Toolbar .........
(iii) A shimmering box will appear around the cells that Excel believes to be
the correct range and the range reference is entered automatically into the
cell.
10
© Pearson Education Limited 2002
Burton, Carrol and Wall 2nd edition – Instructor’s Manual
(iv) If the range is correct, press Enter. If the range is incorrect, amend it, either
with the mouse or type the correct range.
(b) Calculate the sum of FiX i and FiXi2.
(i)
Make cell F9 and G9 active.
(ii)
Click on the AutoSum icon.
Your results should appear as:
2
3
4
5
6
7
8
9
B
C
D
Price of item (£) Nº of items sold
LCB
UCB
Fi
1.5
2.5
15
2.5
3.5
2
3.5
4.5
19
4.5
5.5
10
5.5
6.5
14
60
I have used the Border icon
of the spreadsheet.
E
F
Xi
FiXi
2
3
4
5
6
G
30
6
76
50
84
246
H
FiXi^2
Cu Freq
60
15
18
17
304
36
250
46
504
60
1136
on the formatting toolbar to improve the readability
2.2 Calculating the descriptive statistics of grouped data
The arithmetic mean
F X
F
Mean 
i
i
i
From our table we know that FiX i is contained in cell F9 and that F i is contained in
cell D9. Therefore, the value for the mean is obtained by simply dividing cell F9 by D9.
Mean 
F X
F
i
i

i
F 9 246

D9 60
In Excel the above formula equates to: =F9/D9
Thus, in an empty cell (I have used C12) calculate the mean price.
You should find that the mean price is £4.10.
11
© Pearson Education Limited 2002
Burton, Carrol and Wall 2nd edition – Instructor’s Manual
The variance
F X
Variance 
F
i
2
i
i
  Fi X i

 F
i





2
Once again we have all the above values – FiX i2 is in cell G9 and F i is in cell D9.
The expression to the right of the minus sign is the same as the mean squared.
F X
Variance 
F
i
2
 X 
2
i
i
Using our table (B2:H8) in Excel this formula becomes:
In cell C13: ....................................................... =G9/D9-C12^2
The result is 2.123333333 giving a variance of 2.12 square £s. In the business context,
this calculation has little use and is rarely interpreted. However, it does form the basis
of the standard deviation.
The standard deviation
The standard deviation is simply the square root of the variance:
Standard deviation 
F X
F
i
i
2
i
  Fi X i

 F
i





2
We can use the result of the variance calculation to obtain the standard deviation:
In cell C14: ................................................ =SQRT(C13)
OR
In cell C14: ................................................ =C13^0.5
resulting in 1.4571662.
Subscript notation
Many students of quantitative methods find the subscript notation confusing so, using
the current example – the table of prices, let us examine the concept.
We have five classes or groups, so j = 5. If we number each class 1 to 5 then:
when j =1:
when j =2:
when j =3:
the class is £1.5 to 2.5;
the class is £2.5–3.5;
the class is £3.5–4.5;
12
© Pearson Education Limited 2002
Burton, Carrol and Wall 2nd edition – Instructor’s Manual
when j =4:
when j =5:
the class is £4.5–5.5;
the class is £5.5–6.5.
Thus, when we consider the lower class boundary (LCB) of the fourth class we can
express it in mathematical shorthand as LCB4. Additionally, when we have the notation
j–1 then if j=4, j –1 must be 3, or the third class.
The median and the quartiles
For grouped data, the formula for calculating the median is:
number of observatio ns to median position
number of observatio ns in the median class interval
Median = LCB + class width 
Often, we will also need to calculate the upper and lower quartiles and, since the
median is also the second quartile, the formula can be changed to:
number of observatio ns to quartile position
number of observatio ns in the quartile class interval
Quartile = LCB + class width 
The number of observations to the quartile position is found by subtracting the previous
cumulative frequency (CF):
Quartile position – previous CF
To express this mathematically:
 quartile position  CF j 1 

Quartile = LCB j + class width j 


F
j


where: j = the number of the class interval.
As with ungrouped data, the first step in calculating the quartiles (the median being
the second quartile or Q2) is to find the quartile positions using the following formulae:
1
Median  Q2  (n  1)
2
3
Upper quartile  Q3  ( n  1)
4
1
Lower quartile  Q1  (n  1)
4
We have found that it is easier to use decimal values rather than fractions in Excel, so
we can find the quartile position as follows:
13
© Pearson Education Limited 2002
Burton, Carrol and Wall 2nd edition – Instructor’s Manual
Q2  0.5(Fi  1)
Q3  0.75(Fi  1)
Q1  0.25(Fi  1)
1 Find the quartile positions.
(a) Construct a table in cells E11:G16:
E
11
12
13
14
15
16
Median
Q3
Q1
Iqrange
Q dev
F
G
Position VALUE
30.5 4.210526
45.75 5.482143
15.25
1.625
3.857143
1.928571
(b) Calculate the quartile positions:
In cell F12........................................... =0.5*($D$9+1)
Multiplying Fi + 1 (the contents of cell D9 plus 1, i.e. 60 + 1) by 0.5 gives the
median position. Since D9 is an absolute reference, when the formula is copied
or moved, the contents of D9 (Fi) will always be used. Thus:
Copy the formula down to cell F14.
In cell F13 .......................................... amend 0.5 to 0.75
In cell F14 .......................................... amend 0.5 to 0.25
You should have the following values: the median position = 30.5;
Q3 position = 45.75;
Q1 position = 15.25.
2 Calculate the quartile values:
(a) Identify the class containing the quartile position.
To identify the class containing the quartile position we will use our cumulative
frequencies – previously calculated in column H.
We see from column H that the total number of observations up to the second
class (i.e. below 3.5) was 17. The total number of observations up to the third
class (i.e. below 4.5) was 36. As the median position is 30.5 it is higher than
17 but less than 36 and it must, therefore, occur in the third class (3.5 to 4.5) or
where j = 3.
(b) Calculate the median value where j=3.
 quartile position  CF31 

Median = LCB3 + class width 3 
F
3


14
© Pearson Education Limited 2002
Burton, Carrol and Wall 2nd edition – Instructor’s Manual
From your spreadsheet you will find that when j = 3:
Cell B6 contains: LCB3 = 3.5:
Class width3 = 1: (UCB3 – LCB3 = 4.5 – 3.5 =1)
Cell F12 contains: Quartile position = 30.5
Cell H5 contains: CF3–1 = 17
Cell D6 contains: F3 = 19
Substituting these values gives:
 30.5  17 
Median = 3.5 + 1

 19 
Since we are calculating the median value in cell G12 and considering
mathematical precedence, enter:
In cell G12: ......................................... =B6+1*(F12-H5)/D6
(c) Calculate the upper quartile (Q3).
(i)
Identify the class containing Q3 by using step (a) above.
(ii)
Calculate the value of Q3 by using step (b) above.
(d) Calculate the lower quartile (Q1).
(i)
Identify the class containing Q1 by using step (a) above.
(ii)
Calculate the value of Q1 by using step (b) above.
The interquartile range and quartile deviation
This measure is found in the same way as ungrouped data, i.e.
Interquartile range = Q3 – Q1
Since cell G13 contains the value pertaining to Q3 and cell G14 contains the lower
quartile value, the interquartile range is found:
In cell G15: ................................................ =G13-G14
and the quartile deviation, which is half the interquartile range, by:
In cell G16: ................................................ =G15/2
15
© Pearson Education Limited 2002
Burton, Carrol and Wall 2nd edition – Instructor’s Manual
Your final table should now resemble:
E
11
12
13
14
15
16
F
G
Position VALUE
30.5 4.210526
45.75
5.475
15.25
2.625
2.85
1.425
Median
Q3
Q1
Iqrange
Q dev
As the values are in pounds sterling, format rows 12 to 16 to 2 decimal places.
All the statistics for grouped data can now be interpreted as for ungrouped data.
Exercise 2.2 Consumer profile
For a company to devise an appropriate strategy, it needs to understand its customers.
The strategy of a fast food company is based on offering a higher quality of product than
the opposition but charging higher prices. Consequently, data on income, spending and
age are important factors if the company wishes to pursue this strategy. To this end, a
questionnaire is designed which is to produce a profile of the average consumer to
ascertain:
 how much they spend on the average fast food meal;
 their income bracket;
 the age of the typical customer.
The questionnaire addressed the first point by asking:
Approximately how much do you spend on an average fast food meal?
1
2
3
4
5
6
Less than £4
£4–£5
£5–£6
£6–£7
£7–£8
More than £8
16
© Pearson Education Limited 2002
Burton, Carrol and Wall 2nd edition – Instructor’s Manual
The questionnaire addressed the second point by asking:
What is your approximate family income before tax?
1
under £10,000
2
£10,000 to £14,999
3
£15,000 to £19,999
4
£20,000 to £24,999
5
£25,000 to £29,999
6
£30,000 to £34,999
7
£35,000 to £39,999
8
£40,000 and over
The questionnaire addressed the third point by asking:
What age are you?
15–19
20–24
25–29
30–34
35–39
40–49
50–59
60–69
70 or over
1
2
3
4
5
6
7
8
9
The respondent indicates their expenditure range by circling the appropriate number;
for example, if a customer spends an average of £6.50 they mark the number 4. On
examination of the data, you will find that the numbers 1 to 6 indicate the number of
each class and therefore equate to the subscript j. Thus, j = 1 to 6.
The answers from the all the completed questionnaires are then entered into a
spreadsheet for you to analyse. The raw data is saved under the file name Profile.
1 Load the file Profile from the network.
2 Use row 2 for your labels, starting at G2.
3 Enter the values of j in cells G3:G8.
4 Enter your lower and upper class boundaries in columns H and I.
5 Construct frequency distributions for expenditure using the COUNTIF function.
(a) In cell J3: .............................................. click on the Function Wizard and select
the COUNTIF function.
17
© Pearson Education Limited 2002
Burton, Carrol and Wall 2nd edition – Instructor’s Manual
(b) The second step dialogue box appears:
In the range box: ................................ specify where the raw data for
expenditure is stored (column A).
In the criteria box .............................. specify what you wish to count.
For this exercise we wish to count the
number of occurrences of 1, then 2 and
up to 6; or each occurrence of j, and
since j=1 is stored in cell G3, enter G3.
(c) Click on the
icon: .............. You should find that only four people
spent less than £4.
(d) Copy the function to include all the classes.
6 Calculate the measures of central location and dispersion of the distribution from
first principles (i.e. do not use the Function Wizard).
7 Calculate the measures of central location and dispersion of the income data.
8 Calculate the measures of central location and dispersion of the age data.
What are your conclusions from this analysis?
Exercise 2. 3
Load the file basedata. For this exercise we will use the data for the revenues of firms
A and B.
Comparing two values
The range
The range is the difference between the maximum and minimum value,
i.e. range = maximum – minimum.
In the construction of a frequency table, we used functions to calculate the range.
There, one cell was used to find the maximum value, another to find the minimum and a
third to calculate the difference between the two. This is a common technique because
the two statistics are routinely calculated as part of an overall analysis of a data set.
To extend our expertise with Excel, we will use only one cell to calculate the range
and examine a technique often called point and click. Point and click provides a quicker
and more efficient method of entering references into your formulae and functions by
using the mouse to point to the relevant cell or range then clicking with the left mouse
button. The reference is entered into the formula.
Note: After you have entered the opening bracket of your function, you can point to
the column heading relevant to your data (in this case column A), then click
with the left mouse button. The range A:A is entered as your argument.
18
© Pearson Education Limited 2002
Burton, Carrol and Wall 2nd edition – Instructor’s Manual
In an empty cell (perhaps G3) enter ................. =MAX(A:A)-MIN(A:A)
It should be no surprise to see that the range has the value £233.09.
The range is often used in conjunction with the arithmetic mean and mode and has
the advantage of being simple to calculate. It is frequently used in the stock market to
show the variability of share prices over a given period. For example, if two companies
have a similar average share price but the range statistic of one company is greater, then
the larger range indicates greater volatility in share price, i.e. it is less stable.
By using the range A:A, instead of A2:A107, flexibility is added to your analysis.
This technique allows you to update (or amend) errors in your raw data without having
to change the arguments in your formula. For example, change the value in any cell in
column A to 1500 and you will see that the range changes automatically to 1067.46.
Restore your data set to its original value by clicking on the undo icon
– it
reverts to £233.09.
From this demonstration you should now observe a major limitation of using the
range as a measure of spread – the statistic is easily distorted by one value. Generally, it
is advisable not to use the range as the only measure of spread, but can be informative if
used as a supplement to other measures of spread such as the standard deviation or
semi-interquartile range.
Remember that labels improve the clarity of your work.
The interquartile range
The interquartile range is the absolute difference between the upper quartile (Q3 … the
three-quarter point) and the lower quartile (Q1 … quarter point). Expressed symbolically
we have:
Interquartile range = Q3 – Q1
The calculation of the interquartile range uses a similar approach to that of the median,
except that the data array is divided into four equal portions or quartiles instead of two.
The values forming the lowest 25% and the highest 25% of the array are ignored –
leaving only the central 50%.
Once the data set was sorted into an array, the median divided the array into two
equal portions. Thus, exactly half the data values were less than the median value
(547.91) and exactly half were greater. Figure 2.1 shows that the interquartile range
contains only the middle 50% of the array.
19
© Pearson Education Limited 2002
Burton, Carrol and Wall 2nd edition – Instructor’s Manual
figure 2.1 The relationship of the quartiles
We will examine two procedures to calculate the interquartile range – both similar to the
methods used to find the range:
 Method 1 – calculating the quartiles
1 In cell G4 find the upper quartile (Q3) ..................... =QUARTILE(A:A,3)
2 In cell G5 find the lower quartile (Q1) ..................... =QUARTILE(A:A,1)
3 In cell G6 find the interquartile range (Q3 – Q1)...... =G4-G5

Method 2 – combining functions
In cell G7 enter: ............................................ =QUARTILE(A:A,3)-QUARTILE(A:A,1)
Either method should provide the same result of 75.97.
As the interquartile range is only slightly affected by extreme scores, it provides a
good measure of spread for skewed distributions but is seldom used for data that are
approximately normally distributed.
For a full description and syntax of the QUARTILE function use the help facility;
then add its contents to your library of functions.
Semi-interquartile range
The semi-interquartile range, or quartile deviation, is simply half the interquartile range
and has similar advantages and disadvantages as the interquartile range. The formula for
semi-interquartile range is therefore:
Interquart ile range 
Q3  Q1
2
20
© Pearson Education Limited 2002
Burton, Carrol and Wall 2nd edition – Instructor’s Manual
In Excel this is calculated as: ............ =(QUARTILE(A:A,3)- QUARTILE(A:A,1))/2
You will note that an extra set of brackets have been used to ensure that the numerator is
calculated before the denominator.
Now calculate the quartile deviation in cell G8.
Continuity correction
Excel’s QUARTILE function ignores continuity correction so, strictly speaking, we
should make slight adjustments when finding the quartile values of a distribution (see
page 13 in your textbook). However, for most business purposes the value returned by
the function is adequate. Figure 2.2 illustrates the above calculations.
figure 2.2: Measuring the dispersion by comparing two points
Comparing all the values
When using only two points in a data set to measure of dispersion then the odd high or
low value gives a false impression. A better approach, therefore, would be to use every
value in the data set then compare it against some benchmark. The general approach is
to use the arithmetic mean as a benchmark and then calculate how far each variable is
from the benchmark. In other words, we calculate how far a value deviates, or varies,
from the mean. The sum of squared deviations of scores from their mean is lower than
their squared deviations from any other number.
Expressed mathematically: X i  X , where Xi are the variables and X is the
arithmetic mean. Therefore, if there are 1000 variables in a data set of one column, we
must first calculate the arithmetic mean of the data and then find the difference from the
mean of all 1000 variables. Fortunately, the use of a spreadsheet means we need only
21
© Pearson Education Limited 2002
Burton, Carrol and Wall 2nd edition – Instructor’s Manual
insert the formula once and then copy it down the column to make the other 999
calculations – the spreadsheet makes the appropriate adjustments to the formulae.
However, before we can exploit the spreadsheet’s capabilities we need to understand
how formulae and functions are adjusted when they are moved or copied.
Types of reference
When a formula is copied from one location to another in the worksheet, Excel adjusts
the cell references in the formula relative to their new positions. It is, therefore,
important to select the correct type of reference. Excel recognizes three types of
references:
1 relative;
2 absolute;
3 mixed.
Relative references
A relative reference can be compared with giving someone directions that explain how
to reach a location from a specific starting point. For example, if we were to say ‘Mrs.
Smith’s home is the third house down from this one on the other side of the road’ then it
is only valid from one location. Therefore, if you move location the directions will need
to be different.
If a formula containing a relative reference is copied, then that relationship is also
copied. For example, if cell B9 contains the formula =B4+B5+B6+B7 and is then
copied to cell C9, Excel would automatically change the formula to =C4+C5+C6+C7.
Absolute references
Frequently, you will not want the cell references to be adjusted when a formula is copied
– thus the reference will need to be made absolute. Using an absolute reference is
similar to providing a street address – ‘Mrs Smith lives at 19 Railway Cutting’. In this
form of reference, the final destination will always be that particular house in Railway
Cutting – no matter where the directions are given from.
In Excel, an absolute reference is designated by adding a dollar sign ($) before the
column letter and row number. A reference such as $D$4 will always refer to cell D4 no
matter where the formula is copied to. For example, if the formula in B10 is =B8/$D$4
and then copied to cell C10, the formula will become =C8/$D$4.
Mixed references
A mixed reference is designated by adding a dollar sign ($) before the column letter or
row number. For example, in the mixed reference $A4 the column reference ($A) is
absolute but the row reference (4) is relative.
22
© Pearson Education Limited 2002
Burton, Carrol and Wall 2nd edition – Instructor’s Manual
Absolute vs relative:
A relative reference:
is a cell reference in a formula that is adjusted when the
formula is copied.
An absolute reference: is a cell reference in a formula that does not change when
copied to a new location.
A mixed reference:
is a reference that is only partially absolute, such as A$2 or
$A2. When a formula that uses a mixed reference is copied to
another cell, only part of the cell reference is adjusted.
Changing the reference type
1
Select the cell where the results of your calculation are to appear.
2
Enter the formula containing the reference.
3
Position the cursor within the relevant reference in the formula bar.
4
Press the F4 key once and a $ sign will appear with the cell reference ($A$l).
Each time you press the F4 key the reference changes in the following order:
5
Relative (A1); Absolute ($A$1); Mixed (A$1); Mixed
Relative(A1).
($A1);
Copying formulae using the mouse and standard toolbar
1
Select the cell containing the formula that you wish to copy.
2
Click on the copy icon
on the standard toolbar – a shimmering outline
will appear around the selected cells.
3
Select the cell(s) into which you wish to copy the formula.
4
Click on the paste icon
on the standard toolbar.
Copying formulae by dragging
1
Select the cell containing the formula that you wish to copy (make it active).
2
Move to the cell handle – the cursor changes to a thin black cross.
3
Using the mouse, drag the cursor to the last cell in the range.
4
You will find that you will only be able to copy the formula horizontally or
vertically. To move diagonally, copy in one direction first (either horizontally
or vertically) and then drag in the other direction.
23
© Pearson Education Limited 2002
Burton, Carrol and Wall 2nd edition – Instructor’s Manual
We will continue using the revenue data of firm A in the file basedata. Your
spreadsheet should contain your earlier measures of dispersion.
To understand the logic involved in variation, we will calculate from first principles
then examine the various functions offered by the spreadsheet.
So add 4 extra rows above row 2.
Inserting rows or columns
Inserting rows
1
Click on the row heading (in this case row 2).
2
Select Insert on the menu bar.
3
On the drop-down menu: select Rows.
4
A new row will be added ABOVE the selected row.
5
Should you need to add further rows, hold down the control key and press the
Y key for each additional row.
Inserting columns
1
Click on the column heading.
2
Select Insert on the menu bar.
3
On the drop-down menu: select Columns.
4
A new column will be added to the LEFT of the selected row.
5
Should you need to add further columns, hold down the control key and press
the Y key for each additional column.
Now we can move on to calculating the remaining measures of dispersion.
Mean deviation
The mean deviation is simply the arithmetic mean of all the deviations. It is expressed in
mathematical symbols as:
Mean deviation 
X
i
X
n
Which is translated as: sum () all the absolute deviations X i  X and divide by the
number of deviations (n). The straight brackets (modulus) specify that the sign is
ignored, i.e., an absolute value.
24
© Pearson Education Limited 2002
Burton, Carrol and Wall 2nd edition – Instructor’s Manual
1.
Calculate X (arithmetic mean) of the raw data.
2.
In cell G14: ........................................... Use the AVERAGE function to
calculate the mean of your raw data.
Calculate all the absolute deviations X i  X .
Calculate the first absolute deviation from the mean:
In cell B7 enter:..................................... =ABS(A7-$G$14)
This is equivalent to: 480.46  542.2921 . The absolute function ensures the
returned value is positive.
Copy the formula down to cell B111.
3.
Since A7 is a relative reference, as it is copied down it becomes A8 then A9 up
to A111. However, since G14 is an absolute reference, denoted by the $’s, it
will always refer to G14 – the mean.
In cell B6 label your new data set: ........ Abs Devs
Calculate the sum of all the absolute deviations.
4.
In cell B4:.............................................. Use the Sum function to find the sum of
all the deviations (contained in cells
B7:B111).
Find the mean of the deviations.
In cell G15: ........................................... Use the COUNT function to find n.
In cell G17 enter: .................................. =B4/G15
X
Since B4 equals  X i  X and G15 equals n, G17 
X
 42.35 .
n
As a result of this statistic we can now say that on average the revenue of Firm A is
£542.29 plus or minus £42.35.
The function =AVEDEV(A:A) returns the mean deviation in one cell and thus does
away with the need for the above four steps.
Variance
 X
Variance 
1
 X
2
i
n
Calculate X (arithmetic mean) of the raw data.
We already have this value in cell G14
25
© Pearson Education Limited 2002
i
Burton, Carrol and Wall 2nd edition – Instructor’s Manual
2
Calculate the square of each deviation X i  X  .
2
Calculate the first square deviation from the mean:
In cell C7 enter: .................................... =(A7-$G$14)^2
The ^ means to the power of.
Here we have subtracted the mean of the raw data from the first variable (X1) and
squared the result
Copy the formula down to cell C111.
3.
Thus we have subtracted the mean of the raw data from the each successive
variable (X2, X3, X4… X105), squaring each result.
In cell C6 label your new data set: Sq Devs
Calculate the sum of all the squared deviations.
4.
In cell C4: ............................................. Use the Sum function to find the sum of
all the squared deviations (contained in
cells C7:C111).
Find the mean of the deviations.
Use cell G15 in which you have already found n.
In cell G18 enter: .................................. =C4/G15
Since C4 equals
G18 
X
i
n
 X
 X
 X  and G15 equals n, then,
2
i
 2687.87 to 2 decimal places .
From our calculation we can say that the variance of Firm A’s revenue is £2687.872 –
which has little meaning. As the result is given in square units, the variance is rarely
used in business statistics.
The function =VARP(A:A) returns the variance.
Standard deviation
Although variance is a useful measure, results measured in squares of numbers make it
tricky to relate the variance to the original data. This problem is easily remedied by
simply finding the square root of the variance and calling the measure standard
deviation.
The standard deviation is the most commonly used measure of dispersion. If a data
set contains an odd extreme high and/or low value, the standard deviation imparts a
more practical measurement than the range. It is also mathematically flexible e.g., many
formulae in inferential statistics use the standard deviation.
26
© Pearson Education Limited 2002
Burton, Carrol and Wall 2nd edition – Instructor’s Manual
The formula for calculating the standard deviation is:
Standard deviation 
 X
 X
2
i
n
which is the square root of the variance. Consequently, we follow the same procedure
for calculating the variance but, in addition, we can either use the square root function or
raise the original equation to the power of 0.5 as follows:
Either,
In cell G19 enter: ....................................... =SQRT(C4/G15)
or,
In cell G19 enter: ....................................... =(C4/G15)^0.5
Both methods produce the same result of 51.84464224, as would the function
=STDEVP(A:A).
Co-efficient of variation
Where two data sets with markedly different means are compared, the standard
deviation is used in conjunction with the mean to calculate the coefficient of variation
(see page 46 of the textbook).
The standard deviation is not a good measure of spread in highly-skewed
distributions and should be supplemented in those cases by the semi-interquartile range.
Now calculate all the measures of dispersion for firm B on Sheet 2.
You should now be capable of using Excel to illustrate your data and to calculate the
measures of central tendency and of dispersion.
Exercise 2.4
Practice your techniques by examining the data on Sheet 3 of the file Basedata, which
shows the output in kilograms of two production machines measured over 84 shifts. Use
your illustrative and statistical techniques to decide if one machine performs better than
the other.
You need to calculate all the statistical measures you have been taught to date and
then decide on the most appropriate measure of central tendency and dispersion,
depending on the way the data is distributed.
27
© Pearson Education Limited 2002
Burton, Carrol and Wall 2nd edition – Instructor’s Manual
Regression and correlation
By the end of this chapter you should be able to:
 calculate the regression equation:

calculate the co-efficient of determination (R2).
3.1 Regression analysis
Spreadsheets or statistical packages, with their abundance of functions, will provide all
the necessary statistics simply and quickly. However, an understanding of the basic
principles will provide a foundation for developing regression further. Consequently,
we will start by using the first principles of the coding formula (see textbook, p. 55) to
calculate the necessary statistics.
The data below is based upon Luigi’s ice cream business and is used to find the
regression equation or line of best fit using a spreadsheet. In this example, it is assumed
that the daily temperature influences the number of ice creams sold, as follows:
Temp (ºC)
15
16
20
18
21
19
Sales
272
282
315
289
320
301
table 3.1: Papa Luigi’s Ices
28
© Pearson Education Limited 2002
Burton, Carrol and Wall 2nd edition – Instructor’s Manual
Scatter diagram showing Sales to Temperature
350
300
Sales
250
200
150
100
50
0
0
5
10
15
20
25
Temperature
figure 3.1 Scatter diagram of Papa Luigi’s Ices
A scatter diagram using the data in Table 3.1 gives the graph shown in Figure 3.1.
It is obvious from the diagram that the six points do not lie on one straight line.
However, it is reasonable to try to obtain the straight line that comes as close as possible
to as many of the points as possible. This could be attempted by drawing a straight line
onto the graph that appears to come as close to the points as possible. Such an approach
would plainly be quite subjective, with different people drawing (slightly) different
straight lines.
This can be achieved mathematically by calculating the parameters of the straight line
(m and c) that comes closest to these points. A technique to minimise the sum of the
squared deviations of the observations from the line is used. For this reason, it is called
the method of least squares – using an approach similar to that used in the calculations
of grouped data.
Exercise 3.1 Calculating the regression equation from first principles
Remember that we require an equation such that:
Y = mX + c
We use the coding formula to obtain values for m and c of the linear equation linking
Sales to Temperature. First, calculate the slope, m.
29
© Pearson Education Limited 2002
Burton, Carrol and Wall 2nd edition – Instructor’s Manual
m
x y
x
i
i
2
i
Where:
xi  X i  X
yi  Yi  Y
This value is then used to calculate c.
1 Assign 7 columns of your spreadsheet.
2 Label the columns:............................. X.... Y.... x ... y ... xy .... x^2 ....y^2
3 Enter the Temperature (X) and the Sales (Y) data into the appropriate columns.
4 Calculate X and Y in 2 empty cells above the X and Y values.
5 Complete the rest of the first row of the table. Appropriate use of the $ symbol will
allow the formulae to be copied down the length of the table.
(a) In the cell below the label x: ........ Enter a formula to find the value of x
where: x  X  X .
(Consider using a mixed reference.)
(b) In the cell below the label y: ........ Enter a formula to find the value of y
where: y  Y  Y .
(c) In the column labelled xy enter a formula which multiplies the calculated value
of x by the calculated value of y.
(d) In the column labelled x^2, square the values of x.
(e) Copy the above formula one cell to the right (into the column labelled y^2).
6 Complete the rest of the table.
(a) Highlight the cells containing the above 5 calculations.
(b) Copy to the bottom of the table.
7 Sum the last 3 columns.
Your spreadsheet should now resemble Table 3.2.
30
© Pearson Education Limited 2002
Burton, Carrol and Wall 2nd edition – Instructor’s Manual
B
1
2
3
4
5
6
7
8
9
10
11
C
D
Mean X
Mean Y
18.16667
296.5
X
Y
15
16
20
18
21
19
272
282
315
289
320
301
E
F
G
H
x
-3.166667
-2.166667
1.833333
-0.166667
2.833333
0.833333
y
-24.5
-14.5
18.5
-7.5
23.5
4.5
Sums
xy
77.58333
31.41667
33.91667
1.25
66.58333
3.75
214.5
x^2
10.02778
4.694444
3.361111
0.027778
8.027778
0.694444
26.83333
I
y^2
600.25
210.25
342.25
56.25
552.25
20.25
1781.5
table 3.2
The cell references that follow assume a layout as in Table 3.2. If your layout differs,
adjust the following cell references as necessary.
8 Format columns E:I to 4 decimal places.
9 Calculate the m value using the coding formula:
m
x y
x
i
i
2
i
(a) Label cell G1: .............................. m (Slope)
(b) In cell G2: .................................... Enter the formula to calculate the slope.
It should return the value 7.9938
Use this value in conjunction with X and Y to calculate c: where c  Y  mX .
X and Y have already been calculated.
10 Calculate c.
(a) Label cell H1: .............................. c (Intercept)
(b) In cell H2: .................................... Enter the intercept formula.
It should return the value 151.2796
11 Calculate R2 where:
 x y 

 x  y 
2
R2
i
2
i
Again the values for:
G11:I11 in Table 2).
i
2
i
 xy ,  x
2
and
y
2
have all been calculated (cells
(a) Label cell I1: ................................ R^2
31
© Pearson Education Limited 2002
Burton, Carrol and Wall 2nd edition – Instructor’s Manual
(b). In cell H2: ................................... Enter the intercept formula.
It should return the value 0.9625.
Interpretation
The value of m (in this case 7.9938) is also known as the X coefficient. So a negative
coefficient would show an inverse relationship: as one variable increases, the other
decreases.
However, the value of m is positive, demonstrating a positive relationship between
the two variables: as one increases so does the other. This is what we expected from
these two variables.
As R2 is 0.9625, which is greater than 0.9 and close to one (perfect correlation) there
is an extremely strong relationship between the two variables. It seems reasonable to
conclude that there is a strong positive relationship between sales of ice cream and the
daily temperature. We can also say that 96.25% of the sales of ice cream can be
explained by the daily temperature, whilst 3.75% is the result of other factors.
However, caution needs to be exercised when interpreting R2.
The coefficient measures the strength of the relationship between two variables – it
does not measure cause and effect.
The coefficient measures the strength of the linear relationship between the two
variables. A low value for R2, therefore, implies little evidence of a linear relationship
but the relationship could be curvilinear.
Exercise 3.2
Use a spreadsheet to solve self-check question 3.3 on page 58 of the key text, then
check your calculations with the answers on pages 330–331.
32
© Pearson Education Limited 2002
Burton, Carrol and Wall 2nd edition – Instructor’s Manual
Probability distributions
4.1 The Normal distribution and Z statistic
By the end of this chapter you should be able to:
 examine a data set, graphically, for a normal distribution;

standardize a data set which is normally distributed;

use Excel functions to:
convert Z-values to a probability;
convert a probability to a Z-value.
Calculating the area under the curve of the normal distribution can be achieved by
finding the Z-value (or Z-statistic).
The Z-statistic is simply the number of standard deviations of a variable from its
mean value. For values less than the mean, the sign of the Z-statistic is negative; for
values above the mean, the sign of the Z-statistic is positive.
The process of converting variables to a Z-statistic is known as transformation or
standardizing the normal distribution.
Standardizing a distribution uses the equation:
Z
Xi  

where: Z = the number of standard deviations from the mean
Xi = the value to be standardized
 = the mean
 = the standard deviation
Exercise 4.1
It is known from past experience that the life of a machine component is approximately
normally distributed with a mean () of 200 hours and a standard deviation () of 4
hours. Calculate the probability that an individual component, chosen at random has a
life of:
(a) at least 206 hours;
(b) less than 198 hours;
(c) between 204 and 208 hours.
33
© Pearson Education Limited 2002
Burton, Carrol and Wall 2nd edition – Instructor’s Manual
200
206
Solving the problem with Excel is straightforward where all we need to do is to
substitute the given values into the equation.
1 Calculate the Z-statistic for 206 hours.
(a) In cell B3 enter the label: ......................................... X-value
(b) In cell B4 enter the X value: .................................... 206
(c) In cell C3 enter the label:......................................... Z-value
(d) In cell C4 enter the following equation as an Excel formula:
Z
Xi  

where: Z = the number of standard deviations from the mean
Xi = the value to be standardized ............... 206
 = the mean ............................................. 200
 = the standard deviation ............................ 4
If you obtain the value 181 you have obviously made a mistake!
You have entered the formula: =206-200/4 which is incorrect.
Excel formulae follows mathematical precedent so you need to insert brackets
around the numerator, i.e.,
=(206-200)/4, which will return the correct Zvalue of 1.5.
We can now say that in this distribution with a mean of 200 and standard
deviation of 4, 206 has a Z-value of 1.5.
We could also say that 206 is situated 1.5 standard deviations away from the
mean and, since the value is positive, it is to the right of the mean.
34
© Pearson Education Limited 2002
Burton, Carrol and Wall 2nd edition – Instructor’s Manual
The STANDARDIZE(x,mean,standard_dev) function also calculates the Zstatistic where:
X is the value you wish to standardize.
Mean is the arithmetic mean of the distribution.
Standard_dev is the standard deviation of the distribution.
For a full description of the function, use the help facility.
(e) Click on cell C4, enter:
=STANDARDIZE(206,200,4)
Again you should obtain a Z-value of 1.5.
Since cell B4 contains the our X-value of 206 we could simply have
entered =STANDARDIZE(B4,200,4).
2 Convert the Z-statistic to a P value.
(a) Using first principles
(i) Use your tables to look up the probability of obtaining a value of 1.5. The
answer should be (0.9332).
As the tables give the area to the left, there is a 93.32% probability that
values will be 206 or less.
(ii) However, we want 206 or greater, and any value that is greater must be to
the right of the distribution:
Area = 1 – 0.9332 = 0.0668
(b) Using an Excel function
The NORMSDIST(z) function(where z = the Z-value) is equivalent to looking
up the Z-statistic in tables and also gives the area to the left of the distribution.
(i) In cell D3 enter the label: ...................... P-value
(ii) In cell D4 enter: ..................................... =1-NORMSDIST(1.5)
Since cell C4 contains our Z-statistic of 1.5 we could simply have entered
=NORMSDIST(C4)
The answer should be ............................ 0.066807 which gives a probability that 6.6807% of all values in the distribution will be greater than
206.
By subtracting NORMSDIST(z) from 1, we will obtain the probability to
the right of the X value (i.e. values greater than X).
Out of every 100 components, 6.68 (almost 7) have a life of at least 206 hours.
3 Calculate the probability for 198 hours.
35
© Pearson Education Limited 2002
Burton, Carrol and Wall 2nd edition – Instructor’s Manual
198
200
(a) In cell B5 enter the X value: .................................... 198
(b) In cell C5 enter: .............................................. Standardize the X value (see 1(e)
above)
You should obtain a z-value of -0.5.
(c) In cell D5 enter: .............................................. =NORMSDIST(C5)
You should obtain a probability of 0.308538.
Out of every 100 components, 30.85, or almost one-third, have a life of less than 198
hours.
Calculate the probability that an individual component, chosen at random, has a life of
between 204 and 208 hours
4 Calculate the probability for 204 to 208 hours.
204
208
(a) In cell B7 enter the higher X value: ......................... 208
(b) In cell C7 enter: .............................................. Standardise the X value
You should obtain a z-value of 2.
(c) In cell D7 enter: .............................................. =NORMSDIST(C7)
You should obtain a probability of 0.97725.
(d) In cell B8 enter the lower X value: .......................... 204
(e) In cell C8 enter: .............................................. Standardise the X value
36
© Pearson Education Limited 2002
Burton, Carrol and Wall 2nd edition – Instructor’s Manual
You should obtain a z-value of 1.
(f) In cell D8 enter: .............................................. =NORMSDIST(C8)
You should obtain a probability of 0.841345.
(g) In cell D9 subtract the lower p-value (0.841345 in cell D7) from the higher
(0.97725 in cell D8) – a result of 0.135905.
Therefore, 13.59 out of every 100 components have a life of between 204 and 208
hours.
Summary
Whenever we use Excel functions in place of tables the resulting value gives the area to
the left of the distribution. Therefore, if we need to find the value of:
X or less, use NORMSDIST(Z)
X or more, use 1-NORMSDIST(Z)
Whenever the probability between 2 values is required:
1 Find the probability of the higher value.
2 Find the probability of the lower value.
3 Subtract the lower probability from the higher.
4.2 Converting a probability into a Z-statistic
When using standard normal tables we find the Z-statistic from a probability, or  value, by:
(a) locating the relevant  -value in the body of the tables;
(b) in the appropriate row read off the z-score (shown in the row heading)
(c) in the appropriate column read off the decimal value shown in the column
heading and add to part (b) above.
Excel uses the NORMSINV function to convert probability to a Z-value as follows:

Syntax: NORMSINV(probability)
Probability is a probability corresponding to the normal distribution.

Remarks

If probability is non-numeric, NORMSINV returns the #VALUE! error value.

If probability < 0 or if probability > 1, NORMSINV returns the #NUM! error
value.
The result given is for the LEFT of the normal distribution.

37
© Pearson Education Limited 2002
Burton, Carrol and Wall 2nd edition – Instructor’s Manual
Exercise 4.2
1
What is the Z-value where P < 0.908789 ?
As the probability is less than (<), the result is the probability to the LEFT of
the normal distribution:
In a blank cell enter: ........................... =NORMSINV(0.908789)
The result equals
2
1.333335 (i.e. Z = 1.333335)
What is the Z-value where P > 0.235641?
As the probability is greater than (>), the result is the probability to the RIGHT of
the normal distribution, so we need to subtract the probability from 1:
In a blank cell enter: ........................... =NORMSINV(1-0.235641)
0.720395 (i.e. Z = 0.720395)
The result equals:
Exercise 4.3
Using Excel functions, find the value of Z1 for each of the following:
a)
P(Z > Z 1) = 0.3300;
b) P(Z < Z 1) = 0.9732;
c)
P(Z < Z 1) = 0.8212;
d) P(Z < Z 1) = 0.5478;
For answers see page 419 of your textbook.
Exercise 4.4
The life of a certain battery is known to be normally distributed with a mean of 30 hours
and a standard deviation of 30 minutes. If 1.7% of the batteries last longer than K hours,
find the value of K.
Solution
1 In a vertical column of 5 cells enter the labels , Z and K
(I have used cells D4:D8).
2 In the appropriate cells of column E enter the values for:
(a)  and  ;
(b)  as decimal (i.e. 0.017).
3 Find the Z-statistic of the probability ().
38
© Pearson Education Limited 2002
Burton, Carrol and Wall 2nd edition – Instructor’s Manual
As we are interested in 1.7% or longer, the probability given is the area to the right
of the distribution. Excel’s NORMSINV function gives the area to the left, thus we
must subtract the -value from 1. If you have used the same cells as I have then:
=NORMSINV(1-E6)
(a) In cell E7 enter:
4 Use the equation: So far we have known the specific value for the variable X and then
calculated the probabilities with which these values might occur. If we know the
probability and need to find the value of X associated with that probability then we
need to rearrange the equation to solve for K as follows:
Z
Z
Xi  

K

K  Z   
Therefore in cell E8 enter the above equation in Excel format to calculate the value
of K. Your results should resemble:
D
6



7
8
Z
K
4
5
E
30
0.5
0.017
2.120069
31.06003
5 Interpret your findings.
With a mean of 30 hours and a standard deviation of 30 minutes, 1.7% of batteries
last longer than 31.06 hours.
39
© Pearson Education Limited 2002