1 Course review, syllabus, etc. Chapter 1 – Introduction Chapter 2 – Graphical Techniques Quantitative Business Methods A First Course

2 Population and Sample PopulationSample Use parameters to summarize features Use statistics to summarize features Inference on the population from the sample

3 Some Important Definitions…… A ___________________(or universe) is the whole collection of things under consideration. A ______________ is a portion of the population selected for analysis. A PARAMETER is a summary measure computed to describe a characteristic of the population.  µ A STATISTIC is a summary measure computed to describe a characteristic of the sample.   Discuss examples…….. Ω

4 Statistical Methods Descriptive Statistics Inferential Statistics Collecting and describing data. Drawing conclusions and/or making decisions concerning a population based only on sample data.

5 Collect Data –e.g. Survey Present Data –e.g. Tables and graphs Characterize Data –e.g. Sample Mean = Descriptive Statistics

6 Inferential Statistics Estimation e.g. Estimate the population mean using the sample mean. Hypothesis Testing e.g. Test the claim that the population mean weight is 120 pounds. Drawing conclusion and/or making decisions concerning a population based on sample results.

7 1. Write the following in scientific notation (ex x 10 2 or 4.63E02) (you may use as many significant figures as you wish) A B Write the following numbers in standard notation (ie. Not in scientific notation) A E10 B x Perform the following calculations, using only your calculator (try to enter it all in to your calculator). 4. Perform the following calculation without using your calculator. Analytical Skills Inventory Exercise

8 Use the following information for problems 5-9. = 4.6 n = 10 iXiXi

9 1.A x 10 9 or 3.864E09 B x or 6.251E-06 2.A B A. 9 B

10 Graphical Descriptive Techniques Chapter 2

Introduction Descriptive statistics involves the arrangement, summary, and presentation of data, to enable meaningful interpretation, and to support decision making. Descriptive statistics methods make use of graphical techniques numerical descriptive measures. The methods presented apply to both the entire population the population sample

12 2.2Types of data and information A variable - a characteristic of population or sample that is of interest for us. Cereal choice Capital expenditure The waiting time for medical services Data - the actual values of variables Interval data are numerical observations Nominal data are categorical observations

13 Types of data - examples Interval data Age - income Age - income Weight gain Weight gain Nominal Person Marital status 1married 2single 3single.. Person Marital status 1married 2single 3single.. Computer Brand 1IBM 2Dell 3IBM.. Computer Brand 1IBM 2Dell 3IBM..

14 Types of data - examples Interval data Age - income Age - income Nominal data With nominal data, all we can do is, calculate the proportion of data that falls into each category. IBM Dell Compaq OtherTotal % 22% 16% 12% IBM Dell Compaq OtherTotal % 22% 16% 12% Weight gain Weight gain

Graphical Techniques for Interval Data Example 2.1Example 2.1: The monthly bills of new subscribers in the first month after signing on with a telephone company. Collect data Prepare a frequency distribution Draw a histogram

16 Largest observation Collect data (There are 200 data points Prepare a frequency distribution How many classes to use? Number of observations Number of classes Less then , ,000 – 5, , , More than 50, Class width = [Range] / [# of classes] [ ] / [8] = Largest observation Largest observation Smallest observation Smallest observation Smallest observation _______ observation _________ observation Example 2.1Example 2.1: Providing information

17 Draw a Histogram Example 2.1Example 2.1: Providing information

Bills Frequency What information can we extract from this histogram About half of all the bills are small 71+37= =32 A few bills are in the middle range Relatively, large number of large bills =60 Example 2.1Example 2.1: Providing information

19 It is often preferable to show the relative frequency (proportion) of observations falling into each class, rather than the frequency itself. Relative frequencies should be used when the population relative frequencies are studied comparing two or more histograms the number of observations of the samples studied are different Class relative frequency = Class frequency Total number of observations Class frequency Total number of observations Relative frequency

20 There are four typical shape characteristics Shapes of histograms

21 ______________ skewed Negatively skewed Shapes of histograms

22 A modal class is the one with the largest number of observations. A unimodal histogram The modal class Modal classes

23 Modal classes A bimodal histogram A modal class

24 Bell shaped histograms “_____________ ___________”

25 Example 2.3Example 2.3: Comparing students’ performance Students’ performance in two statistics classes. Different in their teaching emphasis  Class A – math analysis and development of theory.  Class B – applications and computer based analysis. The final mark was recorded. Draw histograms and interpret the results. Interpreting histograms

26 Interpreting histograms The mathematical emphasis creates two groups, and a larger spread.

27 Preliminary analysis. Original observations vs. histogram approach. Split each observation into two parts. There are several ways of doing that: Stem Leaf 4219 Stem Leaf 42 A stem and leaf display for Example 2.1 will use this method  Stem and Leaf Display Observation:

28 A stem and leaf display for Example 2.1 (See page 42 for ref) Stem-and-Leaf Display for Bills: unit = 1.0 1|2 represents | | (23) 2| | | | | | | | | | The length of each line represents the _________ of the class defined by the stem. Stem and Leaf Display SG Demo

29 Ogives } } Ogives are cumulative relative frequency distributions. Example continued SG Demo: Freq Tab

30

Graphical Techniques for Nominal data The only allowable calculation on nominal data is to count the frequency of each value of a variable. When the raw data can be naturally categorized in a meaningful manner, we can display frequencies by Bar charts – emphasize frequency of occurrences of the different categories. Pie chart – emphasize the proportion of occurrences of each category.

32 Marketing 25.3% Finance 20.6% General management 14.2% Other 11.1% Accounting 28.9% (28.9 /100)(360 0 ) = The Pie Chart Ex #2.4: The student placement office at a university wanted to determine the general areas of employment of last year school graduates.

33 Rectangles represent each category. The height of the rectangle represents the frequency. The base of the rectangle is arbitrary The Bar Chart SG Demo: Desc- Categ-Tab

Describing the Relationship Between Two Variables The relationship between two interval variables. Example 2.7 A real estate agent wants to study the relationship between house price and house size Twelve houses recently sold are sampled and the size and price recorded Use graphical technique to describe the relationship between size and price. Size Price ……………..

35 Solution The size (independent variable, X) affects the price (dependent variable, Y) We use Excel to create a scatter diagram 2.5 Describing the Relationship Between Two Variables Y X The greater the house size, the greater the price

36 Typical Patterns of Scatter Diagrams Positive linear relationship Negative linear relationship No relationship Negative nonlinear relationship This is a weak linear relationship. A non linear relationship seems to fit the data better. Nonlinear (concave) relationship

37 Graphing the Relationship Between Two Nominal Variables We create a contingency table. This table lists the frequency for each combination of values of the two variables. We can create a bar chart that represent the frequency of occurrence of each combination of values.

38 Example 2.8Example 2.8 (Data: 2.8a) To conduct an efficient advertisement campaign the relationship between occupation and newspapers readership is studied. The following table was created Contingency table