Economics 173 Business Statistics Lectures 1 Fall, 2001 Professor J. Petry

2 Introduction Purpose of Statistics is to pull out information from data –“without data, ours is just another opinion” –“without statistics, we are just another person on data overload” Because of its broad usage across disciplines, Statistics is probably the most useful course irrespective of major. –More data, properly analyzed allows for better decisions in personal as well as professional lives –Applicable in nearly all areas of business as well as social sciences –Greatly enhances credibility

3 Statistics as “Tool Chest” Different types of data, allow different types of analysis Quantitative data –values are real numbers, arithmetic calculations are valid Qualitative data –categorical data, values are arbitrary names of possible categories, calculations involve how many observations in each category Ranked data –categorical data, values must represent the ranked order of responses, calculations are based on an ordering process. Time series data –data collected across different points of time Cross-sectional data –data collected at a certain point in time

4 Statistics as “Tool Chest” Different objectives call for alternative tool usage Describe a single population Compare two populations Compare two or more populations Analyze relationship between two variables Analyze relationship among two or more variables By conclusion of Econ 172 & 173, you will have about 35 separate tools to select from depending upon your data type and objective

5 Describe a single population Compare two populations Compare two or more populations Analyze relationships between two variables Analyze relationships among two or more variables. Problem Objective?

6 Describe a single population Z- test & estimator of p Z- test & estimator of p Central location Variability t- test & estimator of  t- test & estimator of    - test & estimator of  2   - test & estimator of  2 Data type? QuantitativeQualitative TwoTwo or more Type of descriptive measurements? Number of categories?  2 goodness of fit test  2 goodness of fit test

7 Experimental design? Type of descriptive measurements? Compare two populations Data type? Sign test Sign test Central location Variability F- test & estimator of   2 /   2 F- test & estimator of   2 /   2 Experimental design? Continue Wilcoxon rank sum test Wilcoxon rank sum test Independent samples Matched pairs Number of categories Two Two or more Z - test & estimator of p 1 - p 2  2 -test of a contingency table Quantitative Ranked Qualitative Continue

8 Numerical Descriptive Measures Measures of central location –arithmetic mean, median, mode, (geometric mean) Measures of variability –range, variance, standard deviation, coefficient of variation Measures of association –covariance, coefficient of correlation

9 –This is the most popular and useful measure of central location Sum of the measurements Number of measurements Mean = Sample meanPopulation mean Sample sizePopulation size § Arithmetic mean Measures of Central Location Sum of the measurements Number of measurements Mean =

10 Example The mean of the sample of six measurements 7, 3, 9, -2, 4, 6 is given by Example Calculate the mean of 212, -46, 52, -14, 66 54

11 26,26,28,29,30,32,60,31 Odd number of observations 26,26,28,29,30,32,60 Example 4.4 Seven employee salaries were recorded (in 1000s) : 28, 60, 26, 32, 30, 26, 29. Find the median salary. –The median of a set of measurements is the value that falls in the middle when the measurements are arranged in order of magnitude. Suppose one employee’s salary of $31,000 was added to the group recorded before. Find the median salary. Even number of observations 26,26,28,29, 30,32,60,31 There are two middle values! First, sort the salaries. Then, locate the value in the middle First, sort the salaries. Then, locate the value s in the middle 26,26,28,29, 30,32,60, , § The median

12 –The mode of a set of measurements is the value that occurs most frequently. –Set of data may have one mode (or modal class), or two or more modes. The modal class § The mode

13 – Example The manager of a men’s store observes the waist size (in inches) of trousers sold yesterday: 31, 34, 36, 33, 28, 34, 30, 34, 32, 40. What is the modal value? This information seems valuable (for example, for the design of a new display in the store), much more than “ the median is 33.2 in.”. 34

14 Relationship among Mean, Median, and Mode If a distribution is symmetrical, the mean, median and mode coincide If a distribution is non symmetrical, and skewed to the left or to the right, the three measures differ. A positively skewed distribution (“skewed to the right”) Mean Median Mode

15 ` If a distribution is symmetrical, the mean, median and mode coincide If a distribution is non symmetrical, and skewed to the left or to the right, the three measures differ. A positively skewed distribution (“skewed to the right”) Mean Median Mode Mean Median Mode A negatively skewed distribution (“skewed to the left”)