1 BA 555 Practical Business Analysis Housekeeping Review of Statistics Exploring Data Sampling Distribution of a Statistic Confidence Interval Estimation Hypothesis Testing Agenda

2 Definition “Statistics” is the science of data. It involves collecting, classifying, summarizing, organizing, analyzing, and interpreting numerical information. We will learn how to make based on data

3 Fundamental Elements of Statistics A population is a set of units (usually people, objects, transactions, or events) that we are interested in studying. It is the totality of items or things under consideration. A sample is a subset of the units of a population. It is the portion of the population that is selected for analysis. A parameter is a numerical descriptive measure of a population. It is a summary measure that is computed to describe a characteristic of an entire population. A statistic is a numerical descriptive measure of a sample. It is a summary measure calculated from the observations in the sample.

4 Example A manufacturer of computer chips claims that less than 10% of his products are defective. When 1,000 chips were drawn from a large production, 7.5% were found to be defective. What is the population of interest? What is the sample? What is the parameter? What is the statistic? Does the value 10% refer to the parameter or to the statistic? Is the value 7.5% a parameter or a statistic?

5 Statistical Analysis (p.3)

6 Types of Data (p.2) Numerical (Quantitative) Data Regular numerical observations. Arithmetic calculations are meaningful. Age Household income Starting salary Categorical (Qualitative) Data Values are the (arbitrary) names of possible categories. Gender: Female = 1 vs Male = 0. College major

7 Employee Database (class website, EmployeeDB.sf3) QuantitativeQualitative

8 Describing Qualitative Data (p.4)

9 Summarizing Qualitative Data

10 Describing Quantitative Data (p.4) Graphical Methods

11 Quantitative Data: Histogram Histogram applet

12 Describing Quantitative Data (p.4) Descriptive/Summary Statistics

13 Guessing Correlations

14 Correlation: Be Careful Correlation value Scatter plot ? Correlation

15 Example Upper invisible line: 17.4Lower invisible line: 11.8

16 Statgraphics Plus (SG+) Demo (p.1) Questions to ask when describing and summarizing data: Where is the approximate center of the distribution? Are the observations close to one another, or are they widely dispersed? Is the distribution unimodal, bimodal, or multimodal? If there is more than one mode, where are the peaks, and where are the valleys? Is the distribution symmetric? If not, is it skewed? If symmetric, is it bell-shaped?

17 The Empirical Rule (p.5)

18 Example The average salary for employees with similar background/skills/etc. is about $120,000. Your salary is $122,000. Is it a big deal? Why or why not? What additional information is required to answer this question?

19 What to do next? Generalize the results from the empirical rule. Justify the use of the mound-shaped distribution.

20 Example: Warranty Level Mean = 30,000 miles STD = 5,000 miles Q1: If the level of warranty is set at 15,000 miles, about what % of tires will be returned under warranty? Q2: If we can accept that up to 2.5% of tires can be returned under warranty, what should be the new warranty level?

21 Example: Warranty Level Mean = 30,000 miles STD = 5,000 miles Q1: If the level of warranty is set at 12,000 miles, about what % of tires will be returned under the warranty? Q2: If we can accept that up to 3.0% of tires can be returned under warranty, what should be the warranty level?

22 Normal Probabilities

23 Sampling Distribution (p.6) The sampling distribution of a statistic is the probability distribution for all possible values of the statistic that results when random samples of size n are repeatedly drawn from the population. When the sample size is large, what is the sampling distribution of the sample mean / sample proportion / the difference of two samples means / the difference of two sample proportions?  NORMAL !!!

24 Central Limit Theorem (CLT) (p.6) Sample: X 1, X 2, …, X n

25 Central Limit Theorem (CLT) (p.6)CLT Sample: X 1, X 2, …, X n

26 Standard Deviations

27 Statistical Inference: Estimation Research Question: What is the parameter value? Sample of size n Population Tools (i.e., formulas): Point Estimator Interval Estimator

28 Confidence Interval Estimation (p.7)

29 Example A random sampling of a company’s monthly operating expenses for a sample of 12 months produced a sample mean of $5474 and a standard deviation of $764. Construct a 95% confidence interval for the company’s mean monthly expenses.

30 Statistical Inference: Hypothesis Testing Research Question: Is the claim supported? Sample of size n Population Tools (i.e., formulas): z or t statistic

31 Hypothesis Testing (p.9)

32 Example A bank has set up a customer service goal that the mean waiting time for its customers will be less than 2 minutes. The bank randomly samples 30 customers and finds that the sample mean is 100 seconds. Assuming that the sample is from a normal distribution and the standard deviation is 28 seconds, can the bank safely conclude that the population mean waiting time is less than 2 minutes?

33 Margin of Error (B) What does B tell us about the point estimator? How do we reduce the value of B?

34 Relations among B, n, and  B (margin of error)N (sample size) Confidence Level (e.g., 90%, 95%) How to reduce B?

35 Estimation in Practice Determine a confidence level (say, 95%). How good do you want the estimate to be? (define margin of error) Use formulas (p.8) to find out a sample size that satisfies pre-determined confidence level and margin of error.

36 Accuracy Gained by Increasing the Sample Size (p.8)