1 STAT 500 – Statistics for Managers STAT 500 Statistics for Managers

2 STAT 500 – Statistics for Managers Estimation Confidence Intervals

3 STAT 500 – Statistics for Managers Learning Objectives Define a probability interval. Calculate and interpret a probability interval for a specific random variable. Define a confidence interval. Calculate and interpret a confidence interval for the population mean.

4 STAT 500 – Statistics for Managers Learning Objectives (cont.) Calculate and interpret one-sided confidence intervals, and Apply confidence intervals to understand the mean of 0,1 population

5 STAT 500 – Statistics for Managers Factors Affecting Interval Width Data dispersion –Measured by  Sample size   X =  /  n Level of confidence (1 -  ) –Affects Z Intervals extend from  X - Z   X to  X + Z   X © T/Maker Co.

Confidence Interval Mean (  Unknown) Assumptions –Population standard deviation is unknown –Population must be normally distributed Use Student’s t distribution Confidence interval estimate

Student’s t Distribution 0 t (df = 5) Standard normal t (df = 13) Bell- shaped Symmetric ‘Fatter’ tails Note: As d.f. approach 120, Z and t become very similar

Student’s t Table Assume: n = 3 df= n - 1 = 2  =.10  /2 = t values  / 2.05

Degrees of Freedom Number of observations that are free to vary after sample statistic has been calculated Example –Sum of 3 numbers is 6 X 1 = 1 (or any number) X 2 = 2 (or any number) X 3 = 3 (cannot vary) Sum = 6 degrees of freedom = n -1 = 3 -1 = 2

Estimation Example Mean (  Unknown) A random sample of n = 25 has  X = 50 & S = 8. Set up a 95% confidence interval estimate for .

11 STAT 500 – Statistics for Managers You’re a time study analyst in manufacturing. You’ve recorded the following task times (min.): 3.6, 4.2, 4.0, 3.5, 3.8, 3.1. What is the 90% confidence interval estimate of the population mean task time?

12 STAT 500 – Statistics for Managers  X = 3.7 S = n = 6, df = n - 1 = = 5 S /  n = /  6 = t.05,5 = (2.015)(1.592)  (2.015)(1.592)  6.908

Confidence Interval for the Mean The Middle of the C.I. is the Sample Mean The Width of the C.I. is Determined by: –The Confidence Desired Higher Confidence  Wider Interval -z.025 z.025  -z.005 z.005 

14 STAT 500 – Statistics for Managers Confidence Interval for the Mean The Middle of the C.I. is the Sample Mean The Width of the C.I. is Determined by: –The Confidence Desired Higher Confidence  Wider Interval –The Variability of the Data: Standard Deviation Greater Variability  Wider Interval

15 STAT 500 – Statistics for Managers Confidence Interval for the Mean The Middle of the C.I. is the Sample Mean The Width of the C.I. is Determined by: –The Confidence Desired Higher Confidence  Wider Interval –The Variability of the Data: Standard Deviation Greater Variability  Wider Interval –The Sample Size, n Larger Sample Size  Narrower Interval

16 STAT 500 – Statistics for Managers Two Common Interpretations If many samples were taken and a 95% confidence interval computed from each, the population mean would be contained in about 95% of them. With 95% confidence, the population mean lies within the 95% confidence interval endpoints.

17 STAT 500 – Statistics for Managers Confidence Interval for the Proportion

18 STAT 500 – Statistics for Managers How to compute a confidence interval for a population proportion

19 STAT 500 – Statistics for Managers Pre-Election Poll in Anywhere, USA For prop 1565% Against prop 1535% What is the percentage of all voters who favor Prop 15 ? How much uncertainty is there in the estimated percentage ?

20 STAT 500 – Statistics for Managers Population Parameter p  = ??? 1 Inductive Inference

21 STAT 500 – Statistics for Managers Population Parameter Sample Statistic p = ??? p s = Inductive Inference

Population Parameter Sample Statistical Analysis Statistic Inference p  = ??? p s = Inductive Inference

Binomial Probabilities : Application to Opinion Polling Assumptions n independent repeatable trials one of two mutually exclusive outcomes p = Pr(success) remains constant (the same) on each trial (Population Size is VERY Large) k = # of successes in the n trials Chance Situation or “Trial” Repeated n Times

24 STAT 500 – Statistics for Managers Central Limit Theorem for Binomial Proportions If Independent Observations Sample Size is Sufficiently Large Then

Confidence Interval Derivation z-z 1-  0

1. Estimate p : Confidence Interval for a Proportion

1. Estimate p : 2. Estimate SE : Confidence Interval for a Proportion

1. Estimate p : 2. Estimate SE : 3. Obtain the z Value (Normal Table) e.g. 95% Confidence Interval, z = 1.96 Confidence Interval for a Proportion

1. Estimate p : 2. Estimate SE : 3. Obtain the z Value (Normal Table) e.g. 95% Confidence Interval, z = Calculate Confidence Interval for a Proportion

30 STAT 500 – Statistics for Managers Latest Poll Suppose n = 1,500 and p s =.55 95% Confidence Interval

31 STAT 500 – Statistics for Managers THANK YOU