Two Population Means Hypothesis Testing and Confidence Intervals With Known Standard Deviations

SITUATION: 2 Populations Population 1 Mean =  1 St’d Dev. =  1 Population 2 Mean =  2 St’d Dev. =  2 Salaries in ChicagoSalaries in St. LouisWomen’s Test ScoresMen’s Test ScoresLakers AttendanceClippers AttendanceAnaheim SalesIrvine Sales

KEY ASSUMPTIONS Sampling is done from two populations. –Population 1 has mean µ 1 and variance σ 1 2. –Population 2 has mean µ 2 and variance σ 2 2. –A sample of size n 1 will be taken from population 1. –A sample of size n 2 will be taken from population 2. –Sampling is random and both samples are drawn independently. –Either the sample sizes will be large or the populations are assumed to be normally distribution.

The Problem  1 and  2 are unknown  1 and  2 may or may not be known (In this module we assume they are known.)OBJECTIVES Test whether  1 >  2 (by a certain amount) –or whether  1   2 Determine a confidence interval for the difference in the means:  1 -  2

Key Concepts About the Random Variable. is the difference in two sample means. mean 1.Its mean is the difference of the two individual means: variance 2.If the variables are independent (which we assumed), the variance (not the standard deviation) of the random variable of the differences = the sum (not the difference) of the two variances: standard deviation standard deviation 3.Thus its standard deviation is:standard deviation distribution 4.Its distribution is: Normal if σ 1 and σ 2 are known t if σ 1 and σ 2 are unknown

Hypothesis Test Statistics for Difference in Means, Known σ’s We will be performing hypothesis tests with null hypotheses, H 0, of the form: From the general form of a test statistic, the required test statistic will be: H 0 : µ 1 - µ 2 = v

Confidence Intervals for µ 1 - µ 2 Known σ’s Recall the general form of a confidence interval is: Thus when the σ’s are known this becomes: (Point Estimate) ± z α/2 (Appropriate Standard Error)

EXAMPLE Hypothesis Test:  1,  2 Known Test whether starting salaries for secretaries in Chicago are at least $5 more per week than those in St. Louis. GIVEN: Salaries assumed to be normal Standard Deviations known: Chicago $10; St. Louis $15 Sample Results Sampled 100 secretaries in Chicago; 75 secretaries in St. Louis Sample averages: Chicago - $550, St. Louis -$540

Hypothesis Test Hypothesis Test H 0 :  1 -  2 = 5 H A :  1 -  2 > 5 Use  =.05 Reject H 0 (Accept H A ) if z > z.05 = 1.645

Calculating z Remember

Conclusion Since 2.5 > –It can be concluded that the average starting salary for secretaries in Chicago is at least $5 per week greater than the average starting salary in St. Louis. The p-value: –The area above z= 2.5 on the normal curve = =.0062 –Since.0062 is low (compared to α), it can be concluded that the average starting salary for secretaries in Chicago is at least $5 per week greater than the average starting salary in St. Louis.

EXAMPLE EXAMPLE Confidence Interval:  1,  2 Known EXAMPLE Construct a 95% confidence for the difference in average between weekly starting salaries for secretaries in Chicago and St. Louis. $10 ± $3.92 $6.08 ↔ $13.92

Excel Approach Excel Approach Suppose, as shown on the next slide the data for Chicago is given in column A (A2:A101) and the data for St. Louis is given in column B (B2:B76). The analysis can be done using an entry from the Data Analysis Menu: z-test: Two Sample for Means

Go to Data Then Data Analysis Select z-Test: Two Sample For Means Go to Data Then DataAnalysis Go to Data Then Data Analysis Select z-Test: Two Sample For Means

For 1-tail tests, input columns so that the test is a “>” test. Enter Hypothesized Difference Enter Variances Not Standard Deviations Check Labels Enter Beginning Cell For Output

p-value for “>” test p-value for “  ” test

=(E4-F4)-NORMSINV(0.975)*SQRT(E5/E6+F5/F6) Highlight formula in cell E15—press F4. Drag to cell E16 and change “-” to “+”.

Estimating Sample Sizes Usual Assumptions: –Same sample size from each pop.: n 1 = n 2 = n –Standard deviations,      known Calculate n from the “±” part of the confidence interval for known  1 and  2

Example How many workers would have to be surveyed in Chicago and St. Louis to estimate the true average difference in starting weekly salary to within  $3?

Review Mean and standard deviation for  X 1 -  X 2 Assumptions for tests and confidence intervals z-tests for differences in means when  1 and  2 are known: –By Formula –By Excel data analysis tool Confidence intervals for differences in means when  1 and  2 are known: –By Formula –By Excel data analysis tool Estimating Sample Sizes