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

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 Chicago. Women’s Test Scores. Lakers Attendance. Anaheim Sales.

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

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

  2. SITUATION: 2 Populations Population 1 Mean = 1St’d Dev. = 1 Population 2 Mean = 2St’d Dev. = 2 Salaries in Chicago Women’s Test Scores Lakers Attendance Anaheim Sales Salaries in St. Louis Men’s Test Scores Clippers Attendance Mission Viejo Sales

  3. KEY ASSUMPTIONS Sampling is done from two populations. • Population 1 has mean µ1 and variance σ12. • Population 2 has mean µ2 and variance σ22. • A sample of size n1 will be taken from population 1. • A sample of size n2 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.

  4. 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

  5. Key Concepts About the Random Variable . • is the difference in two sample means. • Its mean is the difference of the two individual means: • 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: • Thus its standard deviation is: • Its distribution is: • Normal if σ1 and σ2 are known • t if σ1 and σ2 are unknown

  6. Hypothesis Test Statistics forDifference in Means, Known σ’s • We will be performing hypothesis tests with null hypotheses, H0, of the form: • From the general form of a test statistic, the required test statistic will be: H0: µ1 - µ2 = v

  7. Confidence Intervals for µ1 - µ2Known σ’s • Recall the general form of a confidence interval is: Thus when the σ’s are known this becomes: (Point Estimate) ± zα/2(Appropriate Standard Error)

  8. EXAMPLEHypothesis 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

  9. Hypothesis Test H0: 1 - 2 = 5 HA: 1 - 2 > 5 Use  = .05 Reject H0 (Accept HA) if z > z.05 = 1.645

  10. Calculating z Remember

  11. Conclusion • Since 2.5 > 1.645 • 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 = 1 - .9938 = .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.

  12. EXAMPLEConfidence Interval: 1, 2 Known • 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

  13. 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

  14. Go to Tools Data Analysis Go to Tools Data Analysis Select z-Test: Two Sample For Means

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

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

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

  18. Estimating Sample Sizes • Usual Assumptions: • Same sample size from each pop.: n1 = n2 = n • Standard deviations, s1, s2known • Calculate n from the “±” part of the confidence interval for known 1and 2

  19. 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?

  20. Review • Mean and standard deviation for X1 -X2 • Assumptions for tests and confidence intervals • z-tests for differences in means when 1 and 2 are known: • By hand • By Excel • Confidence intervals for differences in means when 1 and 2 are known: • By hand • By Excel • Estimating Sample Sizes

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