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Chapter 13

Chapter 13. Inference About Comparing 2 Populations. Comparing 2 Populations. Recall : Previously we looked at techniques to estimate & test parameters for ‘ 1 ’ population: Population Mean µ Population Proportion p Now : looking at ‘ 2 ’ populations

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Chapter 13

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  1. Chapter 13 Inference AboutComparing2 Populations

  2. Comparing 2 Populations • Recall: Previously we looked at techniques to estimate & test parameters for ‘1’ population: • Population Mean µ • Population Proportionp • Now: looking at ‘2’populations • Focus: The difference between 2 means.

  3. Population 1 Sample, size: n1 Statistics: Parameters: Difference between 2 Means (1) • How test and estimate the difference between 2 population means ?  Draw random samples from each of 2 populations. • (Likewise, we consider for Pop 2) • Statistics : • unbiased and consistent estimator of µ1- µ2.

  4. Sampling Distribution of • is normally distributed if the original populations are normal –or– approximately normal • If the populations are non-normal and the sample sizes are large (n1, n2 > 30) • The expected value of is µ1- µ2 • The variance of is • The standard error is:

  5. Making Inferences About μ1-μ2 • In practice, the z statistic is rarely used since the population variances are unknown. • Instead we use a t-statistic • Consider 2 cases for the unknown population variances i.e. when we believe: • they are equal • they are Unequal ??

  6. “RambangMata…” – t stat μ1-μ2 UNEQUAL VARIANCE EQUAL VARIANCE T-Statistic with PVE

  7. “RambangMata…” – CI estimator μ1-μ2 UNEQUAL VARIANCE EQUAL VARIANCE With PVE

  8. Testing the Population Variances • Testing the Population Variances H0: σ12 / σ22 = 1 H1: σ12 / σ22 ≠ 1 • Test statistic: s12 / s22 • F-distributed • df ν1 = n1– 1 and ν2 = n2 −2. • Required condition: the same as t-test of µ1 - µ2  both populations are normally distributed • 2-tail test so the rejection region is Checking the equality OR

  9. Example 13.1 (Mutual Fund Profitability) (1) • Mutual funds can be purchased either directly or through brokers, who charge a fee. • Question: can investors do better by buying mutual funds directly compared to through brokers? • Randomly sampled the annual returns (AR) (Xm13-01) • acquired directly; • bought through brokers (net of all chargeable fees) • Can we conclude at the 5% significance level that directly-purchased mutual funds outperform mutual funds bought through brokers?  AR (directly-purchased) > AR (Broker)

  10. Example 13.1 (Mutual Fund Profitability) (2) • How? – Compares the population of mutual funds AR bought from direct and through broker • Parameter to be tested: the difference between 2 means  µ1- µ2. • Claim: AR (directly-purchased) > AR (Broker) • H1: µ1- µ2 > 0 & H0: µ1- µ2 = 0 • To decide which of the t-tests of µ1 - µ2 to apply we conduct the F-test of σ12/ σ22 • Reason: to check the equality of variance!!!

  11. Example 13.1 (Mutual Fund Profitability) (3) • From the data we calculated the following statistics. s12 = 37.49 and s22 = 43.34 n = 50 • Test statistic: F = 37.49/43.34 = 0.86 • H0: σ12 / σ22 = 1 &H1: σ12 / σ22 ≠ 1 • Rejection region: • Since F< then we fail to reject Ho • Hence, equal variance B-15 1.75

  12. Example 13.1 (Mutual Fund Profitability) (4) Click Data, Data Analysis, and F-Test Two Sample for Variances

  13. Example 13.1 (Mutual Fund Profitability) (5) • Test statistic is F = 0.86 • Excel outputs the one-tail p-value • Since a two-tail test, we double that value • Thus, the p-value is 2 X 0.3068 = 0.6136 • There is insufficient evidence to infer that the population variances differ  apply the equal-variances t-test of µ1- µ2

  14. Example 13.1 (Mutual Fund Profitability) (6) Click Data, Data Analysis, t-Test: Two-Sample Assuming Equal Variances

  15. Example 13.1 (Mutual Fund Profitability) (7)

  16. Example 13.1 (Mutual Fund Profitability) (8) • Test statistic is 2.29 & One-tail p-value is 0.0122 • p-value is small & the test statistic falls into the rejection region • Since H1: µ1- µ2 > 0 H0: µ1- µ2 = 0 • ……and p-value < then reject H0 • Conclusion: There is sufficient evidence to infer that on average directly-purchased mutual funds outperform broker-purchased mutual funds • i.e. AR (directly-purchased) > AR (Broker)

  17. Example 13.1 – CI Estimator (1) • 95% CI estimate of the difference between the AR on directly purchased mutual and otherwise • Apply the equal-variances estimator • Use the t-Estimate_2 Means (Eq-Var) worksheet in the Estimators workbook or manually Interpretation: We estimate that the return on directly purchased mutual funds is on average between 0.39 and 5.43 percentage points larger than broker-purchased mutual funds.

  18. Example 13.2 (CEO succession) (1) • What happens to the family-run business when the boss’s son or daughter takes over? • Does the business do better after the change if the new boss is the offspring of the owner or does the business do better when an outsider is made CEO? • n = 140 firms (1994 to 2002), 30% passed appointed offspring, 70% appointed outsider. • Data: Operating income scaled by assets in the year before and after the new CEO took over. Xm13-02 • Question: Does the effect of making an offspring CEO is different from the effect of hiring an outsider as CEO?

  19. Example 13.2 (CEO succession) (2) • Objective: to compare 2 populations • Population 1: Operating income of companies whose CEO is an offspring of the previous CEO Population 2: Operating income of companies whose CEO is an outsider • Data type = interval (operating incomes) • Testable Parameter = µ1- µ2 • Test proposition: µ1 different from µ2 µ1- µ2≠ 0. • H1: µ1- µ2 ≠ 0 H0: µ1- µ2 = 0 SL=5% • Offspring=42[30%X140) Outsider=98[70%X140] • 1st step: (un) equal-variances ?

  20. Example 13.2 (CEO succession) (3) • F-test of σ12/ σ22 s12 = 3.79 and s22 = 8.03 • Test statistic: F = 3.79/8.03 = 0.47 • Rejection region: • Or…. = 0.57 • Since then reject Hounequal var • Excel  Click Data, Data Analysis, and F-Test Two Sample for Variances 1.64

  21. Example 13.2 (CEO succession) (4) • F = 0.47 p-value = 2X0.0040 = 0.0080 < (0.025) • Reject Ho apply the unequal-variances t-test of µ1- µ2.

  22. Example 13.2 (CEO succession) (5) Click Data, Data Analysis, t-Test: Two-Sample Assuming Unequal Variances p-value = 0.0017 < α (0.025) Hence, Reject H0 We conclude that there is sufficient evidence to infer that the profitability differs between the 2 groups.

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