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TWO SAMPLE STATISTICAL INFERENCES Additional material 1 How can we compare two means? • Suppose that we have two populations. Furthermore, suppose that the first population has a mean 1 and the second population a mean 2. • We would like to use inferential statistics to compare 1 and 2 by using either a confidence interval and/or a significance test. 2 Independent Samples • Independent Samples occur when observations are gathered independently from two different populations. • For example, suppose we want to compare starting salaries for males and females at a certain bank. We collect a random sample of males and a random sample of females and measure their starting salaries. 3 Paired Samples • Paired Samples occur when observations are gathered in pairs, where each pair consists of an observation from population 1 and an observation from population 2, and these two observations are related. • Although there are other types of paired samples, a common type is a before-after sample. 4 Paired Samples • For example, suppose we want to determine the effectiveness of a new drug in reducing blood pressure. • For each individual in the sample a measurement is taken prior to the medication and another measurement is taken after using the medication for a fixed period of time. • These two observations (for each individual) are related. This constitutes a paired sample. 5 Outline • Two sample comparisons: • Paired Samples (dependent samples) • Independent Samples – Population Standard Deviations Not Assumed to be Equal (σ1≠ σ2) • Independent Samples – Population Standard Deviations Assumed to be Equal (σ1= σ2) • All methods assume random samples from normal populations with unknown means and unknown standard deviations. • For small sample sizes, it is critical that our observations are normal or approximately normal. • However, for sufficiently large sample sizes (from the C.L.T.) the assumptions of normality can be relaxed. 6 PAIRED SAMPLES 7 Paired (dependent) Samples 8 Paired (dependent) Samples 9 Paired (dependent) Samples t-score = di invttail(n-1,α/2) 10 Paired (dependent) Samples 11 INDEPENDENT TWO SAMPLES 12 Independent Two Samples • We gather independent and random samples from each population. 13 Independent Two Samples 14 Independent Two Samples 15 Independent Two Samples: Not Assuming Equal Standard Deviations (σ1≠ σ2) ~ tdf=v, Note: Unless we obtain our degrees of freedom using a computer, we will round v down to the nearest integer. 16 Independent Two Samples: Not Assuming Equal Standard Deviations (σ1≠ σ2) • A (1-α)×100% CI is ( x1 x2 ) t * df v , / 2 2 1 2 2 s s n1 n2 * where tdf v, / 2 a critical value that makes the right tail probability equal to α/2 Stata: di invttail(df, α/2) 17 Independent Two Samples: Not Assuming Equal Standard Deviations (σ1≠ σ2) 18 Exercise: Does alcohol affect males and females differently? A study involving males and females with similar physical characteristics was conducted. In a controlled setting, each individual was asked to consume 4 ounces of alcohol. One hour after consumption each participant took a Breathalyzer to measure his or her blood alcohol level. The following results were obtained: •Is there a real difference in blood alcohol levels between males and females? Carry out an appropriate test to answer this question. •Calculate a 99% confidence interval for the difference in blood alcohol levels between males and females. 19 Exercise • Stata command and outputs: • ttesti n1 u1 s1 n2 u2 s2, unequal level(#) 20 Independent Samples: Assuming Equal Standard Deviations (σ1= σ2) 21 Independent Samples: Assuming Equal Standard Deviations (σ1= σ2) 22 Independent Samples: Assuming Equal Standard Deviations (σ1= σ2) 23 Independent Samples: Assuming Equal Standard Deviations (σ1= σ2) 24 To be pooled, or not to be pooled? Equal S.D. or not equal? • We could base our decision on an informal rule (usually works and is much less complicated). • If no sample standard deviation is twice the other, (i.e 0.5 < s1/s2 < 2), then the assumption of equal standard deviations should be ok. • We could perform a graphical analysis and look at the box-plots for the samples to informally assess the equal standard deviations assumption. • There are formal tests to assess the evidence against equal population standard deviations • Variance Ratio Test • Levene’s Test 25 Variance Ratio Test • Ho: σ12 = σ22 vs. Ha: σ12 ≠ σ22 • Test statistic: F= s12 / s22 ~ Fdf1=n1-1, df2=n2-1 • p-value: 26 Variance Ratio Test • Stata command: sdtesti n1 . sd1 n2 . sd2 • Example: Test σ1= σ2 when observed n1=75, sd1=6.5, n2=65, and sd2= 7.5 • sdtesti 75 . 6.5 65 . 7.5 27 Exercise • Two machines are used to fill plastic bottles with dishwashing detergent. Two random samples are taken and the following results are obtained: • Is there a real difference between their average fills? Carry out an appropriate test to answer this question. • Calculate a 95% confidence interval for the difference in the mean fill between the two machines 28 Exercise • s1=√0.113=0.336; s2= √0.125 =0.354 • Testing for Equal Variances 29 Exercise • Stata command for two-sample t test with equal variances • ttesti N1 u1 s1 N2 u2 S2, level(#) Compared with machine 2, machine 1 had a statistically significantly lower mean fill. The difference in mean fill was -0.41 (95%CI: -0.59 to -0.23) for machine 2 relative to machine 1 (p<0.001). 30