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Stat 151 Ch.22 -1/5 Comparing Several Means (ANOVA) (Chapt.22) Heart-rate (beats/min.) while performing a stressful task in the presence of a friend, pet and in a control situation. Control 80.37 87.45 90.02 99.05 75.48 87.23 91.75 87.79 77.80 62.65 84.74 84.88 73.28 84.52 70.88 Friend 99.69 83.40 102.15 80.28 88.02 86.99 92.49 91.35 100.88 101.06 97.05 81.60 89.82 98.20 76.91 Pet 69.17 70.17 75.99 86.45 68.86 64.17 97.54 85.00 65.45 72.26 58.69 79.66 69.23 69.54 70.08 Is there any difference in mean stress level (i.e., heart-rate) of the three groups? Stat 151 Ch.22 -2/5 Recall that we used a two sample t-test to test for a difference between two population means. That is, we wished to test: H 0 : m1 = m 2 We can extend this procedure to test whether any differences exist between more than two means. That is, we wish to test: H 0 : m1 = m 2 = m 3 = L = m k vs Ha : not all m1 , m 2 , K, m k are equal where k is the number of groups (populations) involved in the study. This test is called the Analysis of Variance test, or ANOVA. Stat 151 Ch.22 -3/5 ANOVA (F-test) Some definitions we will need: The Mean Squared Error: ( n1 - 1) s12 + ( n2 - 1) s22 + L + ( nk - 1) sk2 MSE = N -k where N is the total number of observations in the sample ni is the number of observations in the ith group, and si is the sample standard deviation of the ith group. Note: MSE is the pooled sample standard deviation, and is an estimate of the standard deviation s, that is assumed to be the same for all groups. The Mean Squared Group: 2 2 2 n1 ( x1 - x ) + n2 ( x2 - x ) + L + nk ( xk - x ) MSG = k -1 where x is the overall sample mean and and xi is the sample mean of the ith group. Stat 151 Ch.22 -4/5 The test statistic we use to test H 0 : m1 = m 2 = m 3 = L = m k vs Ha : not all m1 , m 2 , K, m k are equal is MSG F= MSE where under H0, F has a F distribution on k - 1 and N - k degrees of freedom. Anova: Single Factor SUMMARY Groups Pet Friend Control Count 15 15 15 Sum Average Variance 1237.86 82.52 85.41 1369.88 91.33 69.57 1102.25 73.48 99.40 ANOVA Source of Variation SS Between Groups 2387.69 Within Groups 3561.30 df 2 42 Total 44 5948.99 MS 1193.84 84.79 F P-value 14.08 0.00 Stat 151 Ch.22 -5/5 ANOVA Assumptions · The k samples, one from each of the k populations, are independent. · Each of the k samples is drawn from a population that has a Normal distribution. · All of the k populations have the same standard deviation, s. Confidence Intervals from ANOVA Output We can obtain confidence intervals for the ith population mean, m i from the usual form using the pooled sample standard deviation as the estimate of s. That is, a confidence interval for m i is of the form xi ± t * SE ( xi ) sp MSE = . where SE ( xi ) = ni ni