Independent Samples t-test Mon, Apr 12 th

t Test for Independent Means wComparing two samples –e.g., experimental and control group –Scores are independent of each other wFocus on differences betw 2 samples, so comparison distribution is: –Distribution of differences between means

Hypotheses wHo: or  1 -  2 = 0 (no group difference) or  1 =  2 wHa =  1 -  2 not = 0 (2 tailed) or  1 -  2 > or < 0 (if 1-tailed) wIf null hypothesis is true, the 2 populations (where we get sample means) have equal means wCompare T obs to T critical (w/Df= N-2) wIf |Tobs| > |T crit|  Reject Null

Pooled Variance wT observed will use concept of pooled variance to estimate standard error: –Assume the 2 populations have the same variance, but sample variance will differ… – so pool the sample variances to estimate pop variance –Then standard error used in denom of T obs wNote – I’ll show you 2 approaches (you decide which to use based on what data is given to you)

Finding Estimated Standard Error using SS (lab approach) wPooled variance (S 2 p ) S 2 p = (SS 1 + SS 2 ) / df 1 + df 2 Where, df1 = N 1 -1 and df2= N 2 -1 and SS 1 and SS 2 are given to you wThen use this to estimate standard error: (S xbar 1 –xbar2 ) = sqrt [(S 2 p /N 1 ) + (S 2 p /N 2 )] Estimated Standard Error (using x notation)

Estimated Standard Error using S 2 y (sample variances – book & HW approach) wSkip calculating pooled variance and just estimate standard error: S ybar1 – ybar2 = sqrt [((N 1 -1)S 2 y 1 ) + ((N 2 -1)S 2 y 2 )] (N 1 +N 2 ) – 2 * Sqrt [(N 1 + N 2 )/ N 1 N 2 ] Estimated Standard Error (using y notation) Note: See p. 480 in book for better representation of this formula!!

T observed wOnce you’ve estimated standard error, this will be used in T obs: T obs = (xbar 1 – xbar 2 ) – (  1 -  2 ) S xbar1 – xbar2 Always = 0) Estimated Standard Error (doesn’t matter if use x or y notation!)

Example – using S 2 y info wGroup 1 – watch TV news; Group 2 – radio news; difference in knowledge? wHo:  1 -  2 = 0; Ha:  1 -  2 not = 0 –ybar1 = 24, S 2 1 = 4, N1 = 61 –ybar2 = 26, S 2 2 = 6, N2 = 21 –Alpha =.01, 2-tailed test, df tot = N-2 = 80 –S ybar1-ybar2 = sqrt[((60)4) +((20)6)] ( ) - 2 = 2.12 * sqrt [( ) / 61 *21] = = 2.12 *.253 =.536

wT obs = (24-26) – = wt criticals, alpha =.01, df=80, 2 tailed –2.639 and –2.639 w|T observed | > |T critical| (3.73 > 2.639) wReject null – there is a difference in knowledge based on news source –(check means to see which is best)…radio news was related to higher knowledge. wNote: in lab 22 you’ll use other approach (find SS first, then standard error for T denom); this ex. is how HW will look…

SPSS example wAnalyze  Compare Means  Independent Samples t –Pop up window…for “Test Variable” choose the variable whose means you want to compare. For “Grouping Variable” choose the group variable –After clicking into “Grouping Variable”, click on button “Define Groups” to tell SPSS how you’ve labeled the 2 groups

(cont.) –Pop up window, “Use Specified Values” and type in the code for Group 1, then Group 2, hit “continue” · For example, can label these groups anything you’d like when entering data. Are they coded 0 and 1? 1 and 2?…etc. Specify it here. –Finally, hit OK –See output example in lab for how to interpret

GSS data example wHo:  male -  female = 0, –Ha: difference not = 0 wDV = # siblings wMale xbar = 4.00, female xbar = 3.98 wIn output, 1 st look at Levene’s test of equality of variances (2 lines) Ho: equal variances: –Equal variances assumed  look at sig value –Equal variances not assumed –If ‘sig value” <.05  reject Ho of equal variances and look at ‘equal variance not assumed’ line –If ‘sig’ value >.05  fail to reject Ho of equal variances and use ‘equal variance assumed’ line

(cont.) wHere, fail to reject Ho, use equal variances assumed –Next, using that line, look for ‘sig 2-tailed’ value  this is the main hypothesis test of mean differences –If ‘sig’ <.05  reject Ho of no group differences –Here, >.05, so fail to reject Ho, conclude # sibs doesn’t differ between male/female