Download Introduction to Statistics - Study Guide | PSY 230 and more Exams Statistics in PDF only on Docsity!
Okun PSY 230 Study Guide # t Tests for Hypotheses about Two Population Means I. Distinguishing Between Independent and Related Samples
- What are two limitations of the 1-sample t test?
- What does it mean for two samples to be independent? Independent samples. We have independent samples when we (a) use a separate sample for each condition or group; (b) each person is in only one condition or group; and (c) the selection of any one participant into one of the two samples has no implications for the selection of other participants into the first or second sample.
- How can two independent samples be created?
Study Participant Study ID # Group Assignment via Table of Random Numbers
Andy 1 Lecture Brandy 2 Acupuncture Candy 3 Lecture Dan 4 Acupuncture Earl 5 Acupuncture Fred 6 Acupuncture Georgia 7 Acupuncture Helen 8 Lecture Irma 9 Acupuncture Jessica 10 Lecture Kari 11 Lecture Laura 12 Lecture Mary 13 Lecture Nikki 14 Acupuncture Olivia 15 Acupuncture Peter 16 Lecture
Treadmill Time In Minutes to Run a Marathon from Blood Doping versus No Blood Doping: Independent Samples Design Name Not Blood Doped Name Blood Doped Quentin 224 Rachel 225 Steven 227 Todd 224 Ursula 228 Venus 228 Warren 227 Xavier 226 Yvette 229 Zack 227 Mx = 227 Mx = 226 Sx = 1.87 Sx = 1.
- How are related samples created? There are two ways to create related samples. First, we can test each participant prior to and after exposure to the treatment. This approach is known as repeated measures because we repeat the administration of our measure of the dependent variable before and after each participant is exposed to the treatment. Repeated Measures Study of Acupuncture: A Related Samples Design
Study Participant Study ID # Posttest Pretest Difference Score (Post-Pre)
Brandy 2 119 161 - Dan 4 63 140 - Earl 5 70 168 - Fred 6 15 119 - Georgia 7 154 210 - Irma 9 21 126 - Nikki 14 98 74 + Olivia 15 84 154 -
A second type of related samples design involves matching pairs of participants. With a matched samples design, samples are created by pairing up participants and then by assigning one member of each pair to one group and the other member of each pair to the other group. Matching Study of Blood Doping: A Related-Samples Design Rankings based upon best time to complete the Boston Marathon Name Rank Todd 1 Quentin 2 Rachel 3 Steven 4 Warren 5 Xavier 6 Ursula 7 Zack 8 Venus 9 Yvette 10 Not Blood Doped Blood Doped Difference Score Name Rank Time Name Rank Time Not Doped-Doped Quentin 2 224 Todd 1 224 0 Steven 4 227 Rachel 3 225 + Warren 5 227 Xavier 6 226 + Ursula 7 228 Zack 8 227 + Yvette 10 229 Venus 9 228 + MD = +1 SD = 0.71 npairs = 5
t = -66/15.17 = -4. df = np – 1 where np = number of pairs of observations. If we set = .01 and df = 8-1 or 7, the CV of t = plus or minus 3.449. Decision: Reject the null hypothesis. The number of cigarettes smoked after the acupuncture treatment is significantly ( p < .01) less than the number of cigarettes smoked before the acupuncture treatment.
- How can we test the null hypothesis that the difference between two population means from related samples equals zero from summary data? Treadmill Time in Minutes from Blood Doping Matched-Participants Design Not Blood Doped Blood Doped Difference Score Name Rank Time Name Rank Time Not Doped-Doped Quentin 2 224 Todd 1 224 0 Steven 4 227 Rachel 3 225 + Warren 5 227 Xavier 6 226 + Ursula 7 228 Zack 8 227 + Yvette 10 229 Venus 9 228 + When working with summary data, we will be given D, SD, and npairs. MD = +1 SD = 0.71 npairs = 5 First, df = 5-1 or 4. If alpha = .01, with 4 degrees of freedom, CV = /4.604/. Second, we must compute S_ D
S_ = = 0.71/ 5 = 0.71/2.24 = 0. D Third, we compute t. t = +1/0.32 = +3.125. Decision: Retain the null hypothesis. Blood doping does not appear to significantly ( p > .01) lower the time to run a marathon.
III. Transition to Independent Samples t test
- How is the structure of the related samples t-test similar to the structure of the 1- sample t-test? 1-Sample t test : t = MX-HO: X / S_ where X __ S_ = Sx / n X Related Samples t test t = MD-HO: D / S_ = MD / S_ where D D
S_ = SD / np D S_ X (^) tells us how much, on average, we should expect a sample mean to deviate from the hypothesized value of the population mean due to sampling fluctuation. S_ D (^) tells us how much, on average, we should expect a sample mean computed from difference scores to deviate from zero due to sampling fluctuation.
IV. Computing the Independent Samples t- test
- What does the sample variance (Sx^2 ) equal? (Xi-Mx)^2 SS Sx^2 = _________ = _____ n-1 n-
- How can we derive a formula for S _ _? X1-X If n 1 = n 2 :
S _ _ = S 12 / n 1 + S 22 / n 2 = S 12 + S 22 / ng X1-X where ng = number of participants in each group. If n 1 does not = n 2 , you must first compute the pooled variance (S^2 pooled). (N 1 -1)S 12 + (N 2 -1)S 22 S^2 pooled = ____________________ N 1 + N 2 -
S _ _ = S^2 pooled / n 1 + S^2 pooled / n 2 X1-X
- How can we test the null hypothesis that the difference between two population means from independent samples equals zero from raw data? Mx1 - M (^) x t = _________ S _ _ X1-X Cigarettes Smoked from Acupuncture versus Lecture Independent Samples Design
Acupuncture # of cigarettes smoked Xi-Mx (Xi- Mx)^2 Sx^2 = SS/n- Alice 119 41 1681 Bill 63 -15 225 Dave 70 -8 64 Hal 15 -63 3969 Ivanna 154 76 5776 June 21 -57 3249 Nikki 98 20 400 Paula 84 6 36
Mx = 624/8 = 78 SS = 15400 Sx^2 = 15400/7 = 2200
Lecture # of cigarettes smoked Xi- Mx (Xi- Mx)^2 Sx^2 = SS/n- Carl 161 17 289 Erin 140 -4 16 Felicia 168 24 576 Gina 119 -25 625 Kari 210 66 4356 Laura 126 -18 324 Mary 74 -70 4900 Oliver 154 10 100
Mx = 1152/8 = 144 SS = 11186 Sx^2 = 11186/7 = 1598
- When is it appropriate to use each of the three t tests used to test null hypotheses about population means? Summary of three types of t tests
- What are the advantages and disadvantages of the repeated measures related-sample design, the matched samples related samples design, and the independent samples design?
- How is the value of the independent samples t test affected by the (a) the difference in the sample means; (b) standard deviations; and (c) sample sizes (n)?
- How can the strength of the effect of the independent variable on the dependent variable be computed in the case of the independent samples t test? When we reject the null hypothesis that two independent population means are equal, it is useful to report an index of how strong the effect of the independent variable is on the dependent variable. The index that we will use is the square of the point-biserial correlation coefficient. r^2 point biserial = t^2 / [t^2 + df] For the cigarette smoking studying, t = -3.02 and df = 14 r^2 point biserial = -3.02^2 / [-3.02^2 + 14] = 9.1204/[9.1204 + 14] = 9.1204/23.04 = .39.
