Comparing Means: t-tests for Two Independent Samples

Comparing ean: t-tet for Two Independent Sample

Independent-eaure Deign t-tet for Two Independent Sample Allow reearcher to evaluate the mean difference between two population uing data from two eparate ample. The identifying characteritic of the independent-meaure or between-ubject deign i the exitence of two eparate and independent ample. Thu, an independent-meaure deign can be ued to tet for mean difference between two ditinct population (uch a men veru women) or between two different treatment condition (uch a drug veru no-drug).

Independent-eaure Deign t-tet for Two Independent Sample

Independent-eaure Deign t-tet for Two Independent Sample The independent-meaure deign i ued in ituation where a reearcher ha no prior knowledge about either of the two population (or treatment) being compared. In particular, the population mean and tandard deviation are all unknown. Becaue the population variance are not known, thee value mut be etimated from the ample data.

The t Statitic for an Independent-eaure Deign The purpoe of the independent-meaure t tet i the ame a that for hypothei tet we dicued previouly: determine whether the ample mean difference obtained in a reearch tudy indicate a real mean difference between the two population (or treatment) or whether the obtained difference i imply the reult of ampling error. A with the other hypothei tet, you mut tate the null and reearch hypothee before doing anything ele. In general form thee will be: Null Hypothei H 0 : µ = µ or µ - µ = 0 Reearch Hypothei H : µ µ or µ - µ 0

The Independent-Sample t-tet: Step. Ue t ditribution table to find critical t-value() repreenting rejection region (denoted uing t crit or t α ). Compute t-tatitic For data in which I give you raw core, you will have to compute the ample mean and ample tandard deviation for both ample 3. ake a deciion: doe the t-tatitic for your ample fall into the rejection region?

t-statitic for Independent Sample t-tet for Two Independent Sample Though the baic form of the t-tatitic i imilar to that ued for the ingle-ample tet, Some of the detail are different. In particular, note that now we mut etimate the tandard deviation and mean for both population via ample tatitic: t df df??

Sampling Ditribution of Difference Between ean To compute the t-tatitic, we mut firt characterize the tandard error of the difference between mean under the null hypothei From the variance um law: The variance of the um or difference of two independent variable i the um of their variance. Therefore, the tandard error of the difference between the mean of two independent ample i the quare root of the um of their quared tandard error: n n n n

Etimating the Standard Error of the Difference Between ean Under the null hypothei, the ditribution are equivalent. Thi mean that the population variance for the two ample hould be equal: σ = σ = σ Thi i called homogeneity of variance and it i an important aumption underlying independent-ample t-tet Additionally, ince we don t have the population tandard deviation, we etimate the tandard error of the mean uing the ample tandard deviation So now we want to etimate a ingle population variance (σ ) uing two etimate ( & ). How do we do thi?

t-tet for Two Independent Sample Etimating the Standard Error of the Difference Between ean We compute a weighted average called the pooled variance Why? df df SS SS p, df n df n df df df df where and Sample tatitic computed with more degree of freedom are le variable (more reliable). Therefore, when averaging etimate acro ample, we want to give more weight to ample with more degree of freedom Note that if df = df, thi reduce to a imple average: df df df p

Etimating the Standard Error of the Difference Between ean Finally, we plug the pooled variance (our etimate of the population variance) into the equation for the tandard error of the difference between mean: n n p p, with p df df df df Note that the above reduce to: p p n n n n

t-statitic for Independent Sample t-tet for Two Independent Sample t df df df df n n n n n n p p

The Independent-Sample t-tet: Example Doe family therapy affect the weight gained by anorexic girl? Note that the repeated-meaure verion of thi example contained a poible confound x : weight gain for family treatment group n 7.6; 7.6; 7 x : weight gain for control group 0.45; 7.99; n 6 Null Hypothei H 0 : µ FT = µ C Reearch Hypothei H : µ FT µ C Do we retain or reject the null hypothei? Aume a two-tailed tet, with α = 0.05

t-ditribution Table α t One-tailed tet α/ α/ -t t Two-tailed tet Level of ignificance for one-tailed tet 0.5 0. 0.5 0. 0.05 0.05 0.0 0.005 0.0005 Level of ignificance for two-tailed tet df 0.5 0.4 0.3 0. 0. 0.05 0.0 0.0 0.00.000.376.963 3.078 6.34.706 3.8 63.657 636.69 0.86.06.386.886.90 4.303 6.965 9.95 3.599 3 0.765 0.978.50.638.353 3.8 4.54 5.84.94 4 0.74 0.94.90.533.3.776 3.747 4.604 8.60 5 0.77 0.90.56.476.05.57 3.365 4.03 6.869 6 0.78 0.906.34.440.943.447 3.43 3.707 5.959 7 0.7 0.896.9.45.895.365.998 3.499 5.408 8 0.706 0.889.08.397.860.306.896 3.355 5.04 9 0.703 0.883.00.383.833.6.8 3.50 4.78 0 0.700 0.879.093.37.8.8.764 3.69 4.587 0.697 0.876.088.363.796.0.78 3.06 4.437 0.695 0.873.083.356.78.79.68 3.055 4.38 3 0.694 0.870.079.350.77.60.650 3.0 4. 4 0.69 0.868.076.345.76.45.64.977 4.40 5 0.69 0.866.074.34.753.3.60.947 4.073 6 0.690 0.865.07.337.746.0.583.9 4.05 7 0.689 0.863.069.333.740.0.567.898 3.965 8 0.688 0.86.067.330.734.0.55.878 3.9 9 0.688 0.86.066.38.79.093.539.86 3.883 0 0.687 0.860.064.35.75.086.58.845 3.850 0.686 0.859.063.33.7.080.58.83 3.89 0.686 0.858.06.3.77.074.508.89 3.79 3 0.685 0.858.060.39.74.069.500.807 3.768 4 0.685 0.857.059.38.7.064.49.797 3.745 5 0.684 0.856.058.36.708.060.485.787 3.75 6 0.684 0.856.058.35.706.056.479.779 3.707 7 0.684 0.855.057.34.703.05.473.77 3.690 8 0.683 0.855.056.33.70.048.467.763 3.674 9 0.683 0.854.055.3.699.045.46.756 3.659 30 0.683 0.854.055.30.697.04.457.750 3.646 40 0.68 0.85.050.303.684.0.43.704 3.55 50 0.679 0.849.047.99.676.009.403.678 3.496 00 0.677 0.845.04.90.660.984.364.66 3.390

Compute t-statitic for Diff. of ean n 7.6 7.6 7 67.6 57.99 0.45 6 5 7. 99 46.5 n 6 58.933 4 df df df n 6 n 5 df df 4 Compute Pooled Variance: df df p df df Compute df: Etimate Standard Error: n p p n 58.933 58.933 7 6 5.73.39 Compute t-tatitic: t df t 4 7.6 ( 0.45).39 7.7 3. 3. 39

The Independent-Sample t-tet: A Full Example Return to our original hypothei teting example (exam core). Thi time, aume that we randomly aigned tudent into the coure of two different intructor (Dr. & Dr. K) and that do not know the population mean or SD for either cla x : ample core from Dr. cla x {65,70,7,63,69} x : ample core from Dr. K cla x {74,67,69,77,84} Null Hypothei H 0 : µ K = µ Reearch Hypothei H : µ K µ Do we accept or reject the null hypothei? Aume a two-tailed tet, with α = 0.05

Compute ample mean and SD: X X 65 45 70 4900 7 504 63 3969 69 476 Sum 338 896 X X 74 5476 67 4489 69 476 77 599 84 7056 37 77 x 338 x 67.6 n 5 n 37 74. 5 SS x 338 x n 896 47. 5 SS x 37 x n 77 8.80 5 SS 47. 3.44 n 4 SS 8.80 6.76 n 4

Compute t-statitic for Diff. of ean SS n SS n 67.6 47. 5 74. 8. 8 5 Compute Pooled Variance: SS SS p df df 47. 8.8 8 8.75 Compute t-tatitic: t df t 8 67.6 74. 3.39.95 Compute df: Etimate Standard Error: df df df 4 4 df df 8 n p p n 8.75 8.75 5 5.50 3. 39

Repeated-eaure Veru Independent-eaure Deign Advantage of repeated-meaure deign: Becaue a repeated-meaure deign ue the ame individual in both treatment condition, it uually require fewer participant than would be needed for an independent-meaure deign. The repeated-meaure deign i particularly well uited for examining change that occur over time, uch a learning or development. The primary advantage of a repeated-meaure deign, however, i that it reduce variance and error by removing individual difference.

Repeated-eaure Veru Independent-eaure Deign Advantage of repeated-meaure deign (continued): Recall that the firt tep in the calculation of the repeated-meaure t tatitic i to find the difference core for each ubject. The proce of ubtracting to obtain the D core remove the individual difference from the data. I.e., the initial difference in performance from one ubject to another are eliminated. Removing individual difference tend to reduce the variance. Thi create a maller tandard error and increae the likelihood of a ignificant t tatitic.

Repeated-eaure Veru Independent-eaure Deign Diadvantage of repeated-meaure deign: There are potential diadvantage to uing a repeatedmeaure deign intead of independent-meaure Becaue the repeated-meaure deign require that each individual participate in more than one treatment, there i a rik that expoure to the firt treatment will caue a change in the participant that influence their core in the econd treatment

Repeated-eaure Veru Independent-eaure Deign Advantage of repeated-meaure deign (continued): For example, practice in the firt treatment may caue improved performance in the econd treatment. Thu, the core in the econd treatment may how a difference, but the difference i not caued by the econd treatment. When participation in one treatment influence the core in another treatment, the reult may be ditorted by order effect or carry-over effect; thi can be a eriou problem in repeated-meaure deign.

Clicker Quetion We would be leat likely to ue a repeated-meaure or matched-ubject deign when a) there are ubtantial individual difference. b) there are minimal individual difference. c) we want to control for difference among ubject. d) we want to compare huband and wive on their level of marriage atifaction.

Comparion of t-statitic for Different Tet Tet Sample Data Hypotheized Population Parameter z-tet µ Etimated Standard Error σ n Etimated Variance σ Single-ample t-tet µ n = SS df Relatedample t-tet D, where D = x x μ D =0 D n D D = SS D df D Independentample t-tet μ μ =0 p + p p = SS + SS n n df + df

z and t-tet Deciion Flowchart t-tet for Two Independent Sample ye z-tet one I σ provided? no One-ample t-tet Number of Sample two Are core matched acro ample? ye no Related ample t-tet Independent ample t-tet