Independent Samples T- test

Transcription

1 Independent Sample T- tet With previou tet, we were intereted in comparing a ingle ample with a population With mot reearch, you do not have knowledge about the population -- you don t know the population mean and tandard deviation INDEPENDENT SAMPLES T-TEST: Hypothei teting procedure that ue eparate ample for each treatment condition (between ubject deign) Ue thi tet when the population mean and tandard deviation are unknown, and eparate group are being compared Example: Do male and female differ in term of their exam core? Take a ample of male and a eparate ample of female and apply the hypothei teting tep to determine if there i a ignificant difference in core between the group

2 Formula: t ( x x ) ( µ µ ) x x We are intereted in a difference between population (female, µ, and male, µ ) and we ue ample (female, x, and male, x ) to etimate thi difference ESTIMATED STANDARD ERROR OF THE DIFFERENCE: Give u the total amount of error involved in uing ample mean to etimate population mean. It tell u the average ditance between the ample difference (x -x ) and the population difference (µ -µ ) A we ve done previouly, we have to etimate the tandard error uing the ample tandard deviation or variance and, ince there are ample, we mut average the two ample variance.

3 POOLED VARIANCE: The average of the two ample variance, allowing the larger ample to weighted more heavily Formulae: ( df) + ( df ) pooled df + df df df for t ample; n - df df for nd ample; n - OR SS pooled df + + SS df Etimated Standard Error of the Difference pooled x x + n n pooled x x SS + SS n + n n + n book formula Degree of freedom (df) for the Independent t tatitic i n + n - or df +df 3

4 Hypothei teting uing an Independent Sample t-tet: Example: Do male and female differ in their tet core for exam? The mean tet core for female i 7. (.57, n9), and the mean tet core for male i 6.7 (3.63, n0) Step : State the hypothee H 0 : µ -µ 0 (µ µ ) H : µ -µ 0 (µ µ ) Thi i a two-tailed tet (no direction i predicted) Step : Set the criterion α? df n +n -? Critical value for the t-tet? Step 3: Collect ample data, calculate x and From the example we know the mean tet core for female i 7. (.57, n9), and the mean tet core for male i 6.7 (3.63, n0) 4

5 Step 4: Compute the t-tatitic t where ( x x ) ( µ µ ) x x pooled x x + n n pooled Calculate the etimated tandard error of the difference ( df) + ( df ) pooled df + df pooled (8).57 + (9) ( 8) (9)

6 Compute the tandard error (continued) t pooled x x + n n pooled x x 9 0 Calculate the t tatitic ( x x ) ( µ µ ) x x (7. 6.7) t *Thi alway default to 0 Step 5: Make a deciion about the hypothee The critical value for a two-tailed t-tet with df37 (approx. 40) and α.05 i.0 Will we reject or fail to reject the null hypothei? 6

7 Aumption for the Independent t-tet: Independence: Obervation within each ample mut be independent (they don t influence each other) Normal Ditribution: The core in each population mut be normally ditributed Homogeneity of Variance: The two population mut have equal variance (the degree to which the ditribution are pread out i approximately equal) Repeated Meaure T-tet Ue the ame ample of ubject meaured on two different occaion (within-ubject deign) Ue thi when the population mean and tandard deviation are unknown and you are comparing the mean of a ample of ubject before and after a treatment We are intereted in finding out how much difference exit between ubject core before the treatment and after the treatment 7

8 DIFFERENCE SCORE (or D) The difference between ubject core before the treatment and after the treatment It i computed a x -x, where x i the ubject core after the treatment and x i the ubject core before the treatment We ue the ample of difference core to etimate the population of difference core (µ D ) Example: Doe alcohol affect a peron ability to drive? A reearcher elect a ample of 5 people and et up an obtacle coure. Each ubject drive the coure and the number of cone he or he knock over i counted. Next, the reearcher ha each ubject drink a ix-pack of beer, then drive the coure again, counting the number of cone each ubject knock over. NOTE: Theory ha hown that alcohol decreae motor and cognitive kill 8

9 Step : State the hypothee H 0 : µ D 0 H : µ D 0 Step : Set the criterion One-tail tet or two-tail tet? α? dfn- Critical value for t? Step 3: Collect ample data, calculate D Once the difference core are obtained, all further tatitic are calculated uing thee core intead of the pretet / pottet or before / after core Before After D Subject (x ) (x ) (x - x ) D 5 9

10 Find the mean (average) difference core (D) D D n 6 + D The average difference of the number of cone knocked down from before drinking to after drinking i 5 cone. Remember, we are hypotheizing the difference to be zero. Step 4: Calculate the t-tatitic Formula: etimated td. deviation of diff. core t D D where D D and D n D n etimated td.error of mean diff. core the mean difference core 0

11 Compute the etimated tandard deviation of the difference core ( D ) D D D (D D) D SS D 0 σ SS n SS n -.58 The average deviation of the difference core (D) about the mean difference core (D) i.58 cone Compute the etimated tandard error of the mean difference core D.58 D. 707 n D 5 The average deviation of the ample mean difference core (D) from the population mean difference core (µ D ) i.707 cone Compute the t-tatitic t D 5 t D.707

12 Step 5: Make a deciion The critical value for a one-tailed t-tet with df4 and α.05 i.3 Will we reject or fail to reject the null hypothei? Advantage and Diadvantage of the Repeated Meaure t-tet: Advantage: Control for pre-exiting individual difference between ample (becaue only ample of people are being ued More economical (fewer ubject are needed) Diadvantage: Subject to practice effect - the ubject are performing the meaurement tak (i.e. driving the obtacle coure, taking an exam) twice - core may improve due to the practice

13 Aumption of the Repeated Meaure t- Tet: Independent Obervation: The core from before and after the treatment mut not be related (no practice effect) Normal Ditribution: The population of difference core mut be normally ditributed Summary of Hypothei Teting through t-tatitic We have looked at four inferential tatitic: z-core tatitic ingle ample t-tatitic independent ample t-tatitic repeated meaure t-tatitic, or matched ubject the generic formula for thee tatitic i: z or t ample tatitic - population parameter tandard error 3

14 Summary of Hypothei Teting through t-tatitic z-core tatitic compare a ample to a population when the population.d. i known t-tatitic compare a ample to a population when the population.d. i unknown independent ample t-tatitic compare independent ample repeated meaure t-tatitic compare ample meaured on occaion 4