What if.... The two samples have different sample sizes (n)

Results Psychology Sociology

Results Psychology Sociology

If samples have unequal n All the steps are the same! Only difference is in calculating the Standard Error of a Difference

Standard Error of a Difference When the N of both samples is equal If N 1 = N 2 : Sx 1 - x 2 =

Standard Error of a Difference When the N of both samples is not equal If N 1 = N 2 : N 1 + N 2 - 2

Results Psychology Sociology  X 1 = 535  X 1 2 = N 1 = 4  X 2 = 265  X 2 2 = N 2 = 3

N 1 + N  X 1 = 535  X 1 2 = N 1 = 4  X 2 = 265  X 2 2 = N 2 = 3

N 1 + N  X 1 = 535  X 1 2 = N 1 = 4  X 2 = 265  X 2 2 = N 2 =

N 1 + N  X 1 = 535  X 1 2 = N 1 = 4  X 2 = 265  X 2 2 = N 2 =

 X 1 = 535  X 1 2 = N 1 = 4  X 2 = 265  X 2 2 = N 2 =

5  X 1 = 535  X 1 2 = N 1 = 4  X 2 = 265  X 2 2 = N 2 =

5  X 1 = 535  X 1 2 = N 1 = 4  X 2 = 265  X 2 2 = N 2 = (.58)

5  X 1 = 535  X 1 2 = N 1 = 4  X 2 = 265  X 2 2 = N 2 = = 10.69

Practice I think it is colder in Philadelphia than in Anaheim (  =.10). To test this, I got temperatures from these two places on the Internet.

Results Philadelphia Anaheim

Hypotheses Alternative hypothesis –H 1 :  Philadelphia <  Anaheim Null hypothesis –H 0 :  Philadelphia = or >  Anaheim

Step 2: Calculate the Critical t df = N 1 + N df = = 6  =.10 One-tailed t critical =

Step 3: Draw Critical Region t crit = -1.44

Now Step 4: Calculate t observed t obs = (X 1 - X 2 ) / Sx 1 - x 2

6  X 1 = 275  X 1 2 = N 1 = 5 X 1 = 55  X 2 = 219  X 2 2 = N 2 = 3 X 2 = = 3.05

Step 4: Calculate t observed = ( ) / 3.05 Sx 1 - x 2 = 3.05 X 1 = 55 X 2 = 73

Step 5: See if t obs falls in the critical region t crit = t obs = -5.90

Step 6: Decision If t obs falls in the critical region: –Reject H 0, and accept H 1 If t obs does not fall in the critical region: –Fail to reject H 0

Step 7: Put answer into words We Reject H 0, and accept H 1 Philadelphia is significantly (  =.10) colder than Anaheim.

SPSS

So far.... We have been doing independent samples designs The observations in one group were not linked to the observations in the other group

Example Philadelphia Anaheim

Matched Samples Design This can happen with: –Natural pairs –Matched pairs –Repeated measures

Natural Pairs The pairing of two subjects occurs naturally (e.g., twins)

Matched Pairs When people are matched on some variable (e.g., age)

Repeated Measures The same participant is in both conditions

Matched Samples Design In this type of design you label one level of the variable X and the other Y There is a logical reason for paring the X value and the Y value

Matched Samples Design The logic and testing of this type of design is VERY similar to what you have already done!

Example You just invented a “magic math pill” that will increase test scores. On the day of the first test you give the pill to 4 subjects. When these same subjects take the second test they do not get a pill Did the pill increase their test scores?

Hypothesis One-tailed Alternative hypothesis –H 1 :  pill >  nopill –In other words, when the subjects got the pill they had higher math scores than when they did not get the pill Null hypothesis –H 0 :  pill < or =  nopill –In other words, when the subjects got the pill their math scores were lower or equal to the scores they got when they did not take the pill

Results Test 1 w/ Pill (X) Mel3 Alice5 Vera 4 Flo3 Test 2 w/o Pill (Y) 1 3 2

Step 2: Calculate the Critical t N = Number of pairs df = N = 3  =.05 t critical = 2.353

Step 3: Draw Critical Region t crit = 2.353

Step 4: Calculate t observed t obs = (X - Y) / S D

Step 4: Calculate t observed t obs = (X - Y) / S D

Step 4: Calculate t observed t obs = (X - Y) / S D X = 3.75 Y = 2.00

Step 4: Calculate t observed t obs = (X - Y) / S D Standard error of a difference

Step 4: Calculate t observed t obs = (X - Y) / S D S D = S D / N N = number of pairs

S =

Test 1 w/ Pill (X) Mel3 Alice5 Vera 4 Flo3 Test 2 w/o Pill (Y) 1 3 2

S = Test 1 w/ Pill (X) Mel3 Alice5 Vera 4 Flo3 Test 2 w/o Pill (Y) Difference (D) 2 1

S = Test 1 w/ Pill (X) Mel3 Alice5 Vera 4 Flo3 Test 2 w/o Pill (Y) Difference (D) 2 1  D = 7  D 2 =13 N = 4

S = Test 1 w/ Pill (X) Mel3 Alice5 Vera 4 Flo3 Test 2 w/o Pill (Y) Difference (D) 2 1  D = 7  D 2 =13 N = 4 7

S = Test 1 w/ Pill (X) Mel3 Alice5 Vera 4 Flo3 Test 2 w/o Pill (Y) Difference (D) 2 1  D = 7  D 2 =13 N =

S = Test 1 w/ Pill (X) Mel3 Alice5 Vera 4 Flo3 Test 2 w/o Pill (Y) Difference (D) 2 1  D = 7  D 2 =13 N =

S = Test 1 w/ Pill (X) Mel3 Alice5 Vera 4 Flo3 Test 2 w/o Pill (Y) Difference (D) 2 1  D = 7  D 2 =13 N =

.5 = Test 1 w/ Pill (X) Mel3 Alice5 Vera 4 Flo3 Test 2 w/o Pill (Y) Difference (D) 2 1  D = 7  D 2 =13 N =

Step 4: Calculate t observed t obs = (X - Y) / S D S D = S D / N N = number of pairs

Step 4: Calculate t observed t obs = (X - Y) / S D.25=.5 / 4 N = number of pairs

Step 4: Calculate t observed 7.0 = ( ) /.25

Step 5: See if t obs falls in the critical region t crit = 2.353

Step 5: See if t obs falls in the critical region t crit = t obs = 7.0

Step 6: Decision If t obs falls in the critical region: –Reject H 0, and accept H 1 If t obs does not fall in the critical region: –Fail to reject H 0

Step 7: Put answer into words Reject H 0, and accept H 1 When the subjects took the “magic pill” they received statistically (  =.05) higher math scores than when they did not get the pill

SPSS

Practice You just created a new program that is suppose to lower the number of aggressive behaviors a child performs. You watched 6 children on a playground and recorded their aggressive behaviors. You gave your program to them. You then watched the same children and recorded this aggressive behaviors again.

Practice Did your program significantly lower (  =.05) the number of aggressive behaviors a child performed?

Results Time 1 (X) Child118 Child211 Child319 Child46 Child510 Child614 Time 2 (Y)

Hypothesis One-tailed Alternative hypothesis –H 1 :  time1 >  time2 Null hypothesis –H 0 :  time1 < or =  time2

Step 2: Calculate the Critical t N = Number of pairs df = N = 5  =.05 t critical = 2.015

Step 4: Calculate t observed t obs = (X - Y) / S D

1.21 = (D)  D = 8  D 2 =18 N = Time 1 (X) Child118 Child211 Child319 Child46 Child510 Child614 Test 2 (Y)

Step 4: Calculate t observed t obs = (X - Y) / S D.49=1.21 / 6 N = number of pairs

Step 4: Calculate t observed 2.73 = ( ) /.49 X = 13 Y = S D =.49

Step 5: See if t obs falls in the critical region t crit = t obs = 2.73

Step 6: Decision If t obs falls in the critical region: –Reject H 0, and accept H 1 If t obs does not fall in the critical region: –Fail to reject H 0

Step 7: Put answer into words Reject H 0, and accept H 1 The program significantly (  =.05) lowered the number of aggressive behaviors a child performed.

SPSS

New Step Should add a new page Determine if –One-sample t-test –Two-sample t-test If it is a matched samples design If it is a independent samples with equal N If it is a independent samples with unequal N

Thus, there are 4 different kinds of designs Each design uses slightly different formulas You should probably make up ONE cook book page (with all 7 steps) for each type of design –Will help keep you from getting confused on a test

Practice A research study was conducted to examine whether or not there were differences between older and younger adults on perceived life satisfaction. A pilot study was conducted to examine this hypothesis. Ten older adults (over the age of 70) and ten younger adults (between 20 and 30) were give a life satisfaction test (known to have high reliability and validity). Scores on the measure range from 0 to 60 with high scores indicative of high life satisfaction; low scores indicative of low life satisfaction. Older Adults= 44.5; S = 8.68; n = 10 Younger Adults = 28.1; S = 8.54; n = 10

Practice Dr. Willard is studying the effects of a new drug (Drug-Y) on learning improvement following traumatic brain injury. To study this, Dr. Willard takes six rats and lesions a part of the brain responsible for learning. He puts each rat in a maze and counts the number of times it takes each rat to navigate through the maze without making a mistake. Dr. Willard then puts each rat on a regimen of Drug-Y for one week. After one week, he places each rat in a similar maze and counts the number of times it takes each rat to navigate through the maze without a mistake. Examine if Drug-Y had a positive impact on rats performance.

tTime 1Time 2 Ben2824 Splinter2926 George3022 Jerry3330 Fievel3429 Patches3228

Practice A sample of ten 9th grades at James Woods High School can do an average of 11.5 pull-ups (chin-ups) in 30 seconds, with a sample standard deviation of s = 3 The US Department of Health and Human Services suggests that 9th grades be able to do a minimum of 9 pull-ups in 30 seconds, if not, they're watching too much Family Guy. Is this sample of 9th grades able to do significantly (alpha =.01) more pull-ups than the number recommended by the US Department of Health and Human Services?

t obs = t crit = There age is related to life satisfaction.

t = ( )/0.764 = 4.500/0.764 = Because the obtained t-Value is larger that the critical t- Value, the mean difference between the number of maze navigation's at Time 1 and Time 2 is statistically significant. Thus, we can conclude that Drug-Y lead to a statistically significant decrease in the number of times it took rats to navigate through a maze without making a mistake.

The obtained t-Value is ( )/1 = 2.5/1 = ) Because the obtained t-Value (2.500) is less than the critical t-Value (2.689), the difference between the mean number of pull-ups that 9th grades from James Woods High School can do is not significantly greater than the number of pull-ups recommended by the US Department of Health and Human Services.

Cookbook