Practice You wonder if psychology majors have higher IQs than sociology majors ( = .05) You give an IQ test to 4 psychology majors and 4 sociology majors

Results Psychology 110 150 140 135 Sociology 90 95 80 98

Step 1: Hypotheses Alternative hypothesis Null hypothesis H1: psychology > sociology Null hypothesis H0: psychology = or < sociology

Step 2: Calculate the Critical t df = N1 + N2 - 2 df = 4 + 4 - 2 = 6  = .05 One-tailed t critical = 1.943

Step 3: Draw Critical Region tcrit = 1.943

Now Step 4: Calculate t observed tobs = (X1 - X2) / Sx1 - x2

Sx1 - x2 = X1= 535 X12= 72425 N1 = 4 X1 = 133.75 X2= 363 33129 4 4 4 (4 - 1)

X1= 535 X12= 72425 N1 = 4 X1 = 133.75 X2= 363 X22= 33129 N2 = 4 X2 = 90.75 9.38 = 363 535 72425 33129 4 4 4 (4 - 1)

Step 4: Calculate t observed 4.58 = (133.75 - 90.75) / 9.38 Sx1 - x2 = 9.38 X1 = 133.75 X2 = 90.75

Step 5: See if tobs falls in the critical region tcrit = 1.943 tobs = 4.58

Step 6: Decision If tobs falls in the critical region: Reject H0, and accept H1 If tobs does not fall in the critical region: Fail to reject H0

Step 7: Put answer into words We Reject H0, and accept H1 Psychology majors have significantly ( = .05) higher IQs than sociology majors.

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

Results Psychology 110 150 140 135 Sociology 90 95 80 98

Results Psychology 110 150 140 135 Sociology 90 95 80

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 N1 = N2: Sx1 - x2 =

Standard Error of a Difference When the N of both samples is not equal If N1 = N2: N1 + N2 - 2

Results Psychology 110 150 140 135 Sociology 90 95 80 X1= 535 N1 = 4 X2= 265 X22= 23525 N2 = 3

X1= 535 X12= 72425 N1 = 4 X2= 265 X22= 23525 N2 = 3 N1 + N2 - 2

X1= 535 X12= 72425 N1 = 4 X2= 265 X22= 23525 N2 = 3 265 535 N1 + N2 - 2

X1= 535 X12= 72425 N1 = 4 X2= 265 X22= 23525 N2 = 3 265 535 23525 72425 N1 + N2 - 2

X1= 535 X12= 72425 N1 = 4 X2= 265 X22= 23525 N2 = 3 265 535 23525 72425 3 4 4 3 4 + 3 - 2

X1= 535 X12= 72425 N1 = 4 X2= 265 X22= 23525 N2 = 3 265 535 23525 23408.33 72425 71556.25 3 4 .25+.33 4 3 5

X1= 535 X12= 72425 N1 = 4 X2= 265 X22= 23525 N2 = 3 265 535 23525 23408.33 72425 71556.25 197.08 (.58) 3 4 .25+.33 4 3 5

= 10.69 X1= 535 X12= 72425 N1 = 4 X2= 265 X22= 23525 N2 = 3 23408.33 72425 71556.25 114.31 3 4 .25+.33 4 3 5 = 10.69

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

Results Philadelphia 52 53 54 61 55 Newport 77 75 67

Hypotheses Alternative hypothesis Null hypothesis H1: Philadelphia < Newport Null hypothesis H0:  Philadelphia = or >  Newport

Step 2: Calculate the Critical t df = N1 + N2 - 2 df = 5 + 3 - 2 = 6  = .10 One-tailed t critical = - 1.44

Step 3: Draw Critical Region tcrit = -1.44

Now Step 4: Calculate t observed tobs = (X1 - X2) / Sx1 - x2

Standard Error of a Difference When the N of both samples is not equal If N1 = N2: N1 + N2 - 2

X1= 275 X12= 15175 N1 = 5 X1 = 55 X2= 219 X22= 16043 N2 = 3 X2 = 73 219 275 16043 15175 3 5 5 3 5 + 3 - 2

= 3.05 X1= 275 X12= 15175 N1 = 5 X1 = 55 X2= 219 X22= 16043 15987 15175 15125 3 5 .2 + .33 5 3 6 = 3.05

Step 4: Calculate t observed -5.90 = (55 - 73) / 3.05 Sx1 - x2 = 3.05 X1 = 55 X2 = 73

Step 5: See if tobs falls in the critical region tcrit = -1.44 tobs = -5.90

Step 6: Decision If tobs falls in the critical region: Reject H0, and accept H1 If tobs does not fall in the critical region: Fail to reject H0

Step 7: Put answer into words We Reject H0, and accept H1 Philadelphia is significantly ( = .10) colder than Newport.