Question

Trustworthiness and eye color. One group of researches decided to study the relationship between eye color and trustworthiness. In their experiment the researchers took photographs of 80 students (20 males with brown eyes, 20 males with blue eyes, 20 females with brown eyes, 20 females with blue eyes), each seated in a front of a white background looking directly at the camera with a neutral expression. The photos were cropped so the eyes were horizontal and at the same height. They then recruited 105 participants to judge the trustworthiness of each student photo using a 10-poit scale, where 1 meant untrustworthy and 10 very trustworthy. The 80 scores were converted to z-scores and the average z-score of each photo was used. Eye color n ?̅ s Brown 40 0.55 1.68 Blue 40 −0.38 1.53 At the 5% significance, can we conclude from these data, sufficient evidence that brown-eyed students appear more trustworthy compared to their blue-eyed counterparts? Perform the five steps for two-mean t-test using critical or p-value

Answer 1

For Brown :

x̅1 = 0.55, s1 = 1.68, n1 = 40

For Blue :

x̅2 = -0.38, s2 = 1.53, n2 = 40

Null and Alternative hypothesis:

H₀: μ_Brown = μ_Blue

H_a: μ_Brown > μ_Blue

df = ((s1²/n1 + s2²/n2)²)/[(s1²/n1)²/(n1-1) + (s2²/n2)²/(n2-1) ] = 77.3275 = 77

Critical value :

Right tailed critical value, t_c = ABS(T.INV(0.05, 77)) = 1.665

Reject Ho if t > 1.665

Test statistic:

t = (x̅1 - x̅2)/√(s1²/n1 + s2²/n2) = (0.55 - -0.38)/√(1.68²/40 + 1.53²/40) = 2.5885

p-value :

Right tailed p-value = T.DIST.RT(2.5885, 77) = 0.0058

Decision:

p-value < α, Reject the null hypothesis

Conclusion:

There is enough evidence to conclude that brown-eyed students appear more trustworthy compared to their blue-eyed counterparts at 0.05 significance level.