Various studies have shown that facial appearance affects so

Various studies have shown that facial appearance affects social interactions. One recent study looked at the relationship between eye color and trustworthiness.

In this study, there were 229 participants, 72 with brown eyes and 157 with blue or green eyes. Each participant was asked to rate a set of student photos in terms of trustworthiness on a 10-point scale, where 1 means very trustworthy and 10 very untrustworthy. All photos showed a student who was seated in front of a white background and looking directly at the camera with a neutral expression. The photos were cropped so that the eyes were at the same height on each photo and a neckline was visible.
Suppose that for the population of all brown-eyed participants, a photo of a blue-eyed female student has a mean score of 5.6 and a standard deviation of 2.6. That same photo for the population of all blue-eyed participants has a mean score of 6.5 and a standard deviation of 2.2.

(a) What are the means (±0.001) and standard deviations (±0.001) of the distribution of the difference (blue - brown) between the mean scores in the two groups?

(b) Use software to find the probability (±0.001) that the mean score for the brown-eyed group is less than the mean score for the other group.

Solution

a) the mean of distribution of difference between brown and blue is mean of blue -mean of brown and sd is sqrt(var of blue +var of brown)

mean=6.5-5.6=0.9

sd(blue-brown)=sqrt(2.2^2+2.6^2)=3.4058

b) here we use the two means test.

z=xbar1-xbar2/sqrt(var1/n1+var2/n2)

Since p is less than 0.05, we reject the null hypothesis and conclude that the mean score for brown eu=yed group is less than the mean score for the other group.