In a survey, 19 males and 19 females were asked whether they preferred an anonymous person's look before or after plastic surgery, or were indifferent. Values of X=1, X=-1 and X=0 were assigned for respondings preferring the subject's new look, old look and being indifferent between the 2 respectively.
1) Calculate Xm Bar (How much a random male will like her new look)
From the data: Xm Bar = -0.26, Standard Deviation = 0.81
(Interpretation: The average guy was mostly indifferent towards the surgery, with a slight tendency to dislike the new look.)
2) Calculate Xf Bar (How much a random female will like her new look)
From the data: Xf Bar = 0.16, Standard Deviation = 0.90
3) Argue that Xi Bar (where i = m or f) represents the sample supporting rate of the surgery
Xi can take values of 0, 1 or -1 which respectively represent indifference, a preference for the post-surgical look and a preference for the pre-surgical look. Xi Bar, being the average value of Xi, thus represents not only the expected response of each respondent, but how supportive the sample as a whole is of the surgery (ie the sample supporting rate).
4) Is Xm Bar less than 0 at the 10% significance level? How about Xf bar? (Are the guys indifferent? Are the girls?)
H0: Xm Bar >= 0 vs
H1: Xm Bar < 0
Since sample size = 19, we must use Student's t distribution with n-1 = 18 degrees of freedom
Critical value: -1.33
T-statistic: -0.26/(0.81)/sqrt19 = -1.40
-1.40 < -1.33, therefore we reject the null hypothesis at the 10% significance level and conclude that most males think she looks worse after the plastic surgery. [Ed: Rejecting the null hypothesis at the 10% significance level means there's a less than 10% chance of rejecting the null hypothesis if it is true.]
H0: Xf Bar <= 0 vs
H1: Xf Bar > 0
[Ed: Some may note that my null here is different. This is because I wanted to test my personal hypothesis that guys prefer the old look and girls the new.]
Since sample size = 19, we must use Student's t distribution with n-1 = 18 degrees of freedom
Critical value: 1.33
T-statistic: 0.16/(0.90)/sqrt19 = 0.77
0.77 /> 1.33, therefore we cannot reject the null hypothesis at the 10% significance level. However, at the 25% significance level, the critical value is 0.69, so we can conclude at the 25% significance level that most females think she looks better post-surgery.
Normally this would be a weak statistical result from which to draw a conclusion, but since this is a frivolous little study, we can conclude relatively certainly that females prefer her new look.
5) Is there a difference between the sample supporting rates for males and females? (Does being male or female affect how likely you are to prefer the results of the plastic surgery?)
H0: Xm Bar - Xf Bar = 0
H1: Xm Bar - Xf Bar /= 0
Xm Bar - Xf Bar = -0.42
Standard Error: sqrt( 0.81^2/19 + 0.90^2/19 ) = 0.28
Since 19 + 19 = 38, we can use the normal distribution.
Critical value: 1.64 (10%)
T-statistic: -0.42/0.28 = -1.5
Therefore we cannot conclude that males and females have different opinions of the surgery at the 10% significance level. However, we can do so at a roughly 13.3% significance level (p-value = 0.133)
6) Give a 90% confidence interval for the population supporting rate, regardless of gender (ie How favourable the population, regardless of gender, is towards the surgery). Interpret this.
(-0.05 +- 1.64[0.28]) = (-0.51,0.41)
We are 90% sure that the true population supporting rate lies within this interval.
Discussion:
It is reasonable to conclude that respondents' answers were independent and identical, since I doubt any were bored or clairvoyant enough to collaborate to screw up my study.
There was possible selection bias: My sample was not random since respondents were young, and internet savvy. Also, some people didn't want to take part (they thought I would show them disgusting pictures). Of those who took part, not everyone saw all of the "Before" pictures (most saw only one).
The suvey did not take into account factors other than gender which might have affected preference for the pre or post operative look (eg sexual orientation, age, religion). This is both because I haven't learnt multiple regression yet and because I can't afford to waste any more time on this trivial study.
Most of the "after" pictures are of the subject in question at an older age with makeup, nicer clothes and a new look. Unfortunately it is impossible to divorce the impact of these factors on her looks from the impact of the surgery.
Lastly, the survey did not measure the strength of preferences (ie How much the respondents preferred or detested the subject's new look), but that would've introduced a whole host of new problems (ie Of subjectivity).
Johnny Malkavian: holy fuck. you must be the only blogger i've read EVER that blogs statistics. you've raised the geekbar so high, it looks like a dot