# Stat 202: Lecture 13 (student review)

Nathan VanHoudnos
10/17/2014

### Agenda

1. Checkpoint results
3. Review

### Homework Results

These are off by plus or minus 10/160 = .0625

mean(hw4.grades/160)

[1] 0.7055556

median(hw4.grades/160)

[1] 0.84375


Your grades are likely off by 10 pts (Question #4 part E):

• I distributed an incorrect solution set to both the TA and BlackBoard, and
• did not catch the error until after the the TA had graded your assignments (Thanks Stephanie!).

The correct solutions are now on BlackBoard, and in the hands of the TA.

Aaron will re-grade Question #4 part E on Monday.

My apologies!

### Homework: Larry Summers

Larry Summers President of Harvard (2001-06)

• Gave a speech claiming that aptitude differences between men and women explained why men are over represented in science faculty positions.

• The speech was met with intense criticism

• Resigned within a year
• Cost him the US Treasury Secretary position in Obama's administration

### Teachable Moment: Larry's claim

From Question #5:

Can all three of the following statements be true?

1. The average woman is smarter than the average man.
2. In fact, 97.5% of women are smarter than the average man.
3. And yet, the top 2.5% of men are smarter than 99.85% of women.

### Teachable Moment: Larry's claim

Assume that the IQs of men and women are distributed normally.

• Let the IQs of women be distributed with a mean of 100 and a standard deviation of 10.

### Teachable Moment: Larry's claim

Assume that the IQs of men and women are distributed normally.

• Let the IQs of women be distributed with a mean of 100 and a standard deviation of 10.
• The 68-95-99.7 rule implies that .05/2 = 2.5% of women have an IQ below 80.

### Teachable Moment: Larry's claim

Assume that the IQs of men and women are distributed normally.

• Let the IQs of men be distributed with a mean of 80 and a standard deviation of 25.

### Teachable Moment: Larry's claim

Assume that the IQs of men and women are distributed normally.

• Let the IQs of men be distributed with a mean of 80 and a standard deviation of 20.
• The 68-95-99.7 rule implies that only .05/2 = 2.5% of men have an IQ above 130.

### Teachable Moment: Larry's claim

The 68-95-99.7 rule implies that

• 2.5% of men have an IQ over 130
• 0.003/2 = 0.15% of women have an IQ over 130

### Teachable Moment: Larry's claim

1. The average woman is smarter than the average man.
2. 97.5% of women are smarter than the average man.
3. And yet, the top 2.5% of men are smarter than 99.85% of women.

### Teachable Moment: Larry's claim

“even small differences in the standard deviation [between genders] will translate into very large differences in the available pool substantially out [from the mean]”

If assume IQ is normally distributed and interpret

• “substantially out from the mean” as several standard deviations, and
• have evidence$$^\text{[citation needed]}$$ that the standard deviation of Men's and Women's IQ differs,

then Larry was was correct. (Question #6)

### Lessons from Larry

• Political discussions are rarely settled by mathematics because the assumptions of the two sides often differ.