## Probability assignment (Assignment 2), Psych. 153, Fall 2010

This assignment is very simple. It is designed to get you go look at the data from Assignment 1 and think about them a little.

The items were:

 love prefer love stories to action/adventure movies sports choose to watch football, basketball, or hockey on TV program have programmed a computer tall am 67 inches tall or more (5 feet, 7 inches) dress spend 15 minutes or more getting dressed each day

 me whether the statement is true for you p(y) your judged probability of yes for a student in the class p(y|m) judged probability of yes for a male student p(y|f) judged probability of yes for a female student p(m|y) judged probability of being male given yes p(m|n) judged probability of being male given no

Here are some data, in percents. The proportion of males in the group who did the assignment is 51.8%.

In the following tables: Real=true proportions, Judge=judgments, Final=judgment after calculation, Diff=female-male difference, Bayes=calculated by Bayes's theorem, e.g., p(Y|M) = p(M|Y)*p(Y)|p(M). "|" means "conditional on." Notice that we can apply Bayes' theorem to calculate many things other than p(M|Y), which was what you were asked to calculate.

Answer the questions after each data set.

```          RealY|F RealY|M  Diff JudgedY|F JudgedY|M  Diff BayesY|F BayesY|M  Diff
lovestory    73.6    12.1  61.5      70.9      19.4  51.6     60.0     21.1  38.9
sports       24.5    57.6 -33.0      40.2      79.5 -39.2     27.9    117.9 -90.0
computer      5.7    21.2 -15.6      15.1      34.6 -19.6      7.9     44.5 -36.6
tall67       28.3    93.9 -65.6      30.5      74.8 -44.3     20.7    107.1 -86.3
dress15      64.2    21.2  42.9      76.4      29.3  47.1     68.3     31.7  36.5
```

What can you best conclude about the accuracy of perception of the male/female "stereotype" from these data?
1. Most judgments about gender differences are in the right direction and very roughly of the right magnitude.
2. All male/female differences are exaggerated, showing an exaggerated stereotype.
3. All judged differences are too small, showing a weaker stereotype than is true.

4. Are the judgments of p(y|m) for computer calculated by Bayes more accurate than the initial direct judgments of p(y|m) at capturing the true p(y|m)?

5. Are the values calculated by Bayes's theorem for computer more accurate than the direct judgments at capturing the male/female difference?

Should the values calculated by Bayes's theory always be more accurate (in terms of correspondence)?
6. Yes, because Bayes's theorem is a normative model.
7. Not always, because they are based on other judgments, which could be better or worse than the direct judgments.

Here are more data. The column labeled BayesM|Y is what you were asked for in the very last part of Assignment 1.

```          RealM|Y RealM|N  Diff JudgedM|Y JudgedM|N  Diff BayesM|Y BayesM|N  Diff
lovestory     9.3    67.4 -58.1      17.4      76.5 -59.1     15.7     57.7 -42.0
sports       59.4    25.9  33.4      73.0      18.0  55.1     52.4     18.9  33.5
computer     70.0    34.2  35.8      71.1      25.1  46.0     54.9     32.4  22.5
tall67       67.4     5.0  62.4      74.3      21.7  52.5     55.5     19.1  36.4
dress15      17.1    57.8 -40.7      21.8      71.6 -49.8     19.7     60.4 -40.8
```

How do the Bayes values (BayesM|Y and BayesM|N) - not their differences - compare to the real values?
8.

What is illustrated by the fact that, for "lovestory", BayesM|Y and JudgedM|Y are very close but both very far from RealM|Y (and likewise for "sports" M|N)?
9.

Here are data relative to conditional assessment of the probability of "yes," which you were asked to calculate in the first part of the assignment.

```          TrueY JudgedY CondAssess
lovestory  50.0    45.4       51.1
sports     37.2    61.5       55.3
computer   11.6    23.5       22.6
tall67     53.5    52.6       47.5
dress15    47.7    56.2       58.3
```

How do the values calculated by conditional assessment (third column of numbers) compare to the judged values (second column of numbers) in predicting the true values?
10.

Should the values calculated by conditional assessment always be more accurate?
11. Yes, because conditional assessment is based on a normative model.
12. No, because they are based on other judgments, which could be better or worse than the direct judgments.

Consider the following graph. The dashed lines are the correct proportions based on the class's answers.

How would you describe what it shows about conditional assessment?
13.

IMPORTANT!!!