Probability assignment (Assignment 3), Psych. 153, Fall 2009

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

You answered the following questions about each item:

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 class is 56.5%.

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    63.3    10.3  53.1      69.3      21.6  47.7     77.7     16.6  61.1
sports       23.3    89.7 -66.4      40.3      83.9 -43.7     30.8     87.1 -56.3
computer     20.0    12.8   7.2      20.5      36.1 -15.6     13.3     28.7 -15.4
tall67       26.7    94.9 -68.2      27.8      77.7 -49.8     28.6     77.0 -48.4
dress15      83.3    23.1  60.3      75.6      32.6  43.0     93.5     23.0  70.6

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

4. Are the judgments of p(y|m) for lovestory calculated by Bayes more accurate than the initial direct judgments of p(y|m) at capturing the true p(y|m)?
yes no

5. Are the values calculated by Bayes's theorem for lovestory more accurate than the direct judgments at capturing the difference?
yes no

Should the values calculated by Bayes's theory always be more accurate?
6. true false Yes, because Bayes's theorem is based on a normative model.
7. true false No, because they are indirect and the other judgments are direct.
8. true false No, 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 2.

          RealM|Y RealM|N  Diff JudgedM|Y JudgedM|N  Diff BayesM|Y BayesM|N  Diff
lovestory    17.4    76.1 -58.7      19.9      74.1 -54.2     26.2     78.8 -52.6
sports       83.3    14.8  68.5      78.3      14.6  63.7     75.8     23.7  52.2
computer     45.5    58.6 -13.2      78.0      37.1  40.9     74.7     50.1  24.6
tall67       82.2     8.3  73.9      77.7      23.2  54.4     80.1     26.6  53.5
dress15      26.5    85.7 -59.2      21.2      77.1 -55.9     32.2     82.5 -50.3

How do the Bayes values (BayesM|Y and BayesM|N) - not their differences - compare to the real values?
9. all too large all too small some of each

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  33.3    43.3       42.4
sports     60.9    63.7       65.0
computer   15.9    26.7       29.3
tall67     65.2    55.6       56.0
dress15    49.3    55.3       51.3

How do the values calculated by conditional assessment (third column) compare to the judged values in predicting the true values?
10. CA consistently closer to the real values than are the judged values
CA consistently farther from the real values than are the judged values
CA neither consistently closer nor consistently farther

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

Finally, here are the results for the mean absolute error. The values with the prime (apostrophe) are calculated, either by conditional assessment or Bayes. The others are the judgments you made.

          p(y) p'(y) p(y|f) p'(y|f) p(y|m) p'(y|m) p(m|y) p'(m|y) p(m|n) p'(m|n)
lovestory 12.1  11.1   17.0    22.4   12.9    11.3   12.6    13.3   13.5    14.5
sports    11.5   9.7   19.0    14.5    9.7    18.1    9.9    10.5    7.6    12.8
computer  16.6  18.8   16.9    13.9   25.6    17.8   33.2    29.3   21.6     8.5
tall67    12.0  10.5    9.7    12.7   17.4    22.3    9.6     7.1   15.3    18.6
dress15   14.7  13.6   14.1    22.9   18.5    15.5   12.3    12.8   12.1    11.1

I have no question about these. It seems obvious that nothing interesting is happening. For example, I thought that p(y|m) would be more accurate than p(m|y), but it isn't.

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