Question

In: Statistics and Probability

1. Our brains don’t like losses. Most people dislike losses more than they like gains. In...

1. Our brains don’t like losses. Most people dislike losses more than they like gains. In
money terms, people are about as sensitive to a loss of $10 as to a gain of $20. To
discover what parts of the brain are active in decisions about gain and loss,
psychologists presented subjects with a series of gambles with different odds and
different amounts of winnings and losses. From a subject’s choices, they constructed
a measure of “behavioral loss aversion.” Higher scores show greater sensitivity to
losses. Observing brain activity while subjects made their decisions pointed to
specific brain regions. Here are data for 16 subjects on behavioral loss aversion and
“neural loss aversion,” a measure of activity in one region of the brain:1

d. Find the least-squares line for predicting y from x, leaving out the outlier, and
add the line to your plot.
e. The outlier lies very close to your regression line. Looking at the plot, you
now expect that adding the outlier will increase the correlation but will have
little effect on the least-squares line. Explain why.
f. Find the correlation and the equation of the least-squares line with and
without the outlier. Your results verify the expectations from (e).
Neural -50 -39.1 -25.9 -26.7 -28.6 -19.8 -17.6 5.5
Behave 0.08 0.81 0.01 0.12 0.68 0.11 0.36 0.34
Neural 2.6 20.7 12.1 15.5 28.8 41.7 55.3 155.2
Behave 0.53 0.68 0.99 1.04 0.66 0.86 1.29 1.94

Solutions

Expert Solution

1.

Suppose, random variables X and Y denote neural and behave respectively.

(d)

We leave out the outlier (155.2,1.94).

Least square regression line is given by

We have,

We add the obtained least square regression line for predicting y from x in our plot as follows.

(e)

For x=155.2, predicted value of y is given by

This is very close to our observed value 1.94. Adding any point near to regression line increases value of correlation coefficient. Being very closed to our calculated regression line, it will have little effect on the least-squares line.

So, adding the outlier will increase the correlation but will have little effect on the least-squares line.

(f)

With the outlier-

Correlation coefficient is given by

Least square regression line is given by

We have,

Without the outlier-

Correlation coefficient is given by

Least square regression line is given by

We have,

Clearly, obtained least-squares lines for predicting y from x with and without outlier are very close. Thus these results verifies expectations from (e).


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