Question

Please label your work correctly and make them as visible and coherent as possible.

1)In your own words, explain the Simpsons Paradox.

2)How does the coefficient of determination differ from the correlation coefficient?

3)Explain the difference between the empirical method and the classical method of calculating probabilities.

4) In your own words, describe what it means for two events to be mutually exclusive, or disjoint. Give an example of two events that are mutually exclusive/disjoint.

5) Explain the difference between correlation and causation. When is it appropriate to state that the correlation implies causation?

Answer 1

1) Simpson’s paradox characterizes a treatment that appears effective at the population-level may have adverse consequences within each of the population's subgroups i.e when groups of data show one particular trend, but this trend is reversed when the groups are combined together.

For instance, a higher dosage of medicine may be associated with higher recovery rates at the population-level; however, within subgroups (e.g., for both males and females), a higher dosage may actually result in lower recovery rates. Even though a negative relationship exists between “Treatment Dosage” and “Recovery” in both males and females, when these groups are combined a positive trend appears.

2) or coeff. of determination shows percentage variation in y which is explained by all the x variables together. Higher the better. It is always between 0 and 1. It can never be negative – since it is a squared value.

For example; If we try to predict height by using weight as a single predictor, coefficient of determination is 0.64 (equals to square of correlation coefficient ).this means that 0.64 (or 64%) of variation in height can be explained by weight and remaining 36% of variation in height may be due to other factors which affect height of individuals like age, genetics, food intake etc.

The quantity , called the correlation coefficient, measures the strength and the direction of a linear relationship between two variables. For example; height and weight of individuals are correlated with correlation coefficient

R = 0.8 this means that both height and weight of individuals increase/decrease together (positive) and their relationship (linear) is strong.

3) Classical probability is a simple form of probability that has equal odds of something happening. For example: Rolling a fair die. It’s equally likely you would get a 1, 2, 3, 4, 5, or 6.

Empirical probability, also known as experimental probability,is a probability that is based on historical data. In other words, empirical probability illustrates the likelihood of an event occurring based on historical data.

4) Two events (call them "A" and "B") are Mutually Exclusive/Disjoint it is impossible for them to happen together i.e

P( A and B ) =0.

Example- In a Deck of 52 Cards the occourence of a king and the occourence of a queen are disjoint events.

P(King and Queen) = 0