Definitions:

· Type I Error = rejecting the null hypothesis even though it’s true (e.g., false alarm)

o The probability of a type I error is controlled by the level of significance.

· Type II Error = failing to reject the null hypothesis even though it’s false (e.g., missed opportunity)

o The power of a type II error is often calculated with respect to a specific alternative value for the parameter of interest (e.g., he’s become a 0.300 hitter)

o Several factors impact the probability of a type II error

Recap of Investigation 1.7

Manager believes the player to be a 0.250 hitter, but player claims to now be a 0.300 hitter.

A type II error is committed if the manager fails to decide the player has actually improved.

With the Power Simulation applet,

	This is the distribution of the number of successes assuming = 0.250. By setting the level of significance to 0.05, we find that the rejection region is X > 9, giving us (an exact) P(Type I error) = 0.0409.
	This is the distribution of the number of successes assuming = 0.333. Using the rejection region we found from the null distribution, we find P(X > 9 when = 0.333) = .1897, so the power is equal to 0.1897.

The probability of a Type II error = 1 – power, so in this case, there is about an 80% chance that the manager will decide the player has not improved even though he actually had.

Calculating power is a two-step process

Step one: Determine the rejection region corresponding to the null hypothesis hypothesized value, the direction of the alternative hypothesis, and the level of significance.

Step two: Determine the probability of obtaining an observation in the rejection region for a specific alternative value of the parameter.