## Question 269:

1

There are two ways to make the calculations. The first is using the binomial probability theorem and calculate the exact probabilities. The second way is to use the normal approximation to the binomial, which you can do since the values are reasonably large (n=100 and the smallest cell count is 23).  Since this example gave you a standard deviation, I'm going to assume they want you to use the normal approximation.

So all you need to do is calculate the z-scores given the mean of 20 and standard deviation of 4 (given 100 guesses).

1. For 23 correct guesses, the probability that this is better than chance 20 is (23-20)/4 = .75, which is the z-score. Looking the probability for this value using the z-score to percentile calculator (selecting 1-sided b/c we only want better than chance) provides a probability of  77%.
2. For 27 correct guesses, (27-20/4)= 1.75 or a probability of ~96% or we'd expect someone to get 27 correct just from chance alone around 4% of the time.
3. Finally, at 36 correct guesses (36-20)/4 = a z-score of 4, there is less than a .01% probability of this being by chance alone, or a 99.99% chance this person is not just guessing.

A conclusion would need to include the fact that at some threshold, say around 27 correct guesses we might start to conclude that there is enough evidence that something unusual is going on and chance alone is not explaining things sufficiently. Certainly at 36 guess, the evidence is very strong.

By the way, if you were to use the exact probabilities using the binomial, you'd have 81%, 97% and 99.99% for 1-3 above. As you can see, the normal distribution comes pretty close. To calculate those values in excel type =BINOMDIST(23,100,0.2,TRUE) in any cell. The 1st value in parenthesis is the number correct.