TUTORIAL 4

Question 1

Suppose that you want to settle a dispute with your friend by tossing a coin. Unfortunately both of you do not carry any coin, but there happens to be a bottle cap on the floor. So you suggest to toss the bottle cap, and you both agree to call the event that it lands with the top facing up H, and the other event T. Furthermore, both of you agree that the events are not equally likely, i.e., the respective chances, h and t, are not equal.

a)Which of the following way of tossing the cap twice should be used to guarantee fairness in settling the dispute?

I.Your friend wins if the both land with the same side up. That is, he wins if the outcome is HH or TT. Otherwise, you win.

II.Your friend wins if the outcome is HT (first H, then T); you win if the outcome is TH. Otherwise, you repeat the experiment, and go through the previous rules to decide who wins.

b)Suggest a way to estimate h, the probability of the cap landing H.

Question 2

Mammography is used to screen for breast cancer. A positive outcome indicates the disease is present, and a negative outcome indicates absence, but the indications are not perfect.

In a big city, among 100,000 women with negative mammograms, 20 will be diagnosed with breast cancer within 2 years, whereas 1 woman in 10 with positive mammograms will be diagnosed with breast cancer within 2 years. Suppose that 10% of women in the city will have a positive mammogram.

(a)Construct a contingency table to that reflects the information given. State the base rate, sensitivity and specificity of the mammogram test in the city.

(b)What is the likelihood that (i) a woman who has a positive mammogram does not develop breast cancer over the next 2 years; (ii) a woman who does not develop breast cancer over the next 2 years has a positive mammogram?

Question 3

On Alans first visit to Macau, he decided to try his luck in at the Venetian Casino Resort. He played a game that is supposed to offer a 50% chance of winning. Out of seven independent plays, he won the first 6 games and lost the last game. His friend Brad who always bet against Alan is unhappy and thinks the game has been rigged. Brad approaches this issue by testing a hypothesis.

(a)State Brads null and alternative hypotheses.

(b)If the null hypothesis is true, what is the chance of winning 6 games in a row and losing the next one?

(c)Calculate the P value based on Alans data, to one significant figure.

(d)What can Brad conclude?