Assume that someone claims they can predict if the coin lands heads or tails up
ID: 3332525 • Letter: A
Question
Assume that someone claims they can predict if the coin lands heads or tails up based only on the sound of the coin (not looking at the coin). You want to check this claim and conduct the experiment: the coin is flipped n times, and the “magician” tries to predict the outcome. (a) State the hypothesis for this problem and, assuming that n = 10, find the test with significance level 0.1 (think carefully about the alternative hypothesis).
(b) Find the (approximate) probability of type II error against the alternative that the “magician” can predict the outcome based on the sound with probability 0.75.
(c) Assume that the “magician” guessed 7 out of 10 coins correctly. What is the associated p-value of your test?
(d) What is the minimal number of coins throws necessary to obtain the test with significance level 0.1 and power 0.9 (for the alternative hypothesis as in the previous question)?
Explanation / Answer
Answer to part a)
The hypothesis are as follows:
Null hypothesis : Proportion of correct claims P = 0.5
Alternate hypothesis; Proportion of correct claims P > 0.5
[right tailed test]
.
Part b)
At 0.1 significance level , z = 2.575
Z = (P^ - P) / SE
SE = sqrt(0.5*0.5 /10) = 0.1581
2.575 = (p^ - 0.5) / 0.1581
p^ = 0.9071
.
Thus We get z = (0.9171 -0.75) / sqrt(0.75*0.25/10)
z = 0.1671 / 0.1369 = 1.22
P(z > 1.22) = 0.1112
.
Answer to part c)
If sample proportion P^ = 7/10 = 0.7
z = (0.7-0.5) / 0.1581
z = 1.27
P(z > 1.27) = 0.1020
.
Answer to part d)
In part b, the power value = 1 -0.1112 = 0.8888
We need power > 0.9
For that the sample size may be 11