Question

A coin is tossed 23 times, and the sequence of heads and tails is the outcome. A statistical test is conducted for the following hypotheses. H,: The coin is a fair coin. H,: The chance of obtaining a head is three time as the chance of obtaining a tail. The critical region for the test is the event “more than k heads”. Here k is a positive integer. If we want the power of the test to be at least 80%. Find the largest value of k. a. 16 b. 17 c.18 d. 19

Answer 1

The power of the test has to be at least 80%. The power of the test is computed as the probability of rejecting the null hypothesis given that the null hyopthesis is false that is P(head) = 3P(tail) which means that P(head) = 0.75

Therefore, we have here:
P(X <= k) = 0.8 for p = 0.75 here.

We would be doing this using the hit and trial method here:

For k = 17

We have from EXCEL here:
=BINOM.DIST(17,23,0.75,TRUE)

0.5315 is the output here but as this is < 0.8, we need to look for higher k values here.

For k = 18, we have here from EXCEL:
=BINOM.DIST(18,23,0.75,TRUE)

0.7168 is the output here but as this is < 0.8, we need to look for higher k values here.

For k = 19, we have here from EXCEL:
=BINOM.DIST(19,23,0.75,TRUE)

0.8630 is the output here but as this is > 0.8, we need to look for higher k values here.

Therefore k = 19 is the required value here.