Question

In order to test whether a certain coin is fair, it is tossed ten times and the number of heads (X) is counted. Let p be the "head probability". We wish to test the null hypothesis: p = 0.5 against the alternative hypothesis: p > 0.5 at a significance level of 5%.

(a) Suppose we will reject the null hypothesis when X is smaller than h. Find the value of h.

(b) What is the probability of committing a type I error?

(c) What is the probability of committing a type II error if the true value of p is 0.75?

Answer 1

Here we have n=10 and p=0.5

(a)

Standard deviation of the proportion is:

Since test is right tailed so critical value of z for which we will reject the null hypothesis is 1.645. So critical value of sample proportion for which we will reject the null hypothesis is

So required X is x= 10* 0.8 = 8

That is if X >= 8, we will reject the null hypothesis.

(b)

The  probability of committing a type I error is equal to 0.05. That is it is equal to 0.05.

(c)

Standard deviation of the proportion is:

The z-score for p = 0.75 and is

The type II error is

Answer 2

(a) To find the value of "h," we need to determine the critical value that separates the critical region (where we reject the null hypothesis) from the non-critical region (where we fail to reject the null hypothesis) at a significance level of 5%.

Since the null hypothesis is p = 0.5, the distribution of the number of heads (X) follows a binomial distribution with parameters n = 10 (number of coin tosses) and p = 0.5 (under the null hypothesis).

We need to find the value of "h" such that the probability of getting a result smaller than or equal to "h" is less than or equal to 0.05.

Mathematically, we want to find the smallest integer value of "h" such that:

P(X ≤ h) ≤ 0.05

Using a binomial probability table or a statistical software, we can find that P(X ≤ 4) ≈ 0.0269 and P(X ≤ 5) ≈ 0.0552.

Since we want the probability to be less than or equal to 0.05, the critical value "h" should be 5. Therefore, we will reject the null hypothesis if the number of heads (X) is smaller than 5.

(b) The probability of committing a type I error (α) is the significance level of the test, which is given as 5% or 0.05. It represents the probability of rejecting the null hypothesis when it is actually true. In this case, it means wrongly concluding that the coin is unfair (p > 0.5) when it is, in fact, fair (p = 0.5).

(c) The probability of committing a type II error (β) depends on the true value of "p" under the alternative hypothesis and the critical value "h."

Given that the true value of "p" is 0.75 under the alternative hypothesis (p > 0.5), we want to find the probability of failing to reject the null hypothesis (not reaching the critical region) when "p" is actually 0.75.

Mathematically, we want to find:

P(X ≥ h | p = 0.75)

Using the binomial probability formula:

P(X = x | n, p) = (nCx) * p^x * (1 - p)^(n - x)

where nCx represents "n choose x" (the number of combinations of "n" items taken "x" at a time).

For "h" = 5, we need to calculate:

P(X = 5 | n = 10, p = 0.75) + P(X = 6 | n = 10, p = 0.75) + ... + P(X = 10 | n = 10, p = 0.75)

Using a binomial probability table or a statistical software, we can find these individual probabilities and sum them up to find the probability of committing a type II error. This value will depend on the specific probabilities for each outcome in the binomial distribution.