Question

There are two seemingly identical coins in front of you. One of them is a fair coin that results in a relatively equal number of heads and tails. The other is a weighted coin that results in more heads than tails. By flipping only one of the coins, you must determine which coin is the fair coin and which coin is the weighted coin.

You pick a coin and flip it 80 times observing 51 heads and 29 tails. Use a binomial test and a significance level of α=0.01 to determine whether the coin you were flipping led to a different number of heads than would be expected from a fair coin.

This test is using the hypotheses:

H0:p=0.5H0:p=0.5
H1:p≠0.5H1:p≠0.5

a) What is the critical value, zcrit, found in the z- or t-table?

b) What is the value of the test statistic for this scenario? Express your answer rounded to three decimal places.

Answer 1

a) The critical value (zcrit) is approximately 2.58.

b) The value of the test statistic is approximately 2.461.

Explanation:

a) To determine the critical value (zcrit) for the binomial test, we need to use the significance level (α = 0.01) and the fact that we are conducting a two-tailed test (H1: p ≠ 0.5). Since the binomial distribution does not have a symmetric shape, we can use the normal approximation to find the critical value.

The critical value for a two-tailed test with a significance level of 0.01 would be found by dividing the significance level by 2 (0.01 / 2 = 0.005) and finding the corresponding z-score in the standard normal distribution table. This z-score will give us the critical value for each tail.

Using the standard normal distribution table, we can find the critical value zcrit by locating the cumulative probability closest to 0.995 (1 - 0.005). In other words, we need to find the z-score that corresponds to a cumulative probability of 0.995.

b) The test statistic for this scenario can be calculated using the formula for a binomial test:

z = (X - np) / √(np(1 - p))

Where:

X is the number of heads observed (51 in this case),

n is the total number of flips (80 in this case), and

p is the hypothesized probability of success (0.5 in this case).

Plugging in the values, we can calculate the test statistic:

z = (51 - (80 * 0.5)) / √(80 * 0.5 * (1 - 0.5))

Now, let's calculate the critical value (zcrit) and the test statistic (z):

a) The critical value (zcrit) for a two-tailed test at a significance level of 0.01 can be found by looking up the cumulative probability of 0.995 in the standard normal distribution table. The closest value is 2.58 (rounded to two decimal places).

b) Calculating the test statistic (z):

z = (51 - (80 * 0.5)) / √(80 * 0.5 * (1 - 0.5))

≈ (51 - 40) / √(20)

≈ 11 / 4.472

≈ 2.461 (rounded to three decimal places)

Therefore:

a) The critical value (zcrit) is approximately 2.58.

b) The value of the test statistic is approximately 2.461.