Question

3. A coin was tossed 10 times and “heads” appeared exactly 2 times. Is there sufficient evidence that the coin is not fair, that is that the proportion of heads is less than 0.5, at the α = 0.05 significance level? (Note: the sample size is small.) Do the5 step process to show work.

Answer 1

Here, we have to use one sample z test for the population proportion.

The null and alternative hypotheses for this test are given as below:

Null hypothesis: H₀: Coin is fair.

Alternative hypothesis: H_a: Coin is not fair.

H₀: p = 0.5 versus H_a: p < 0.5

This is a lower tailed test.

We are given

Level of significance = α = 0.05

Test statistic formula for this test is given as below:

Z = (p̂ - p)/sqrt(pq/n)

Where, p̂ = Sample proportion, p is population proportion, q = 1 - p, and n is sample size

x = number of items of interest = 2

n = sample size = 10

p̂ = x/n = 2/10 = 0.2

p = 0.5

q = 1 - p = 0.5

Z = (p̂ - p)/sqrt(pq/n)

Z = (0.2 - 0.5)/sqrt(0.5*0.5/10)

Z = -1.8974

Test statistic = -1.8974

P-value = 0.0289

(by using z-table)

P-value < α = 0.05

So, we reject the null hypothesis

There is sufficient evidence to conclude that Coin is not fair.