Introduction to Hypothesis Testing with the Binomial Distribution (Year 12)

Suppose you toss a coin, which you believe to be fair, 10 times.

Let p = proportion of heads obtained.

We say that the Null Hypothesis is p=0.5, the initial belief before conducting the experiment. The correct notation is H₀ : p=0.5

Case 1

Suppose the experiment produces 9 heads from the 10 tosses. We say that 9 is the Test Statistic. Could this be evidence that the coin is biased towards heads? i.e. p > 0.5?

Or is this just a random fluctuation? i.e. The coin is actually fair, so p=0.5.

For this case our hypotheses are H₀ : p=0.5 and H₁ : p>0.5 where H₁ is called the Alternative Hypothesis.

Case 2

Suppose the experiment produces 3 heads from the 10 tosses, so the Test Statistic is 3.

Could this be evidence that the coin is biased against heads? i.e. p < 0.5?

Or is this just a random fluctuation?

For this case our hypotheses are H₀ : p=0.5 and H₁ : p<0.5

Both of these cases are known as One-Tailed Tests.

A Two-Tailed Test would have hypotheses of the form H₀ : p=0.5 and H₁ : p ≠ 0.5

Conducting the Hypothesis Test

Case 1

H₀ : p=0.5 and H₁ : p>0.5

Let X = number of heads from the 10 tosses. Assuming H₀ is true then X~B(10, 0.5).

Now calculate the probability of obtaining the test statistic or any value more extreme; comparing this with a significance level of 5% will decide whether to reject H₀ or not.

i.e. P(X ≥ 9) = 1 - P(X ≤ 8) = 1 - 0.9892578125 = 0.0107421875 < 0.05

So this is a very unlikely event which is smaller than the significance level. We say that the test is significant, so we reject H₀. There is evidence that the coin is biased towards heads.

Case 2

H₀ : p=0.5 and H₁ : p<0.5

As before, let X = number of heads from the 10 tosses, so that under H₀ we have

X~B(10, 0.5).

Now calculate the probability of obtaining the test statistic or any value more extreme and compare this with a significance level of 5%.

i.e. P(X ≤ 3) = 0.171875 > 0.05

This is a small probability but still greater than the significance level. We say that the test is not significant, so we do not reject H₀. There is insufficient evidence that the coin is biased.