Hypothesis Testing

A binomial test is carried out to investigate whether a coin is biased towards heads, where p is the probability of obtaining a head. Which of the following correctly states the hypotheses?

• AH₀: p = 0.5, H₁: p > 0.5✓
• BH₀: p > 0.5, H₁: p = 0.5
• CH₀: p = 0.5, H₁: p ≠ 0.5
• DH₀: p = 0.5, H₁: p < 0.5

Answer: The null hypothesis assumes no bias, p = 0.5. 'Biased towards heads' is a one-tailed alternative in the upper direction, so H₁: p > 0.5. Option C is two-tailed and D tests bias the wrong way.

A teacher claims a six-sided die is biased, but does not say in which direction. A binomial test on the number of sixes is planned. Which type of test is appropriate, and what is H₁?

• AOne-tailed, H₁: p > 1/6
• BTwo-tailed, H₁: p ≠ 1/6✓
• COne-tailed, H₁: p < 1/6
• DTwo-tailed, H₁: p = 1/6

Answer: Because the claim of bias gives no direction, the test must allow deviation either way, so it is two-tailed with H₁: p ≠ 1/6. A one-tailed test would only detect bias in a single direction.

A two-tailed binomial hypothesis test is to be conducted at the 5% significance level. How is the significance level applied to the two tails of the distribution?

• A5% is placed in each tail
• B5% is placed in the upper tail only
• C10% is split as 5% in each tail
• D2.5% is placed in each tail✓

Answer: For a two-tailed test the significance level is split equally between the two tails, so each tail contains 5% ÷ 2 = 2.5%. Option A double-counts the level and B describes a one-tailed test.

For X ~ B(10, 0.25) a one-tailed test of H₀: p = 0.25 against H₁: p > 0.25 is carried out at the 5% level. The observed value is x = 6, and P(X ≥ 6) = 0.0197. What is the conclusion?

• AAccept H₁ is impossible to determine
• BDo not reject H₀; the result is not significant
• CReject H₀; there is no evidence p has increased
• DReject H₀; the result is significant, so there is evidence p > 0.25✓

Answer: Compare the p-value with the significance level: 0.0197 < 0.05, so the result lies in the critical region. We reject H₀ and conclude there is evidence that p > 0.25. Option B wrongly fails to reject; C states the wrong interpretation after rejecting.

A random variable X ~ B(20, 0.3) is used to test H₀: p = 0.3 against H₁: p < 0.3 at the 5% significance level. Given P(X ≤ 2) = 0.0355 and P(X ≤ 3) = 0.1071, state the critical region.

• AX ≤ 2✓
• BX ≤ 3
• CX ≥ 2
• DX ≤ 1

Answer: For a lower-tailed test we need the largest region with probability ≤ 5%. Since P(X ≤ 2) = 0.0355 < 0.05 but P(X ≤ 3) = 0.1071 > 0.05, the critical region is X ≤ 2. Option B exceeds 5%, and D is unnecessarily small.

In a binomial hypothesis test of H₀: p = 0.4 against H₁: p > 0.4, the calculated p-value is 0.08 and the significance level is 5%. Which conclusion is correct?

• AReject H₀: there is strong evidence that p > 0.4
• BDo not reject H₀: there is insufficient evidence that p > 0.4✓
• CAccept H₁: the proportion has definitely increased
• DReject both H₀ and H₁ as inconclusive

Answer: Since the p-value 0.08 > 0.05, the result is not in the critical region, so we do not reject H₀. We conclude there is insufficient evidence to support p > 0.4. Option A rejects despite the p-value exceeding the level.