AS & A Level · Practice questions · Hypothesis tests with binomial and Poisson distributions

State, calculate, compare, conclude.

Six original Cambridge-style Paper 6 questions on hypothesis tests for a binomial proportion or a Poisson mean, with full worked solutions.

Original questions All questions on this page are original work, written in the Cambridge AS & A Level style. They are not from past papers. They test the same concepts and skills the syllabus rewards.

The method

Hypotheses, distribution under \(H_0\), tail, compare, conclude.

Hypotheses: \(H_0: p = p_0\) (or \(\lambda = \lambda_0\)); \(H_1\) with \(<\), \(>\) or \(\ne\) from the wording.
Distribution under \(H_0\): \(B(n, p_0)\) or \(\text{Po}(\lambda_0)\), scaled to the observed period.
Tail probability: observed value or more extreme, in the direction of \(H_1\).
Compare with the significance level (half of it in each tail for a two-tailed test).
Conclude in context, as evidence, never as proof.
Large \(n\) or \(\lambda\): normal approximation with continuity correction.

Recall

[2 marks]

Explain what is meant by (a) the critical region (b) the significance level of a hypothesis test.

(a) The set of values of the test statistic for which \(H_0\) is rejected. ✓
(b) The probability threshold for rejecting \(H_0\); equivalently, the probability of a result in the critical region when \(H_0\) is true. ✓

Calculation

[4 marks]

A coin is tossed 20 times and lands heads 15 times. Test at the 5% significance level whether the coin is biased towards heads.

\(H_0: p = 0.5\); \(H_1: p > 0.5\). Under \(H_0\), \(X \sim B(20, 0.5)\). ✓
\(\text{P}(X \ge 15) = 0.0207\). ✓
\(0.0207 < 0.05\): reject \(H_0\). ✓
There is sufficient evidence at the 5% level that the coin is biased towards heads. ✓

Calculation

[3 marks]

Under \(H_0\), \(X \sim B(25, 0.2)\). Find the critical region for a test of \(H_1: p < 0.2\) at the 10% level, and state the actual significance level.

\(\text{P}(X \le 2) = 0.0982 \le 0.10\); \(\text{P}(X \le 3) = 0.2340 > 0.10\). ✓
Critical region \(X \le 2\); actual significance level \(9.82\%\). ✓

Calculation

[4 marks]

A machine breaks down at random at a mean rate of 3.2 times per month. After a service, it breaks down once in the following 2 months. Test at the 2% level whether the mean rate has decreased.

\(H_0: \lambda = 6.4\) (2 months); \(H_1: \lambda < 6.4\). Under \(H_0\), \(X \sim \text{Po}(6.4)\). ✓
\(\text{P}(X \le 1) = e^{-6.4}(1 + 6.4) = 0.0123\). ✓
\(0.0123 < 0.02\): reject \(H_0\). ✓
There is sufficient evidence at the 2% level that the breakdown rate has decreased. ✓

Analysis

[5 marks]

A shop believes that 40% of its customers pay in cash. In a random sample of 30 customers, 6 pay in cash. Test at the 10% level whether the proportion paying in cash is different from 40%.

\(H_0: p = 0.4\); \(H_1: p \ne 0.4\) (two-tailed: 5% in each tail). ✓
Under \(H_0\), \(X \sim B(30, 0.4)\) with mean 12; 6 is below, so use the lower tail. ✓
\(\text{P}(X \le 6) = 0.0172\). ✓
\(0.0172 < 0.05\): reject \(H_0\). ✓
There is sufficient evidence at the 10% level that the proportion paying in cash is not 40%. ✓

Analysis

[6 marks]

A council claims that 60% of residents support a new park. In a random survey of 150 residents, 78 support it.

(a) Use a suitable approximation to test at the 5% level whether support is lower than the council claims.

(b) Explain why a normal approximation is valid here.

(a) \(H_0: p = 0.6\); \(H_1: p < 0.6\). \(X \sim B(150, 0.6) \approx N(90, 36)\). ✓
\(\text{P}(X \le 78) \approx \text{P}(Y < 78.5)\), \(z = \dfrac{78.5 - 90}{6} = -1.917\). ✓
\(\Phi(-1.917) = 1 - 0.9724 = 0.0276 < 0.05\): reject \(H_0\). ✓
There is sufficient evidence at the 5% level that support is lower than 60%. ✓
(b) \(np = 90\) and \(nq = 60\) are both greater than 5. ✓

Mark this once you have attempted all six and checked your working. It records a Practiced badge on the topic and adds a one-time bonus. Revealing the solutions alone does not count.