Mathematics 9709 · AS & A Level · Hypothesis testing

Hypothesis testing — practice question

Each July, as part of a research project, Rita gathers information on sightings of one particular species of bird. On every day in July she records whether or not she sees this bird, and she counts the number $X$ of days on which it is observed. She represents the distribution of $X$ by $\mathrm{B}(31, p)$, where $p$ is the chance of seeing this bird on a randomly selected day in July. Results from earlier years indicate that $p = 0.3$, but in 2022 Rita thought that the value of $p$ might have fallen. She chose to perform a hypothesis test. During July 2022, she observed this bird on $4$ days.
(a)[5]

Use the binomial distribution to test, at the $5\%$ significance level, whether Rita’s suspicion is supported.

(b)[2]

Calculate the probability of making a Type I error.

(c(i))[3]

Use a suitable approximating distribution to find $\mathrm{P}(Y = 4)$.

(c(ii))[1]

Justify the choice of approximating distribution in this context.

Worked solution & mark scheme

This 11-mark question has a full step-by-step worked solution and mark scheme. One marking point: State the correct hypotheses as $H_0:p=0.3$ and $H_1:p<0.3$.

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