May/June 2026 Paper 62

Eric takes 100 throws and hits the centre $x$ times. He uses this result to find an approximate 95% confidence interval for $p$. The width of this interval is 0.165 correct to 3 significant figures. Find the two possible values of $x$.

Eric takes some more throws. He uses his results to calculate two approximate $\alpha \%$ confidence intervals for $p$ (where $\alpha >90$). It is given that the probability that exactly one of these two intervals contains the true value of $p$ is 0.1128. Find the value of $\alpha$.