A-LevelMathematicsProbability and statistics9709/61 · May/June 2023 · Q46 Marks

A certain train journey takes place every day throughout the year. The time taken, in minutes, for the journey is normally distributed with variance 11.2.

(a)The mean time for a random sample of n of these journeys was found. A 94% confidence interval for the population mean time was calculated and was found to have a width of 1.4076 minutes, correct to 4 decimal places. Find the value of n.[3]
(b)A passenger noted the times for 50 randomly chosen journeys in January, February and March. Give a reason why this sample is unsuitable for use in finding a confidence interval for the population mean time.[1]
(c)A researcher took 4 random samples and a 94% confidence interval for the population mean was found from each sample. Find the probability that exactly 3 of these confidence intervals contain the true value of the population mean.[2]

Mark scheme

Marking points from the official mark scheme (May/June 2023). The mark or mark type is shown in brackets.

[M1] z * √(11.2/n) = 1.4076/2 (Any z, but must be a z.)
[B1] z = 1.881 or 1.882
[A1] n = 80 (Must be a whole number.)
[B1] Jan, Feb and March not typical of whole year. (Or, e.g., weather is different at different times of year.)
[M1] 0.94^3 * 0.06 * 4
[A1] = 0.199 (3 sf)

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