A-LevelMathematicsProbability and statistics9709/63 · May/June 2024 · Q69 Marks

The numbers of green sweets in 200 randomly chosen packets of Frutos are summarised in the table. Number of green sweets 0 1 2 3 > 3 Number of packets 32 50 97 21 0 The manufacturers of Frutos claim that the mean number of green sweets in a packet is 1.65 . Anji believes that the true value of the mean, μ, is less than 1.65. She uses the results from the 200 randomly chosen packets to test the manufacturers' claim.

(a)Calculate an unbiased estimate for the population mean of the number of green sweets in a packet of Frutos, and show that an unbiased estimate of the population variance is 0.783 correct to 3 significant figures.[3]
(b)State suitable null and alternative hypotheses for the test.[1]
(c)Show that the result of Anji's test is significant at the 5% level but not at the 1% level.[4]
(d)It is given that Anji made a Type I error. Explain how this shows that the significance level that Anji used in her test was not 1%.[1]

Mark scheme

Marking points from the official mark scheme (May/June 2024). The mark or mark type is shown in brackets.

[B1] μ̂ = 307/200 or 1.535
[M1] Σx²f = 627, Est(σ²) = 200/199 ('627/200' – '1.535'²) or (1/199) ('627' – '307'²/200')
[A1] = 0.783
[B1] Ho: μ = 1.65 H₁: μ < 1.65
[M1] ('1.535'-1.65) / √(0.783/200)
[A1*] = -1.838 or -1.84
[M1] Φ(0.05) and Φ(0.01) attempted
[DA1] -1.645 > -1.838 > -2.326
[B1] At the 1% level Ho is not rejected

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