Candidates for a certain diploma take two tests. Their marks for the first test and the second test are modelled by the independent variables with distributions $$$\mathrm{N}\left(38.1,3.8^{2}\right)$$$ and $$$\mathrm{N}\left(64.0,6.1^{2}\right)$$$ respectively. The final mark, $$$F$$$, for each candidate is found by doubling the mark in the first test and adding the result to the mark in the second test. Find the probability that the mean, $$$\bar{F}$$$, of the final marks of a random sample of 25 candidates is greater than 143. . . . . . . . . . . . . . . . . . . . . . . . .

Mathematics

IGCSE&ALevel

CAIE

Exam No：9709_s25_qp_65 Year：2025 Question No：5

Answer:

6.2.1.1 more contents

6.2.1.2 more contents

6.2.1.3 more contents

6.2.1.4 if X has a normal distribution then so does aX + b

6.2.1.5 if X and Y have independent normal distributions then aX + bY has a normal distribution

6.2.1.6 if and have independent Poisson distributions then X + Y has a Poisson distribution. (Proofs of these results are not required.)

6.4.3 recognise that a sample mean can be regarded as a random variable, and use the facts that

6.4.4 use the fact that has a normal distribution if X has a normal distribution