Advanced level 2025 Centre Regional Mock Pure mathematics with statistics 3
Advanced level 2025 Centre Regional Mock Pure mathematics with statistics 3
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1. In an institution of higher learning, a random sample of 50 students were chosen and the frequency distribution of their ages were as follows:
| Years (x) | 16-20 | 21-25 | 26-30 | 31-35 | 36-40 | 41-45 | 46-50 | 51-56 |
|---|---|---|---|---|---|---|---|---|
| F | 5 | 10 | 15 | 9 | 5 | 3 | 2 | 1 |
a. Calculate, to the nearest age, the mean and the standard deviation of the student’s ages. b. Draw a cumulative frequency curve to suit these data and use it to estimate the median age of the students. c. Estimate to 1 decimal place, the number of students whose ages are between 29 and 48 years inclusive.
2. a) Two events A and B are such that , and find: i. ii. iii.
b) Two companies X and Y produce respectively, 60% and 40% of the total phones for a certain community. The probability of faulty phones produced by the companies are and respectively. A phone is selected at random from the consignments for the community. Find the probability that: iv. the phone is faulty. v. The phone is faulty or it comes from company Y. vi. The phone is from X given that it is faulty.
3. A random variable Y takes integer value y, with probability P(y), where Find: a. The value of the constant k b. The mean and variance of the distribution c. The mean and the variance of the distribution of the random variable R = 5Y – 8
4. i) The random variable X has a Poisson distribution with parameter λ. Given that , where k is an integer. Show that λ = . (3marks) ii) A manufacturer of carpet knows that faults occur at random in a carpet at an average rate of 0.8 per area. Find, to three significant figures, the probability that in a randomly chosen area of carpet, a) there are no faults b) there are at least 2 faults. An office room of floor area is to be carpeted with the same carpet described above. Find, to 3 significant figures, the probability that the carpet in the room contains exactly 3 faults. iii) It is known that 1% of the teachers in a certain town speaks French as their prime language. Use the Poisson distribution to find the probability that 2 or more teachers speaks French as their prime language.
