Question

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10. (a) In one indoor playground, the number of children playing at the playground has a Poisson distribution with average 12 children in half an hour. Find the mean and variance of the number of children within 12.00pm to 2.30pm 3 marks] In a random survey, 6% of people take their dinner while watching television. If the sample consists of 50 adult and Y is the number of people in this sample who take dinner while watching television, find the mean and standard deviation of the probability distribution of Y. (b) 4 marks] (c) The mean of IELTS score for a group of students was 500 and that 20% of the students scored below 420. The score was normally distributed. Find the standard deviation for this distribution. [4 marks]

Answer 1

10a) In case of Poisson distribution mean and variance are equal to the parameter of the distribution. It is given that on average 12 children play in half an hour.

Hence from 12:00pm to 2:30 pm that is 2.5 hours

Average number of children playing is ​​​​​​

2 × 12 × 2.5 60

b) The random variable Y will follow Binomial distribution as it satisfy all the properties. Taking n=50, p=0.06

We get

Mean ​​​​​​

μ-np-50 x 0.06-3.0

And standard deviation

σ VTp(1-p) = V50 × 0.06 (1-0.06) 1.68

c) Define the standard random variable Z as

. Where X denotes the IELTS score

X-500

It's given that ​​​​​​,

P(X < 420) = 0.20

From normal table know That ​​​​​​

P(Z <-0.84) = 0.20

So

420-5000.84 0.84