According to the latest census, 45% of working women held full time job. In a sa

Question

According to the latest census, 45% of working women held full time job. In a sample of 10 working women, (i) What is the expected number of women working part time? (ii) What is the probability that half of them working full time? (iii) What assumptions do you need to make to find the probabilities? In a sample of 50 working women. (iv) What is the probability of that more than half of them working full time? State clearly any assumptions made. (b) The GO transportation system of buses and commuter trains operates on an honour system. Train travellers are expected to buy their tickets before boarding the train. Only a small number of people will be checked on the train to see whether they bought a ticket. Suppose that a random sample of 400 train travellers was sampled and 68 of them had failed to buy a ticket. (i) Calculate the sample proportion of travellers who do not buy a ticket. (ii) Use the information found in (i) to estimate with 95% confidence the proportion of all tram travellers who do not buy a ticket. Make clear all assumptions made to justify this calculation. (iii) Suppose that before the survey was undertaken, GO wants to estimate the proportion of train travellers who do not buy a ticket to within 5% with 95% confidence. How large a sample does GO need? What would be the sample size if a smaller confidence level is used?

Explanation / Answer

Here it is given that n=10 and p=0.45

i. a. Expected number =np=4.5

b. Half of them means 5, so we need to find P(x=5)=ncxp^x(1-p)^n-x=0.2340

c. assumption is np>10

d. here n=50 so np=22.5>10 so we assume normal to binomial distribution

P(x>25)=P(z>25-22.5/sqrt(np(1-p))=P(z>0.71)=0.2389

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