(a) What do you understand by systematic sampling? Discuss its types. (b) Blood groups of inhabitants of a village were checked. It was found that 600 people possessed blood group A, 650 possessed blood group B, 550 had blood group AB and 200 have blood group O. (i) Calculate the probability of having blood group B. (ii) Calculate the probability of having blood group O. (c) A group of 50 men can construct 20 kilometer road in 40 days. How long will 70 men take to complete same length of road? (d) Zahid left a property worth Rs 1,750,000. His family had to pay off a debt of Rs 150,000. The rest of money was distributed between a son and a daughter. How much did each child receive if share of a son was double than that of a daughter?

(a) Systematic sampling

Systematic sampling is a probability sampling method in which units are selected from an ordered list at a regular interval. The sampling interval is k = N ÷ n, where N is the population size and n the required sample size. A random starting point is chosen between 1 and k, and then every kth unit is taken.

Example: to choose 50 students from a list of 1,000, k = 1,000 ÷ 50 = 20. If the random start is 7, then the sample comprises numbers 7, 27, 47, and so on.

Types

Linear systematic sampling: units are taken in a straight run through the list from the random start, without returning to the beginning. Circular systematic sampling: the list is treated as a circle. After reaching the end, selection continues from the beginning until n units are chosen, which is useful when N is not an exact multiple of n. Multiple-start (replicated) systematic sampling: several random starts are used with a larger interval, and this gives several small samples that help in estimating error.

It is simple and spreads the sample evenly over the population. However, it can be biased if the list contains a repeating pattern that matches the interval.

(b) Blood group probabilities

Total number of people = 600 + 650 + 550 + 200 = 2,000.

(i) P(group B) = 650 ÷ 2,000 = 0.325, or 32.5%.

(ii) P(group O) = 200 ÷ 2,000 = 0.1, or 10%.

(c) Men and road

More men finish the same work in fewer days, so this is inverse proportion, and men × days stays constant.

Total work = 50 men × 40 days = 2,000 man-days.

Days for 70 men = 2,000 ÷ 70 = 28 4/7 days, or about 28.6 days.

(d) Zahid's property

Debts are paid first, so amount to distribute = 1,750,000 − 150,000 = Rs. 1,600,000.

With the son's share double the daughter's, the ratio is 2 : 1, a total of 3 parts. One part = 1,600,000 ÷ 3.

Daughter = Rs. 533,333.33 (approximately) Son = Rs. 1,066,666.67 (approximately)

Check: the two shares add up to 533,333.33 + 1,066,666.67 = Rs. 1,600,000.