4.112 Mean family size You\'d like to estimate the mean size of families in your

Question

4.112 Mean family size You'd like to estimate the mean size of families in your community Explain why you'll tend to get a smaller sample mean if you sample n families than if you sample n individuals (asking them to report their family size). (Hint. When you sample individuals, explain why you are more likely to sample a large family than a small family. To think of this, it may heip to consider the case n 1 with a population of two families, one with 10 tion ng people and one with only 2 people.)

Explanation / Answer

If you sample n families, you will get only 1 sample per family. For eg. let m1, m2 ..... mn be the family sizes of n sampled families. Then the sample mean is,

Sample Mean of n families = (m1 + m2 +.....+ mn)/ n

Now, consider you sample n individuals out (m1 + m2 +.....+ mn) people in n families. There will high chances of more number of people will get sampled from large families as their counts will be more. For e.g. the chances of being sampled of an individual from family of 10 people is 10 / (m1 + m2 +.....+ mn). But the chances of being sampled of an individual from family of 2 people is 2 / (m1 + m2 +.....+ mn). So, in the sample of n individuals, people from family of 10 will be 5 times more than the people from family of 2. All individuals from same family of 10 will report their family size as 10 which was not the case earlier where each family was sampled once and reported its family size only once. So, the total family size of sampled n individuals would increase and hence the sample mean would be bigger.

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