Two dentists survey children in a village to find the state of their teeth. Ther

Question

Two dentists survey children in a village to find the state of their teeth. There are a total of 200 children in the village. Dentist A takes a simple random sample of 20 children and records the number of decayed teeth for each sampled child. The results are as follows:

# Decayed Teeth:        0 1 2 3 4 5 6 7 8 9 10

# Children:                  8 4 2 2 1 1 0 0 0 1 1

Dentist B surveys all 200 children and simply records the number of children with decayed teeth. He finds a total of 60 children with decayed teeth.

a.Estimate the total number of decayed teeth among village children using Dentist A’s information.

b.Estimate the total number of decayed teeth among village children using information from both Dentist A and Dentist B.

c.Which method do you expect to give an estimator with smaller variance?

Explanation / Answer

a) From Dentist A's information we can calculate proportion of children for each number category of decayed teeth. Hence, the total number of decayed teeth among children can be found by multiplying the total no of children 200 with the sample mean of decayed teeth.

and that comes as, (1*4+2*2+3*2+4*1+5*1+9*1+10*1)*200/20=420.

b) From, Dentist B's information we know that,

proportion of children with decayed teeth is 3/10 and proportion of children without decayed teeth is 7/10.

Now, We can use the previous information as,

previously we multiplied the average number of decayed teeth with the total population, but now we have the prior information that 60 children have decayed teeth. Hence combining these two information we can either multiply the estimated number of decayed teeth coming from 1st case with 3/10 or we can multiply sample mean with 60.

Both will lead to an estimate,

(1*4+2*2+3*2+4*1+5*1+9*1+10*1)*60/20 =126.

or 420*3/10= 126.

c) The estimator of the second method has a smaller variance as incorporating more information leads to a more complete estimate than the previous one and multuplying the old estimator with the proportion obtained from new information scale-downs the estimate which results in variance reduction.

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