2010 Population Count: 608,660. If we take a size-10000 IID sample of Seattle re

Question

2010 Population Count: 608,660.

If we take a size-10000 IID sample of Seattle residents at 2010 and count the number of residents who have two or more races. This yields a COUNT statistic. If we divide the COUNT statistic by the sample size, we obtain a proportion statistic. Which of the following statement is correct?

What is the (approximated) probability that the COUNT statistic is greater than 522?

What is the (approximated) probability that the proportion statistic is more than 4.5%?

A:The COUNT statistics will be very close to its expected value because of the Law of Large Number. B: The COUNT statistics will be very close to its expected value because of the Central Limit Theorem. C: The proportion statistics will be very close to its expected value because of the Law of Large Number. D:The proportion statistics will be approximately Normally distributed around its expected value because of the Law of Large Number. Race and Ethnicity Two or More Races Persons of Color: 34% Other 5% Hispanic/Latino Ethnicity 4% (any race): 7% Asian 14 White Black or African American Black or Asian African American Other 8% Two or More Races White 69% Sources: 2010 Census, U.S. Census Bureau

Explanation / Answer

As per the US census bureau ethinicity report, 2010 there are 5% person who are from 2 or more races

So from a sample of 10000, as per data survey expected number of persons who have 2 or more races = 500

so Law of large numbers = the average of the results obtained from a large number of trials should be close to its expected value, and tend to closer as more trials are performed.

Cnetral Limit throrem = The central limit theorem states that the sampling distribution of the mean of any independent, random variable will be normal or nearly normal, if the sample size is large enough.

so by these two definition ,it is clear that option a is correct and The COUNT statistics will be very close to its expected value because of the Law of Large Number.

What is the (approximated) probability that the COUNT statistic is greater than 522?

Now Expected value or mean of COunt statistics = 500

and variance = 10000 * 0.05 * 0.95 = 475

standerd deviation = 21.8

so Z value for X = 522 ; Z = ( 522 - 500) / 21.8 = 1.001

From Z - table relative probabilty value - 0.8438

so probability value for that Z - value = P ( X > = 522; 500; 21.8) = 1 - 0.8438 = 0.1562

so answer is C = 16 %

(approximated) probability that the proportion statistic is more than 4.5% or say count probability more than 450

Z - value for X = 450 = (4.50 - 5.00 ) / 0.218 = -2.294

so Equivalent probabiliyt value for the given z value = 0.0109

so P ( X > 4.5 ; 5; 0.218) = 1 - 0.0109 = 0.9891 = 98.91 %

so answer is B or say 98 %

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