An article reported that 6% of married couples in the United States are mixed ra

Question

An article reported that 6% of married couples in the United States are mixed racially or ethnically, Consider the population consisting of all married couples in the United States. A random sample of n = 150 couples will be selected from this population and p, the proportion of couples that are mixed racially or ethnically, will be computed. What are the mean and standard deviation of the sampling distribution of p? (Round your standard deviation to four decimal places.) mean 9 standard deviation 2.9086 Is it reasonable to assume that the sampling distribution of p is approximately normal for random samples of size n = 150? Explain. Yes, because np 10. No, because np 10. Suppose that the sample size is n = 300 rather than n = 150, as in Part (b) Does the change in sample size change the mean and standard deviation of the sampling distribution of p? What are the values for the mean and standard deviation when n = 300? (Round your standard deviation to four decimal places.) mean 18 standard deviation 4.1134 Is it reasonable to assume that the sampling distribution of p is approximately normal for random samples of sine n = 300? Explain, Yes, because np 10.

Explanation / Answer

Solution

Let represent the population proportion, i.e., = the proportion of couples in the American population who are ‘mixed’. Note that we are given = 0.06(i.e., 6%).

Part (a)

Mean of sample proportion, is: E() = population proportion = 0.06. ANSWER

Standard Deviation of is: {V(()} = {(1 - )/n} = {0.06x0.94/150} = 0.000388

= 0.0197 ANSWER

Part (b)

Distribution of is strictly Binomial with parameter n and . But, in this case n is large and is small, such that n < 10, it can safely be approximated by Normal Distribution.

Part (c)   

Mean of does not depend on n, but Standard Deviation of does change with n.

So,

Mean of sample proportion, is: E() = population proportion = 0.06. ANSWER

Standard Deviation of is: {V(()} = {(1 - )/n} = {0.06x0.94/300} = 0.000194

= 0.0139 ANSWER

Part (d)

Identical to the reasoning given under Part (b), in this case also, it can safely approximated by Normal Distribution.

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