Question

An article reported that 7% 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) A random sample of n = 50 couples will be selected from this population and p̂, 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 p̂? (Round your standard deviation to four decimal places.)

mean
standard deviation

(b) Is it reasonable to assume that the sampling distribution of p̂ is approximately normal for random samples of size n = 50? Explain.

Yes, because np < 10 and n(1 − p) < 10. Yes, because np > 10 and n(1 − p) > 10.     No, because np < 10. No, because np > 10.

(c) Suppose that the sample size is n = 250 rather than n = 50, as in Part (b). Does the change in sample size change the mean and standard deviation of the sampling distribution of p̂? What are the values for the mean and standard deviation when n = 250? (Round your standard deviation to four decimal places.)

mean
standard deviation

(d) Is it reasonable to assume that the sampling distribution of p̂ is approximately normal for random samples of size n = 250? Explain.

Yes, because np < 10. Yes, because np > 10.     No, because np < 10. No, because np > 10.

(e) When n = 250, what is the probability that the proportion of couples in the sample who are racially or ethnically mixed will be greater than 0.08? (Round your answer to four decimal places.)

Answer 1

Solution

Given that,

p = 0.07

1 - p = 1 - 0.07 = 0.93

n = 50

a)   = p = 0.07

=  [p ( 1 - p ) / n] =   [(0.07 * 0.93) / 50 ] = 0.0361

b)  0.07 * 50 = 3.5 No, because np < 10 ,

c) n = 250

= p = 0.07

=  [p ( 1 - p ) / n] =   [(0.07 * 0.93) / 250 ] = 0.0161

d)  250 * 0.07 = 17.5 Yes, because np > 10.

e) P( > 0.08) = 1 - P( < 0.08)

= 1 - P(( - ) / < (0.08 - 0.07) / 0.0161)

= 1 - P(z < 0.62)

Using z table

= 1 - 0.7324

= 0.2676