Suppose a simple random sample of size n = 1000 is obtained from a below populat

Question

Suppose a simple random sample of size n = 1000 is obtained from a below population whose size is N-2,000 000 and whose populat proportion with "peched characteristic spr 045 Complete parts a thou hi (a) Describe the sampling distribution of p Appromately normal, P.045 anda 0157 8. Approximately normal, .045 and, 00004 C Approxinatel norma02 (b) What is the probability ofobtaining x·480 or more ndividuals with te characteristic? Px2480)-0.0283 (Round to four decimal places as needed) (c) What is the probability of obtaining x 430 or fewer individuals with the characteristic? Px s 430)- 0.1018 (Round to four decimal places as needed)

Explanation / Answer

Solution:

a) The shape, mean and standard deviation of the sampling distribution of p of a simple random sample of size n can be determined if the sample size is less is less than or equal to 5% of the population size.

Show that the condition n 0.05 N is satisfied.
Is 1000 0.05(2,00,000) ?
Yes, 1000 10,000
The shape of the sampling distribution of p is approximately nomral provided np(1-p) 10.
Show that the condition np(1-p) 10 is satisfied
Is 1000(1-0.45) 10?
1000(0.55) 10 ?
Yes, 550 10
Therefore, the distribution is approximately normal and the mean of the sampling distribution of
p is p = p.
so, p = 0.45
The standard deviation of the sampling distribution of p is p = sqrt[p(1-p)/n]
Calculate p
= sqrt[0.45(1-0.45)/1000]
= 0.0157
Hence, p 0.0157

b) We know that p = x/n
So, p = 480/1000 = 0.48
Test statistic is
Z = p - p/ sqrt[p(1-p)/n]
= 0.48 - 0.45/sqrt[0.48(1-0.48)/1000]
= 1.89
The proability of more ibdividual is defined as
P(more individual) = P( Z > 1.89)
= 1P(Z<1.89 )
= 1 0.9706 = 0.0294
c) p = 430/1000 = 0.43
Test statistic is
Z = p - p/ sqrt[p(1-p)/n]
= 0.43 - 0.45/sqrt[0.43(1-0.43)/1000]
= -1.27
The proability of more ibdividual is defined as
P(more individual) = P( Z < -1.27)
= 1P ( Z<1.27 )
= 10.898 = 0.102  
=0.0606

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