2. Assume that the population proportion of adults having a college degree is 0.44. A random sample of 375 adults is to be selected to test this claim.
A) What are the shape, mean, and standard deviation of the sampling distribution of the sample proportion for samples of 375?
B) What is the probability that the sample proportion will be less than 0.50?
C) What is the probability that the sample proportion will fall within 4 percentage points(+/- 0.04) of the population proportion?
A)
Given,
n = 375 , p = 0.44
np = 375 * 0.44 = 165 >= 10
nq = 375 ( 1 - 0.44) = 210 >= 10
Since np >= 10 and nq >= 10
The shape of sampling distribution of sample proportion is approximately normal.
Mean = p = 0.44
Standard deviation = sqrt [ p ( 1 - p) / n ]
= sqrt [ 0.44 ( 1 - 0.44) / 375 ]
= 0.0256
B)
Using normal approximation,
P( < p) = P(Z < ( - p) / sqrt [ p ( 1 - p) / n ]
So,
P( < 0.50) = P(Z < ( 0.50 - 0.44) / 0.0256 )
= P(Z < 2.34)
= 0.9904 (From Z table)
c)
P(p - 0.04 < < p + 0.04) = P( 0.44 - 0.04 < < 0.44 + 0.04)
= P( 0.40 < < 0.48)
= P( < 0.48) - P( < 0.40)
= P(Z < ( 0.48 - 0.44) / 0.0256) - P(Z < (0.44 - 0.04) / 0.0256 )
= P( Z < 1.56) - P(Z < -1.56 )
= 0.9406 - 0.0594 (From Z table)
= 0.8812
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