Approximately 60% of all part-time college students in the United States are fem

Question

Approximately 60% of all part-time college students in the United States are female. (In other words, the population proportion of females among part-time college students is p 0.6.) is give the distribution (shape, center, and spread) of the sample proportion of part-time students that are females. b. If a random sample of size 100 were taken, what is the probability that less than 50% are females? C. If a random sample of size 100 were taken, there is a 95% chance that the sample proportion (p) falls between what two values?

Explanation / Answer

Solution:

From the given information
p = 0.6 , q = 1-p = 0.4, n = 100
a) Taking normal approximation to binomial, the sampling distribution of sample proportion follows normal distribution.
mean p = p = 0.6
variance ^2 = p(1-p)/n = 0.6 * 0.4 /100 = 0.0024
Standard deviation p = sqrt(^2) = sqrt(0.6 * 0.4 /100) = 0.0489

b) P(p < 0.5)
Normal distribution, = 0.6, = 0.0489,

Using standard normal distribution z = p - /
= 0.5 - (0.6)/0.0489 = -2.0412
P(p < 0.5) = P( Z < -2.04) = 0.0207

c) Middle 95% range for p
Nomral distribution
= 0.6, = 0.0489,
= 1-0.95 = 0.05
Z/2 = Z0.025 = 1.96

95% range = ± (Z/2)
= 0.6 ± 1.96 *(0.0489)
= (0.5042, 0.6958)

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