Question

According to the CDC, in 2017-2018, the prevalence of obesity among American adults was 42.4%. In a random sample of 80 adults, let ˆp be the proportion that meet the CDC definition of obese.

(a) Describe the sampling distribution of ˆp. Your answer should include the shape of the distribution (and why you can make this conclusion), the mean, and the standard deviation.

(b) What is the probability that ˆp is between .40 and .44.

Answer 1

a)
The distribution is normal with bell shape

Here, μ(pcap) = 0.424, σ(pcap) = sqrt(0.424*(1-0.424)/80) = 0.0553,

b)
x1 = 0.4 and x2 = 0.44. We need to compute P(0.4<= X <= 0.44). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z1 = (0.4 - 0.424)/0.0553 = -0.43
z2 = (0.44 - 0.424)/0.0553 = 0.29

Therefore, we get
P(0.4 <= X <= 0.44) = P((0.44 - 0.424)/0.0553) <= z <= (0.44 - 0.424)/0.0553)
= P(-0.43 <= z <= 0.29) = P(z <= 0.29) - P(z <= -0.43)
= 0.6141 - 0.3336
= 0.2805