Question

2. Assume that the population proportion of adults having a college degree is 0.44. A random sample of 275 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 275 B) What is the probability that the sample proportion will be less than 0.52 C) What is the probability that the sample proportion will al within 4 percentage points+-0.04) of the population proportion?

Answer 1

Solutioin

given : n = 275, p= 0.44

formula: p(1-pl

A) the shape of the distribution is approximately normals large, n > 30)

mean = P = 0.44

standard deviation =Vpilnp) /p(1-p) NT2750 / 0.44 * 0.56 = 0.0299

B) P(p0.52)P P-P 0.52-0.44 p(1-pl

= P(-< 2.67

2.67

0.9962 (from table

C) P(within 0.04) P(0.40 < <0.48) izn

$=P\left ( \frac{0.40-0.44}{\sqrt{\frac{0.44*0.56}{275}}}<\frac{\hat p-p}{\sqrt{\frac{p(1-p)}{n}}}<\frac{0.48-0.44}{\sqrt{\frac{0.44*0.56}{275}}} \right )$

$=P(-1.34<z<1.34)$

required area under normal curve

$=P(z<1.34)-P(z<-1.34)$

$=0.9099-0.0901\;(from\;z-table)$

$={\color{Red} 0.8198}$