Question

Suppose a simple random sample of size 150 is obtained from a population whose size is N15.000 and whose population proportion with a specified characteristic hp 04. Complete parts () through (c) below. (a) Describe the sampling distribution of Choose the phrase that best describes the shape of the sampling distribution below. O A. Approximately normal because 0.05 and 1-10 OB. Approximately normal becausen 00N and re-<10 OG Not normal because 0.06 and 1-12 10 OD. Not normalen SOON and oft - 10 Determine the mean of the samping dibution of (Round to one decimal place as needed) Determine the standard deviation of the sampling debution of (Round to six decimal places as needed) р What is the probability of obtaining xor more individuals with the characteristic? That is what is PD20.44) PD204)Round to four decimal places as needed) (c) What is the probability of obtaining x 48 or lower individuals with the characteristic? That is what is PIDs 32 PISO)-(Round to four decimal places as needed) 2

Answer 1

Here, n=150, N=15,000 and p=0.4

(a) Here

n 150 0.01 0.05 N 15000

n < 0.05N

And

np(1-p) = 150 x 0.4 x (1 – 0.4) = 150 x 0.4 x 0.6 = 36 > 10

option (A) is correct answer.

i.e, Approximately normal because
n< 0.05N and np(1-p) > 10

The mean and standard deviation of $\hat{p}$ is given by,

The mean is,

$\mu_{\hat{p}}=p=0.4$

The standard deviation is

$\sigma_{\hat{p}}=\sqrt{\frac{p(1-p)}{n}}=\sqrt{\frac{0.4\times (1-0.4)}{150}}=\sqrt{\frac{0.4\times 0.6}{150}}=0.04$

So, the distribution of sample proportion is given by,

$\hat{p}\sim N(0.4,0.04)$

(b) We need to find $P(\hat{p}\geq 0.44)$

Solution::

$P(\hat{p}\geq 0.44)$

$=P\left ( \frac{\hat{p}-0.4}{0.04}\geq \frac{0.44-0.4}{0.04} \right )$

$=P(Z\geq 1)$

$=1-P(Z<1)$

$=1-0.8413447$

$=0.1586553$

$\simeq 0.1587$[ round to four decimal place]

The probability is 0.1587

(c) We need to find $P(\hat{p}\leq 0.32)$

Solution::

$P(\hat{p}\leq 0.32)$

$=P\left ( \frac{\hat{p}-0.4}{0.04}\leq \frac{0.32-0.4}{0.04} \right )$

$=P(Z\leq -2)$

$=1-P(Z\leq 2)$

$=1-0.9772499$

$=0.0227501$

$\simeq 0.0228$[ round to four decimal place]

The probability is 0.0228

​​​​​