Question

Suppose 12% of students are veterans. From a sample of 263 students, how unusual would it be to have less than 22 veterans?
Is the success-failure condition of the Central Limit Theorem satisfied?
Yes. Both np and n(1 - p) are > 10.
No. Either np < 10 or n(1 - p) < 10.
The Central Limit Theorem tells us that the distribution of sample proportions approximately follows a ______ distribution with mean _____ and standard deviation normal
Given this knowledge, use technology to compute the probability that, through random selection, one finds a sample proportion that is less than the proportion corresponding to 22 veterans.
Round answer to 4 decimal places.
Is this result unusual?
Yes. There is a less than 5% chance of this happening by random variation.
No. There is at least a 5% chance of this happening by random variation.

Answer 1

Answer:

Yes, both np and n(1-p) are ≥ 10

Mean = 0.12 ; Standard deviation = 0.02004

Yes. There is a less than 5% chance of this happening by random variation. 0.034839

Step-by-step explanation:

Given that :

p = 12% = 0.12 ;

Sample size, n = 263

np = 263 * 0.12 = 31.56

n(1 - p) = 263(1 - 0.12) = 263 * 0.88 = 231.44

According to the central limit theorem, distribution of sample proportion approximately follow normal distribution with mean of p = 0.12 and standard deviation sqrt(p*(1 - p)/n) = sqrt (0.12 *0.88)/n = sqrt(0.0004015) = 0.02004

Z = (x - mean) / standard deviation

x = 22 / 263 = 0.08365

Z = (0.08365 - 0.12) / 0.02004

Z = −1.813872

Z = - 1.814

P(Z < −1.814) = 0.034839 (Z probability calculator)

Yes, it is unusual

0.034 < 0.05 (Hence, There is a less than 5% chance of this happening by random variation.