Question

Separate random samples were collected by a polling agency to investigate the difference in employee satisfaction at​ non-profit organizations and at​ for-profit companies. Data collected from 400 employees at​ non-profit organizations revealed that 365 of them were​ "highly satisfied." From the​ for-profit companies, 431 out of 509 employees reported the same level of satisfaction. Researchers want to test if the proportions of satisfied employees are the same at​ for-profit companies as at​ non-profit companies.

A) What is the difference in the proportions of the two types of​ companies? Assume P1 is the proportion of satisfied​ non-profit employees and P2 is the proportion of satisfied​ for-profit employees. P1 - P2 is ____ (round to 3 decimals).

B) What is the pooled proportion of satisfied employees in both types of companies​ combined?

C) What is the standard error of the difference in part​ a?

D) What is the Z statistic and P Value?

_____ the null hypothesis. There _____ sufficient evidence to conclude that the proportions of satisfied employees in​ non-profit organizations and in​ for-profit companies are different.

Thank you in advance for your help!

Answer 1

a)
p1cap = X1/N1 = 365/400 = 0.913
p1cap = X2/N2 = 431/509 = 0.847

difference = 0.913 - 0.847 = 0.066

b)

pcap = (X1 + X2)/(N1 + N2) = (365+431)/(400+509) = 0.8757

c)
Standard Error, sigma(p1cap - p2cap),
SE = sqrt(p1cap * (1-p1cap)/n1 + p2cap * (1-p2cap)/n2)
SE = sqrt(0.913 * (1-0.913)/400 + 0.847*(1-0.847)/509)
SE = 0.0213

d)

Test statistic
z = (p1cap - p2cap)/sqrt(pcap * (1-pcap) * (1/N1 + 1/N2))
z = (0.913-0.847)/sqrt(0.8757*(1-0.8757)*(1/400 + 1/509))
z = 2.99

P-value Approach
P-value = 0.0028
reject the null hypothesis.

e)

REject the null hypothesis. There is sufficient evidence to conclude that the proportions of satisfied employees in​ non-profit organizations and in​ for-profit companies are different.