Question

To approximate the proportion p of out-state students in KU, n samples are taken in a survey.

(1) Find the mean and standard deviation of sample proportion p .

(2) A survey shows that there are 23 out state students in 100 students. Find the 95% confidence interval for p.

(3) If we require the estimating error is less than 3% with 95% confidence, how many samples are required at least?

(4) Another sample shows that there are 10 out state students in 50 students from KSU. Find the 95% confidence interval for the difference of two proportions between KU and KSU

Answer 1

a)

mean = pcap

std.deviation = sqrt(pcap *(1-pcap)/n)

b)

sample proportion, = 0.23
sample size, n = 100
Standard error, SE = sqrt(pcap * (1 - pcap)/n)
SE = sqrt(0.23 * (1 - 0.23)/100) = 0.0421

Given CI level is 95%, hence α = 1 - 0.95 = 0.05
α/2 = 0.05/2 = 0.025, Zc = Z(α/2) = 1.96

Margin of Error, ME = zc * SE
ME = 1.96 * 0.0421
ME = 0.083

CI = (pcap - z*SE, pcap + z*SE)
CI = (0.23 - 1.96 * 0.0421 , 0.23 + 1.96 * 0.0421)
CI = (0.1475 , 0.3125)

c)
The following information is provided,
Significance Level, α = 0.05, Margin of Error, E = 0.03

The provided estimate of proportion p is, p = 0.23
The critical value for significance level, α = 0.05 is 1.96.

The following formula is used to compute the minimum sample size required to estimate the population proportion p within the required margin of error:
n >= p*(1-p)*(zc/E)^2
n = 0.23*(1 - 0.23)*(1.96/0.03)^2
n = 755.94

Therefore, the sample size needed to satisfy the condition n >= 755.94 and it must be an integer number, we conclude that the minimum required sample size is n = 756
Ans : Sample size, n = 756

d)

Here, , n1 = 100 , n2 = 50
p1cap = 0.23 , p2cap = 0.2

Standard Error, sigma(p1cap - p2cap),
SE = sqrt(p1cap * (1-p1cap)/n1 + p2cap * (1-p2cap)/n2)
SE = sqrt(0.23 * (1-0.23)/100 + 0.2*(1-0.2)/50)
SE = 0.0705

For 0.95 CI, z-value = 1.96
Confidence Interval,
CI = (p1cap - p2cap - z*SE, p1cap - p2cap + z*SE)
CI = (0.23 - 0.2 - 1.96*0.0705, 0.23 - 0.2 + 1.96*0.0705)
CI = (-0.1082 , 0.1682)