Question

A sample of 142 hypertensive people were given an anti-hypertensive drug, and the drug was found to be effective in 40 of those people. (By effective, we mean that their diastolic blood pressure is lowered by at least 10 mm Hg as judged from a repeat measurement taken 1 month after taking the drug.) a)

Find a 91% confidence interval for the true proportion of the sampled population for which the drug is effective. b)

Using the results from the above mentioned survey, how many people should be sampled to estimate the true proportion of hypertensive people for which the drug is effective to within 3% with 99% confidence? c)

If no previous estimate of the sample proportion is available, how large of a sample should be used in (b)?

Answer 1

a)

sample proportion, = 0.2817
sample size, n = 142
Standard error, SE = sqrt(pcap * (1 - pcap)/n)
SE = sqrt(0.2817 * (1 - 0.2817)/142) = 0.0377

Given CI level is 91%, hence α = 1 - 0.91 = 0.09
α/2 = 0.09/2 = 0.045, Zc = Z(α/2) = 1.7

Margin of Error, ME = zc * SE
ME = 1.7 * 0.0377
ME = 0.0641

CI = (pcap - z*SE, pcap + z*SE)
CI = (0.2817 - 1.7 * 0.0377 , 0.2817 + 1.7 * 0.0377)
CI = (0.2176 , 0.3458)

b)

The following information is provided,
Significance Level, α = 0.09, Margin of Error, E = 0.03

The provided estimate of proportion p is, p = 0.2817
The critical value for significance level, α = 0.09 is 1.7.

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 = 0.2817*(1 - 0.2817)*(1.7/0.03)^2
n = 649.75

Therefore, the sample size needed to satisfy the condition n >= 649.75 and it must be an integer number, we conclude that the minimum required sample size is n = 650
Ans : Sample size, n = 650

** if you take proportion = 0.28 upto 2 decimal answer would be change

c)

The following information is provided,
Significance Level, α = 0.09, Margin of Error, E = 0.03

The provided estimate of proportion p is, p = 0.5
The critical value for significance level, α = 0.09 is 1.7.

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 = 0.5*(1 - 0.5)*(1.7/0.03)^2
n = 802.78

Therefore, the sample size needed to satisfy the condition n >= 802.78 and it must be an integer number, we conclude that the minimum required sample size is n = 803
Ans : Sample size, n = 803