Question

QUESTION # 1

A random sample of 5060 permanent dwellings on an entire reservation showed that 1571 were traditional hogans.

(a) Let p be the proportion of all permanent dwellings on the entire reservation that are traditional hogans. Find a point estimate for p. (Round your answer to four decimal places.)


(b) Find a 99% confidence interval for p. (Round your answer to three decimal places.)

lower limit	_________
upper limit	_________

QUESTION # 2

What is the minimal sample size needed for a 95% confidence interval to have a maximal margin of error of 0.1 in the following scenarios? (Round your answers up the nearest whole number.)

(a) a preliminary estimate for p is 0.15


(b) there is no preliminary estimate for p

Answer 1

a)

sample proportion,= 1571/5060 = 0.3105

b)

sample proportion, = 0.3105
sample size, n = 5060
Standard error, SE = sqrt(pcap * (1 - pcap)/n)
SE = sqrt(0.3105 * (1 - 0.3105)/5060) = 0.0065

Given CI level is 99%, hence α = 1 - 0.99 = 0.01
α/2 = 0.01/2 = 0.005, Zc = Z(α/2) = 2.58

CI(Proportion) = (0-2/34, P), P+=2 / 84,7"))

Margin of Error, ME = zc * SE
ME = 2.58 * 0.0065
ME = 0.0168

CI = (pcap - z*SE, pcap + z*SE)
CI = (0.3105 - 2.58 * 0.0065 , 0.3105 + 2.58 * 0.0065)
CI = (0.294 , 0.327)
Lower limit = 0.294
Upper limit = 0.327

2)

a)

The following information is provided,
Significance Level, α = 0.05, Margin of Error, E = 0.1

The provided estimate of proportion p is, p = 0.15
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 = 0.15*(1 - 0.15)*(1.96/0.1)^2
n = 48.98

Therefore, the sample size needed to satisfy the condition n >= 48.98 and it must be an integer number, we conclude that the minimum required sample size is n = 49
Ans : Sample size, n = 49

b)
The following information is provided,
Significance Level, α = 0.05, Margin of Error, E = 0.1

The provided estimate of proportion p is, p = 0.5
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 = 0.5*(1 - 0.5)*(1.96/0.1)^2
n = 96.04

Therefore, the sample size needed to satisfy the condition n >= 96.04 and it must be an integer number, we conclude that the minimum required sample size is n = 97
Ans : Sample size, n = 97