Question

A random sample of 5060 permanent dwellings on an entire reservation showed that 1564 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

Give a brief interpretation of the confidence interval.

1% of the confidence intervals created using this method would include the true proportion of traditional hogans.

99% of all confidence intervals would include the true proportion of traditional hogans.   

1% of all confidence intervals would include the true proportion of traditional hogans.

99% of the confidence intervals created using this method would include the true proportion of traditional hogans.

(c) Do you think that np > 5 and nq > 5 are satisfied for this problem? Explain why this would be an important consideration.

Yes, the conditions are satisfied. This is important because it allows us to say that p̂ is approximately binomial.

No, the conditions are not satisfied. This is important because it allows us to say that p̂ is approximately binomial.

   Yes, the conditions are satisfied. This is important because it allows us to say that p̂ is approximately normal.

No, the conditions are not satisfied. This is important because it allows us to say that p̂ is approximately normal.

Answer 1

a)

sample proportion, = 0.3091

b)

sample size, n = 5060
Standard error, SE = sqrt(pcap * (1 - pcap)/n)
SE = sqrt(0.3091 * (1 - 0.3091)/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

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

CI = (pcap - z*SE, pcap + z*SE)
CI = (0.3091 - 2.58 * 0.0065 , 0.3091 + 2.58 * 0.0065)
CI = (0.292 , 0.326)

lower limit = 0.292    
upper limit = 0.326

c)

99% of all confidence intervals would include the true proportion of traditional hogans.

d)

Yes, the conditions are satisfied. This is important because it allows us to say that p̂ is approximately normal.