Question

For this problem, carry at least four digits after the decimal in your calculations. Answers may vary slightly due to rounding.

A random sample of 5400 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 ?

Give a brief interpretation of the confidence interval.

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.    

1% of the confidence intervals created using this method 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.   

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

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

Answer 1

Solution:- Given that n = 5400, x = 1571
  
p = x/n = 1571/5400 = 0.2909 ,

q = 1-p = 1-0.2909 = 0.7091

99% confidence interval for the Z = 2.576

(a) The point estimate for p = 0.2909

(b) 99% confidence interval for p = p +/- Z*sqrt(pq/n)
= 0.2909 +/- 2.576*sqrt(0.2909*0.7091/5200)
= 0.2747 , 0.3071
= 0.275 , 0.307 (rounded)

(c) option D. 99% of the confidence intervals creted using this method would include the true proportion of traditional hogans.

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