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 5060 permanent dwellings on an entire reservation showed that 1683 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 the confidence intervals created using this method 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.    
○ 1% of all confidence intervals would include the true proportion of traditional hogans.
○ 99% of all confidence intervals 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 normal.    
○ 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.

Answer 1

a) = 1683/5060 = 0.3326

b) At 99% confidence interval the critical value is z_0.005 = 2.58

The 99% confidence interval for population proportion is

+/- z_0.005 * sqrt((1 - )/n)

= 0.3326 +/- 2.58 * sqrt(0.3326(1 - 0.3326)/5060)

= 0.3326 +/- 0.0171

= 0.3155, 0.3497

= 0.316, 0.350

Option - D) 99% of all confidence intervals would include the true proportion of traditional hogans.

c) np = 5060 * 0.3326 = 1682.956

    nq = 5060 * (1 - 0.3326) = 3377.044

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