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 5340 permanent dwellings on an entire reservation showed that 1627 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 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.

1% 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.

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.

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

Answer 1

Solution :

Given that,

n = 5340

x = 1627

(a)

Point estimate = sample proportion = = x / n = 1627 / 5340 = 0.305

1 - = 1 - 0.305 = 0.695

(b)

At 99% confidence level the z is ,

= 1 - 99% = 1 - 0.99 = 0.01

/ 2 = 0.01 / 2 = 0.005

Z_/2 = Z_0.005 = 2.576

Margin of error = E = Z _{/ 2} * (( * (1 - )) / n)

= 2.576 * (((0.305 * 0.695) / 5340)

= 0.017

A 99% confidence interval for population proportion p is ,

- E < p < + E

0.305 - 0.017 < p < 0.305 + 0.017

0.288 < p < 0.322

The 99% confidence interval for the population proportion p is : (0.288 , 0.322)

Lower limit = 0.288

Upper limit = 0.322

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

(c)

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