Question

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

Santa Fe black-on-white is a type of pottery commonly found at archaeological excavations at a certain monument. At one excavation site a sample of 586 potsherds was found, of which 358 were identified as Santa Fe black-on-white.

(a) Let p represent the proportion of Santa Fe black-on-white potsherds at the excavation site. Find a point estimate for p. (Round your answer to four decimal places.)


(b) Find a 95% confidence interval for p. (Round your answers to three decimal places.)

lower limit
upper limit

Give a brief statement of the meaning of the confidence interval.

95% of the confidence intervals created using this method would include the true proportion of potsherds. 5% of the confidence intervals created using this method would include the true proportion of potsherds.     95% of all confidence intervals would include the true proportion of potsherds. 5% of all confidence intervals would include the true proportion of potsherds.

(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 normal. 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 binomial.

Answer 1

Solution:

Given:

At one excavation site a sample of 586 potsherds was found, of which 358 were identified as Santa Fe black-on-white.

thus n = 586 and x = 358

Part a) Let p represent the proportion of Santa Fe black-on-white potsherds at the excavation site. Find a point estimate for p.

Thus

Part b) Find a 95% confidence interval for p.

Formula:

where

We need to find z_c value for c=95% confidence level.

Find Area = ( 1 + c ) / 2 = ( 1 + 0.95) /2 = 1.95 / 2 = 0.9750

Look in z table for Area = 0.9750 or its closest area and find z value.

Area = 0.9750 corresponds to 1.9 and 0.06 , thus z critical value = 1.96

That is : Z_c = 1.96

Thus

Thus

Lower Limit =

Lower Limit =

Lower Limit =

Lower Limit =

and

Upper Limit =

Upper Limit =

Upper Limit =

Upper Limit =

Give a brief statement of the meaning of the confidence interval.

95% of all confidence intervals would include the true proportion of potsherds.

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

q = 1 - p = 1 - 0.6109 = 0.3891

n*p = 586 * 0.6109 = 358 > 5

n*q = 586* 0.3891 = 228 > 5

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