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 572 potsherds was found, of which 353 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 answer to four decimal places.)

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

lower limit

upper limit

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

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

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

5% of all confidence intervals would include the true proportion of potsherds.

95% 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.

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.

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 = 572

x = 353

Point estimate = sample proportion = = x / n = 353/527=0.6698

1 -   = 1-0.6698 =0.3302

At 95% confidence level the z is ,

= 1 - 95% = 1 - 0.95 = 0.05

/ 2 = 0.05 / 2 = 0.025

Z/2 = Z0.025 = 1.96   ( Using z table )

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

= 1.96 (((0.6698*0.3302) / 572)

E = 0.039

A 95% confidence interval for population proportion p is ,

- E < p < + E

0.6698-0.039 < p < 0.6698+ 0.039

0.6308< p < 0.7088

The 95% confidence interval for the population proportion p is : lower limit=0.6308,upper limit=0.7088

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

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