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 584 potsherds was found, of which 354 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. 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. 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. 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.

Answer 1

Solution :

Given that,

n = 584

x = 354

a) Point estimate = sample proportion = = x / n = 354 / 584 = 0.6062

1 - = 1 - 0.6062 = 0.3938

At 95% confidence level

= 1 - 95%

=1 - 0.95 =0.05

/2 = 0.025

Z/2 = Z_{0.025 = 1.960}

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

= 1.96 (((0.6062 * 0.3938) / 584)

= 0.040

A 95% confidence interval for population proportion p is ,

± E

= 0.6062 ± 0.040

= ( 0.566, 0.646 )

lower limit = 0.566

upper limit = 0.646

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

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