Question

Three-circle, red-on-white is one distinctive pattern painted on ceramic vessels of the Anasazi period found at an archaeological site. At one excavation, a sample of 165 potsherds indicated that 70 were of the three-circle, red-on-white pattern. (a) Find a point estimate p̂ for the proportion of all ceramic potsherds at this site that are of the three-circle, red-on-white pattern. (Round your answer to four decimal places.) (b) Compute a 95% confidence interval for the population proportion p of all ceramic potsherds with this distinctive pattern found at the site. (Round your answers to three decimal places.)

lower limit

upper limit

Answer 1

Solution :

Given that,

n = 165

x = 70

Point estimate = sample proportion = $\hat p$ = x / n = 70 / 165 = 0.4242

1 - $\hat p$ = 1 - 0.4242 = 0.5758

At 95% confidence level

$\alpha$ = 1 - 95%

$\alpha$ =1 - 0.95 =0.05

$\alpha$/2 = 0.025

Z$\alpha$/2 = Z_{0.025 = 1.960}

Margin of error = E = Z_{$\alpha$ / 2} * $\sqrt$ (($\hat p$ * (1 - $\hat p$ )) / n)

= 1.96 ($\sqrt$((0.4242 * 0.5758)) / 165 )

= 0.075

A 95% confidence interval for population proportion p is ,

$\hat p$ - E < p < $\hat p$ + E

0.4242 - 0.075 < p < 0.4242 + 0.075

0.3492 < p < 0.4992

0.349 < p < 0.499

The 95% confidence interval for the population proportion p is :

Lower limit : 0.349

Upper limit : 0.499