Three-circle, red-on-white is one distinctive pattern painted on ceramic vessels of the Anasazi period found at the Wind Mountain archaeological site. At one excavation, a sample of 167 potsherds indicated that 68 were of the three-circle, red-on-white pattern.
A. Find a point estimate p for the proportion of all ceramic potsherds at this site that are of the three-circle, red-on-white pattern.
B. Compute a 95 percent confidence interval for the population proportion p of all ceramic potsherds with this distinctive pattern found at the site.
C. Consider the three-circle, red-on-white pattern discussed in the problem above. How many ceramic potsherds must be found and identified if we are to be 95% confident that the sample proportion p-hat of such potsherds is within 6% of the population proportion of three-circle, red-on-white patterns found at this excavation site?

Given that three-circle, red-on-white is one distinctive pattern painted on ceramic vessels of the Anasazi period found at the Wind Mountain archaeological site

Sample size = 167

3 circle =68

Sample proportion = [tex]\frac{68}{167} =0.4072[/tex]

a) a point estimate p for the proportion of all ceramic potsherds at this site that are of the three-circle, red-on-white pattern=0.4072

b) Std error of p = [tex]\sqrt{\frac{pq}{n} } =\sqrt{\frac{0.407(1-0.407)}{167} } \\=0.038[/tex]

Margin of error for 95% = [tex]1.96(0.038) =0.075[/tex]

Confidence interval = [tex](0.407-0.075, 0.407+0.075)\\= (0.332, 0.482)[/tex]

c) Margin of error <6% = 0.06

i.e. [tex]1.96(\sqrt{\frac{0.407(0.593)}{n} } <0.06\\\sqrt{n} >16.05\\n>258[/tex]