Question

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?

Answer 1

Explanation:

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 =
`(68)/(167) =0.4072`

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 =
`\sqrt{(pq)/(n) } =\sqrt{(0.407(1-0.407))/(167) } \\=0.038`

Margin of error for 95% =
`1.96(0.038) =0.075`

Confidence interval =
`(0.407-0.075, 0.407+0.075)\\= (0.332, 0.482)`

c) Margin of error <6% = 0.06

i.e.
`1.96(\sqrt{(0.407(0.593))/(n) } <0.06\\√(n) >16.05\\n>258`