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 352 were identified as Santa Fe black-on-white.

Let p represent the population proportion of Santa Fe black-on-white potsherds at the excavation site. Find a point estimate for p. Find a 95% confidence interval for p. Give a brief statement of the meaning of the confidence interval. Do you think that the conditions np > 5 and nq > 5 are satisfied in this problem? Why would this be important?

- A point estimate for p = 0.603.

- A 95% confidence interval for p = (0.5633, 0.6427)

- Like I explained below, we can say that the confidence interval for the population proportion of Santa Fe black-on-white potsherds at the excavation site is between the range (0.5633) to (0.6427) with a confidence level of 95%

- The conditions np > 5 and nq > 5 are satisfied for this problem.

- This is important as this is the condition for the distribution to approximate a normal distribution and for us to use the z critical value of 1.960 in our calculations instead of the the more strenuous t critical value used when information on the population standard deviation isn't known.

Step-by-step explanation:

- The proportion of Santa Fe black-on-white potsherds at the excavation site for the site where the samples were taken from, can be calculated thus.

p = (352/584) = 0.603.

- Confidence Interval for the population proportion is basically an interval of range of values where the true population proportion can be found with a certain level of confidence.

Mathematically,

Confidence Interval = (Sample proportion) ± (Margin of error)

Sample proportion = 0.603

Margin of Error is the width of the confidence interval about the mean.

It is given mathematically as,

Margin of Error = (Critical value) × (standard Error)

Critical value at 95% confidence interval for sample size of 584 is obtained from the z-tables.

Critical value = 1.960

standard Error is calculated thus.

σₓ = standard error = √[p(1-p)/n]

p = 0.603

n = sample size = 584

σₓ = √(0.603×0.397/584) = 0.02025

95% Confidence Interval = (Sample proportion) ± [(Critical value) × (standard Error)]

CI = 0.603 ± (1.96 × 0.02025)

CI = 0.603 ± 0.0397

95% CI = (0.5633, 0.6427)

95% Confidence interval = (0.5633, 0.6427)

- Like I have explained above, we can say that the confidence interval for the population proportion is between the range (0.5633) to (0.6427) with a confidence level of 95%

- np = 0.603 × 584 = 352 > 5

nq = 0.397 × 584 = 232 > 5

- This is necessary as this is the condition for the distribution to approximate a normal distribution and for us to use the z critical value of 1.960 in our calculations instead of the the more strenuous t critical value used when information on the population standard deviation isn't known.

Hope this Helps!!!

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