Answer:
- 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.
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!!!