Question

5) Determine the point estimate of the population proportion, the margin of error for the following confidence interval, and the number of individuals in the sample with the specified characteristic, x, for the sample size provided. Lower bound = 0.123, upper bound = 0.547, n = 1000

Answer 1

We know that $100(1-\alpha)\%$ confidence interval for true proportion is given by:

$\\(p-Z_{\alpha/2}\sqrt{\frac{p(1-p)}{n}},p+Z_{\alpha /2}\sqrt{\frac{p(1-p)}{n}}) \\ \\\Rightarrow p-Z_{\alpha/2}\sqrt{\frac{p(1-p)}{n}}=0.123,p+Z_{\alpha/2}\sqrt{\frac{p(1-p)}{n}}=0.547$

Solving these two equations, we get,

Required point estiamte = $p=0.335$

Margin of error = $Z_{\alpha/2}\sqrt{\frac{p(1-p)}{n}}=0.212$

So, number of individual in the sample with the specified characteristic = $0.335(1000)=335$