In: Statistics and Probability

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=0.226,upper bound=0.604,n=1200

The point estimate of the population is? round to the nearest thousandth as needed.

the margin error is? round to the nearest thousandth as needed.

the number of individuals in the sample with the specified characteristic is? round to the nearest integer as needed.

Concepts and reason

The concept of the confidence interval, the point estimate, and margin of error is used to solve this problem. Confidence intervals provide a range within which the true value of the unknown population parameter will lie. A point estimate is a value calculated from the sample values and is used to estimate the unknown population parameter. The error margin denotes the maximum amount of permissible error or the deviation of the true parameter value from its estimated value.

Fundamentals

The confidence interval for population proportion at a level of significance \(\alpha\) is calculated by the formula, \(\left(p \pm z_{\alpha / 2} \sqrt{\frac{p(1-p)}{n}}\right)\)

In the formula \(P\) is the sample proportion of a sample of size \(n\) and \({ }^{z} \alpha / 2\) represents the critical value of the z-score at \(\alpha\) level of significance. The average value of the upper and lower bounds of the confidence interval provides the point estimate of the population proportion. The margin of error is computed by the formula, \(M E=z_{\alpha / 2} \sqrt{\frac{p(1-p)}{n}}\)

The lower and upper bound are provided as 0.226 and 0.604, respectively. The point estimate of proportion is,

Point estimate of proportion \(=\frac{0.226+0.604}{2}\)

$$ =0.415 $$

The point estimate of population proportion is \(0.415 .\)

The point estimate is the average of the upper and lower bounds of the confidence interval and is obtained as \(0.415 .\) Thus; the estimated population proportion is \(41.5 \% .\)

The point estimate is calculated as 0.415, and the upper and lower bounds of the confidence interval are 0.604 and 0.226

respectively. So, the margin of error will be \(0.604-0.415=0.189\).

The margin of error is \(0.189 .\)

A margin of error of 0.189 is obtained. This means that a maximum error of \(18.9 \%\) is permissible in estimating the population proportion.

The formula to compute sample proportion is,

\(p=\frac{x}{n}\)

In the formula, \(\mathrm{p}\) is the proportion of people in the sample with a specified characteristic, \(\mathrm{x}\) is the number of individuals in the sample who possess the specified characteristic, and \(\mathrm{n}\) is the total number of people in the sample.

It is estimated as \(p=0.415\) and the sample size \(n=1200\).The number of people in the sample with the specified characteristic is calculated as,

$$ \begin{aligned} p &=\frac{x}{n} \\ 0.415 &=\frac{x}{1200} \\ x &=0.415 \times 1200 \\ x &=498 \end{aligned} $$

The number of individuals in the sample with the specified characteristic are \(498 .\)

There are 498 people with the specified characteristic. That is in a sample of size 1200,498 people possess the specified characteristic.

The point estimate of population proportion is 0.415.

The margin of error is 0.189.

The number of individuals in the sample with the specified characteristic is 498.

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