Chapter 8, Problem 1CR

Statist Literacy In your own words, carefully explain the meanings of the following terms: point estimate, critical value, maximal margin of error, confidence level, and confidence interval.

To determine

To explain: The meanings of terms: point estimate, critical value, maximal margin of error, confidence level, and confidence interval.

Explanation of Solution

Point estimate:

When a parameter is being estimated, the estimate can be either a single number or it can be a range of scores. Point estimate is a single number used to estimate a population parameter.

Critical value:

In hypothesis testing, we compare the value of the some test statistic with a critical value; we accept or reject the null hypothesis depending on which of the values is more extreme. The critical value of the test is the borderline between accepting and rejecting the null hypothesis. The critical value depends on the significance level and type of test (one-sided or two- sided).

Maximal margin of error:

The margin of error using $\overline{x}$ as a point estimate for $\mu$ is $\left|\overline{x}-\mu \right|$. In most practical problems, $\mu$ is unknown, so the margin of error is also unknown. We can compute an error tolerance E, which serves as a bound on the margin of error. We can say that the point estimate $\overline{x}$ differs from the population mean $\mu$ by a maximal margin of error at c% level of confidence.

$E=z_c\frac{\sigma }{\sqrt{n}}$

Confidence level:

The confidence level is the percentage of all possible samples that can be expected to include the true population parameter. For a confidence level c, the critical value $z$ is the number such that the area under the standard normal curve between $-z$ and $z$ equals c. This means that

$P\left(-z<z<z\right)=c$

Confidence interval:

The sample mean x is the best point estimate of the true value of $\mu$. A confidence interval for a parameter is based on a corresponding sample statistic from a random sample of specified sample size n. The confidence interval is a range of likely values for the population parameter based on the point estimate.

For example confidence interval for $\mu$ with $\sigma$ known.

$\overline{x}-E<\mu <\overline{x}+E$

Where, $E=z_c\frac{\sigma }{\sqrt{n}}$ is the maximal margin of error.