A random sample is drawn from a population of known standard...

A random sample is drawn from a population of known standard deviation 22.1. Construct a $95 \%$ confidence interval for the population mean based on the information given (not all of the information given need be used). a. $\quad n=121, x-82.4, s=21.9$ b. $\quad n=81, x-82.4, s=21.9$

A random sample is drawn from a population of known standard deviation 22.1. Construct a $95 \%$ confidence interval for the population mean based on the information given (not all of the information given need be used).
a. $\quad n=121, x-82.4, s=21.9$
b. $\quad n=81, x-82.4, s=21.9$

Key Concepts

Margin of Error

The margin of error represents the maximum expected difference between the true population parameter and a sample estimate. It is calculated by multiplying the critical value from the relevant distribution (in this case, 1.96 from the z-distribution) by the standard error (?/?n). The margin of error determines the width of the confidence interval.

Z-Distribution

The z-distribution, or standard normal distribution, is used in confidence interval construction when the population standard deviation is known or when the sample size is large. For a 95% confidence interval, the critical values are approximately ±1.96. This distribution underpins the calculation of the margin of error in such scenarios.

Sample Size

The sample size (n) is critical in the calculation of the standard error, impacting the precision of the confidence interval. A larger sample size decreases the standard error, thereby yielding a narrower confidence interval and a more precise estimate of the population mean.

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter. In constructing a confidence interval for the population mean, the formula generally used is the sample mean ± (critical value) × (standard error), where the standard error depends on the known or estimated standard deviation and the sample size.

Population Standard Deviation

When the population standard deviation is known, it can be used directly in the calculation of the standard error (?/?n). This known value improves the accuracy of the confidence interval as it removes the uncertainty associated with estimating the standard deviation from the sample.

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