High School

What standard deviation must [tex]\bar{x}[/tex] have so that 99.7% of all samples give a [tex]\bar{x}[/tex] within 1 point of [tex]\mu[/tex]?

Use the 68-95-99.7 rule.

Answer :

Final answer:

To have 99.7% of all sample means within 1 point of the population mean, the required standard deviation of the sample mean (bar(x)) must be 1/3.

Explanation:

The question is asking for the standard deviation that a sample mean (denoted as bar(x)) must have so that 99.7% of all sample means will be within 1 point of the population mean (mu). This scenario is described by the Empirical Rule, specifically the part about 99.7% of the data (or sample means, in this context) being within three standard deviations of the mean.

Since we want bar(x) to be within 1 point of mu, and we know that we're dealing with three standard deviations to cover 99.7% of samples according to the Empirical Rule, we can set up the following equation:

3 * (Standard Deviation of bar(x)) = 1

Solving for the standard deviation (sigma), we get:

Standard Deviation of bar(x) = 1/3