Q 7.78. - Elementary Statistics

Question

A variable of a population is normally distributed with mean $\mu$ and standard deviation $\sigma$. For samples of size $n$, fill in the blanks. Justify your answers.

a. Approximately $68\%$ of all possible samples have means that lie within of the population mean, $\mu$

b. Approximately $95\%$ of all possible samples have means that lie within of the population mean, $\mu$

c. Approximately $99.7\%$ of all possible samples have means that lie within of the population mean, $\mu$

d. $100(1-\alpha )\%$ of all possible samples have means that lie within $\_$of the population mean, $\mu$ (Hint: Draw a graph for the distribution of $x$, and determine the $z-$scores dividing the area under the normal curve into a middle $1-\alpha$ area and two outside areas of$\alpha /2$

Step-by-Step Solution

Answer

Property 1: The mean of$68$ percent of the data points is a standard deviation of one That is, $68\%$ of values are found in the interval $\left(\mu -\frac{\sigma }{\sqrt{n}},\mu +\frac{\sigma }{\sqrt{n}}\right)$

Property 2: Around $95$ percent of the observations fall a standard deviation of two. That is, $95\%$ of values are found in the interval $\left(\mu -\frac{2\sigma }{\sqrt{n}},\mu +\frac{2\sigma }{\sqrt{n}}\right)$

Property 3: There are around$99.7\%$ observations that area standard deviation of three. That is, $99.7\%$ of values lie in the interval $\left(\mu -\frac{3\sigma }{\sqrt{n}},\mu +\frac{3\sigma }{\sqrt{n}}\right)$

Step 1: Given information

The sampling distribution is also normally distributed with sample mean $\mu$ and standard deviation ${\sigma }_{\overline{x}}=\frac{\sigma }{\sqrt{n}}$ for a population with a mean ${\mu }_x$ and standard deviation $\sigma$

Step 2: Concept

$\text{ population mean and standard deviation: }{\mu }_{\overline{x}}=\mu \text{ and }{\sigma }_{\overline{x}}=\sigma /\sqrt{n}\text{. }$

a Step 1: Explanation

The empirical rule is also known as the $68-95-99.7$ rule for the normal distribution. The following is a diagrammatic depiction of the empirical rule:

As a result, roughly $68\%$ of all feasible samples have means within $\frac{\sigma }{\sqrt{n}}$ of the population mean, $\mu$

b Step 1: Explanation

Approximately $95\%$ of all feasible samples have means that are within $\frac{2\sigma }{\sqrt{n}}$ of the population mean, $\mu$, according to empirical rule.

c Step 1: Explanation

Approximately $99.7\%$ of all feasible samples have means that are within the population mean, $\mu$ according to empirical rule.

d Step 1: Explanation

The area under the curve on either side of the shaded zone is symmetric and represents $\frac{\alpha }{2}$, as seen in the graph.

As a result, $100(1-)\%$ of all feasible samples have means that are within $z_{\frac{\alpha }{2}}\frac{\sigma }{\sqrt{n}}$ of the population mean, $\mu$