Question

A random sample of n = 64 observations is drawn from a population with a mean equal to 20 and a standard deviation equal to 16

a. Give the mean and standard deviation of the (repeated) sampling distribution of x.

b. Describe the shape of the sampling distribution of x. Does your answer depend on the sample size?

c. Calculate the standard normal z-score corresponding to a value of x = 15.5.

d. Calculate the standard normal z-score corresponding to x = 23

Short answer

Answer

The standard deviation is commonly utilized as a measurement of an asset's comparative volatility. The standard deviation is determined as the square root of the variation by calculating the departure of every observation point from the mean.

Step-by-step solution

Step-by-Step Solution Step 1: (a) The data is given below

The calculation is given below:

$\begin{array}{c}\text{μ}\overline{X}=20\\ \sigma \overline{X}\text{=}\frac{\sigma }{\sqrt{\text{n}}}\\ =\frac{16}{\sqrt{64}}\\ =2\end{array}$

(b) The data is given below

Step 3: (c) The data is given below of x will have the form of a bell-shaped normally distributed. Yes, it is dependent on the scale of the sample. As the sample size improves the sample size approaches being normal.

(c) The data is given below

The calculation is given below:

$\begin{array}{c}\text{z =}\frac{\overline{\text{X}}-\mu }{\sigma \text{/}\sqrt{n}}\\ =\frac{15.5-20}{2}\\ =-2.25\end{array}$

(d) The data is given below

The calculation is given below:

$\begin{array}{l}\text{z =}\frac{\overline{\text{X}}-\mu }{\sigma \text{/}\sqrt{n}}\\ =\frac{23-20}{2}\\ =1.5\end{array}$