# 5.6: Central Limit Theorem (Worksheet)

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

Work in groups on these problems. You should try to answer the questions without referring to your textbook. If you get stuck, try asking another group for help.

## Student Learning Outcomes

• The student will compute normal probabilities using the Z Table.
• The student will demonstrate an understanding of the Central Limit Theorem.

## I. Calculating Standard Normal Probabilities

Recall that we denote a random variable with the standard normal distribution as $$Z$$, meaning that $$Z\sim N(0,1)$$. Use the Z Table to calculate each of the following probabilities of $$Z$$.

1. $$P(0< Z < 1.07)$$
2. $$P(-1.07 < Z < 0)$$
3. $$P(Z<1.07)$$
4. $$P(Z>1.07)$$
5. $$P(1.07<Z<2.89)$$
6. $$P(-0.43<Z<1.07)$$
7. $$P(-2.89<Z<-0.43)$$
8. Find the value of $$z$$, such that $$P(Z>z) = 0.0250$$.

## II. Calculating Non-Standard Normal Probabilities

Suppose that $$X\sim N(\mu=5, \sigma=2)$$. Use the Z Table to calculate each of the following probabilities of $$Z$$. You will need to convert each probability into an equivalent probability of $$Z$$.

1. $$P(5< X < 6.3)$$
2. $$P(1.7< X < 6.3)$$
3. Find the value of $$x$$, such that $$P(X<x) = 0.0020$$.

## III. Applying the Central Limit Theorem

Let the random variable $$Y=$$ the monthly income of a randomly selected employed adult in Zimbabwe. We can model the distribution of $$Y$$ as exponential with parameter $$\mu =$$ $477.5 USD. 1. Determine the mean and standard deviation of $$Y$$. 2. Suppose a random sample of size $$n=100$$ of employed adults in Zimbabwe is obtained. Use the Central Limit Theorem to approximate the probability that the mean of the sample (that is, the sample mean $$\overline{Y}$$) will be within$25 USD of the population mean that you found in the previous question.

This page titled 5.6: Central Limit Theorem (Worksheet) is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.