Before we explore some detailed applications of the Normal Distribution, you may want to review the calculator skills and core ideas presented in the introductory chapter.
If $X$X is a normal random variable, it is defined by its parameters, the mean $\mu$μ and the standard deviation $\sigma$σ. We write $X\sim N(\mu,\sigma^2)$X~N(μ,σ2).
The Standard Normal variable, denoted by $Z$Z, has a mean of $0$0 and a standard deviation of $1$1. $Z\sim N(0,1)$Z~N(0,1). We can use $z$z scores to analyse and compare different data sets or to find an unknown mean and/or standard deviation.
Recall that a percentile (or a quantile when expressed as a decimal) indicates the proportion of a population that lies below a certain value of the distribution.
Since a normal distribution is a specific type of continuous random variable, a linear change of scale or origin on a normal random variable $X$X, such that $Y=aX+b$Y=aX+b has the following results on mean, variance and standard deviation:
- $E\left[aX+b\right]=aE\left[X\right]+b$E[aX+b]=aE[X]+b
- $Var\left[aX+b\right]=a^2Var\left[X\right]$Var[aX+b]=a2Var[X]
- $\sigma_{aX+b}=a\sigma_X$σaX+b=aσX
Modelling with a normal distribution
Select your brand of calculator below to work through an example using CAS for a normal distribution in an applied setting.
Casio ClassPad
How to use the CASIO Classpad to complete the following tasks involving applications of the normal distribution.
A company produces waffles whose thickness is normally distributed with a mean thickness of $25$25 mm and a standard deviation of $1.7$1.7 mm.
Research shows that waffles that are more than $28$28 mm thick or those that are less than $20$20 mm thick won't heat well at the suggested microwave setting, and should be excluded from packaging.
In a batch of $2000$2000 waffles, how many are expected to be excluded from packaging?
Calculate the thickness exceeded by a fifth of waffles produced on a particular day.
Determine the probability that the thickness of a waffle produced on a particular day is exactly $24$24 mm.
A sample of $80$80 waffles was taken. Determine the probability that no more than $2$2 were less than $21$21 mm thick.
Determine the $0.7$0.7 quantile.
The company wishes to calibrate their production machine to ensure that only $0.5%$0.5% of waffles have a thickness less than $21$21 mm. If they wish to maintain the same mean thickness, what should the new standard deviation be?
TI Nspire
How to use the TI Nspire to complete the following tasks involving applications of the normal distribution.
A company produces waffles whose thickness is normally distributed with a mean thickness of $25$25 mm and a standard deviation of $1.7$1.7 mm.
Research shows that waffles that are more than $28$28 mm thick or those that are less than $20$20 mm thick won't heat well at the suggested microwave setting, and should be excluded from packaging.
In a batch of $2000$2000 waffles, how many are expected to be excluded from packaging?
Calculate the thickness exceeded by a fifth of waffles produced on a particular day.
Determine the probability that the thickness of a waffle produced on a particular day is exactly $24$24 mm.
A sample of $80$80 waffles was taken. Determine the probability that no more than $2$2 were less than $21$21 mm thick.
Determine the $0.7$0.7 quantile.
The company wishes to calibrate their production machine to ensure that only $0.5%$0.5% of waffles have a thickness less than $21$21 mm. If they wish to maintain the same mean thickness, what should the new standard deviation be?
Practice questions
Question 1
Victoria downloads each episode of her favourite TV show as it’s released online. The length of each show is represented by the random variable $T$T, which is approximately normally distributed with a mean length of $50$50 minutes and a standard deviation of $4$4 minutes.
What percentage of her shows are less than $49$49 minutes in length?
Round your answer to one decimal place.
Victoria wants to put a show on her USB drive but only has room for an episode that is $48$48 minutes in length. What is the probability that she won’t be able to fit the show on the drive?
Round your answer to two decimal places.
Of the next five shows that Victoria independently downloads, what is the probability that the first two are less than $49$49 minutes and the last three are more than $49$49 minutes?
Round your answer to three decimal places.
Of the next $5$5 shows that Victoria downloads, what is the probability that exactly two are less than $49$49 minutes?
Round your answer to three decimal places.
Fans of the show have complained that the show length is really inconsistent. Calculate the maximum value of the standard deviation such that the probability of a show being less than $45$45 minutes is no more than $0.2%$0.2%.
Round your answer to two decimal places.
question 2
The marks for an IQ test are normally distributed with a mean of $\mu$μ and a standard deviation of $10$10.
It is known that $13%$13% of participants score above $134$134 marks.
Using a CAS calculator or otherwise, calculate the mean mark for the IQ test correct to three decimal places.
Let $X$X represent the distribution of the IQ test results. If the test is out of $185$185, write a rule in terms of $X$X to scale all the results to a percentage.
question 3
The length of the tail of a domestic cat is normally distributed with a mean of $25$25 cm and a standard deviation of $2.2$2.2 cm.
Use the capabilities of your CAS calculator to answer the following, rounding your answers to three decimal places:
What is the shortest length of a tail in the $70$70th percentile?
What is the shortest length of a tail in the top $15%$15%?
What is the shortest length of a tail in the $0.45$0.45 quantile?
What is the probability that a cat has a tail length less than $23.5$23.5 cm?
Suppose that a cat has a tail length below the $80$80th percentile. Using your rounded result from part (d), what is the probability that their tail length is more than $23.5$23.5 cm?