Before we explore some detailed applications of the Normal Distribution, you may like to review core ideas presented in our 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.
The Standard Normal variable, denoted by $Z$Z, has a mean of $0$0 and a standard deviation of $1$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
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.