Stat 226 - Lecture 10

02/13/14

Suppose $ X \sim N(12, 2^2) $. Find the value of $ x^* $ for each of the following scenarios:
- $ P(X < x^*) = 0.33 $ ($ x^* $ is called the 33rd percentile)
- $ P(X < x^*) = 0.66 $ ($ x^* $ is called the 66th percentile)
- $ P(X < x^*) = 0.01 $ ($ x^* $ is called the 1st percentile)
Suppose monthly revenue in your business market is known to vary according to a normal distribution with mean of $500 thousand and variance of $200 thousand.
- How much revenue do you need next month in order to sell more than 98% of the entire market?
- What monthly revenue figure would imply that 40% of the market receieved a higher revenue?

By now, we are familiar with translating $ X \sim N(\mu, \sigma^2) $ to $ Z \sim N(0,1) $.
We do this by shifting $ X $ by $ \mu $ and scaling by $ 1/\sigma $ (ie, $ Z = \frac{X-\mu}{\sigma} $)
As it turns out, shifting a normal random variable will always give us another normal random variable!
For instance, suppose $ Y = X - \mu $. Then $ Y \sim $
$ N(0, \sigma^2) $
This is a useful idea if we know the distribution of $ X $, but are asked a question about $ Y $.

It is a well known that $ Revenue - Cost $ =
$ Profit! $
Suppose you are a used-car dealership owner. You have the opportunity to buy a car for $1000.
You don't know how much it will sell for, but revenue generated from similar cars can be described by $ X \sim N(1200, 175^2) $.
What is the probability that you will profit off the car?
That is, what is the probability that revenue exceeds cost?
Let $ Y $ be profit. Then $ Y = $
$ X - 1000 $.
Therefore, $ Y \sim N( $
$ 200, 175^2) $
So, the probability of profit is $ P(Y > 0) $!

Everything we've done in Chapter 12 requires that we know a population is normally distributed.
Of course, we never actually know this, but we can use a sample to assess whether the normal assumption is reasonable or not.
The demonstrate how we make this assessment, we will study some baseball data :)

plot of chunk hit_dens

In a baseball game, both teams have several attempts to hit the baseball in play.
I collected data on total # of hits by home and away teams in 2011.
This histogram shows the distribution of hits per game.
Pretty close to a symmetric, bell-shaped distribution. Let's fit a normal distribution to it!

plot of chunk hit_dens2

We can fit a normal distribution to this data by:
- setting the mean parameter $ \mu $ equal to the sample mean $ \bar{x} $ = 8.9
- setting the variance parameter $ \sigma^2 $ equal to the sample variance $ s^2 $ = 12.2

plot of chunk hit_dens3

Think about selecting one game at random. Before selecting, we can make inference using a statistical model for the population of games: $ X \sim N(\mu = 8.9, \sigma^2 = 12.2) $
Note that $ \sigma = \sqrt{12.2} = 3.5 $
If we were to pick one game at random, there is a 68% chance of observing between (8.9 - 3.5, 8.9 + 3.5)
=> (5.4, 12.4)

hits in that game.

plot of chunk qqnorm

If runs/game was perfectly normal, the points would fall exactly on the red line.
Since nothing is perfectly normal, we expect some deviation from this line.
The real question is whether this deviation is statistically significant.
Suppose the deviation here is significant. What does that imply?
See last 3 slides in lecture notes for more examples.