Stat 211 Midterm 2 SOS Session

Ahad Iqbal

What to memorize

• Two types of random variables you learn about in Stat 211: Discrete and Continuous

Very Rudimentary Rules• P(x) 0• Adding up all of the probabilities in the space

gives you 1. (for continuous, it is the area under the curve)

• V = SD^2 (Example 4.6 Chapter 4 Slide)

What to memorize

Discrete Random Variables• Whole Numbers (eg.

Number of people passed the first Stat 211 midterm during different AFM years)

• Expected Value = Mean =

• Variance =

Continuous Random Variables• Not whole numbers (eg.

Speed of your car at specific points in time)

xAll

X xpxμ

xAll

XX xpx 22

Binomial Distribution – Memorize This

• The Binomial Experiment:• 1. Experiment consists of n identical trials• 2. Each trial results in either “success” or “failure”• 3. Probability of success, p, is constant from trial to trial• 4. Trials are independent

x = Number of Successes n = Total Number of Trials p = Chance of Success in one trial q = Chance of Failure in one trail (1-p)

x-nxqp

!x-n!x

!n =xp

npq

npq

np

X

X

X

deviation standard

variance

mean

2

The Binomial Distribution #3

x-nxqp

x-nx

n =xp

!!

!

4-5

• What does the equation mean?• The equation for the binomial distribution consists of the

product of two factors

Number of ways to get x successes and (n–x) failures in n trials

The chance of getting x successes and (n–x) failures in a particular arrangement

L05

Example 4.10 Slide 4-16

Normal Distribution - MemorizeThe Function:

Definition: Mean = Median = Mode

Cumulative Normal Curve

eπσ

xx

2

2

1

2

1=)f(

Z-Scores

• You know that anytime the mean = median = mode we have a normal distribution

• This means that there can be infinite amount of normal distributions

• The table that you get in your exam with numbers on it is only for ND with mean = 0 and SD = 1

• We need to find a way to not need an infinite amount of tables on the exam

• Thus we have z-scores

x

z

THERE ARE TWO TYPES OF TABLESThis is for Normal Tables

• P(b) => LOOK AT THE TABLE for b and go down (Slide 5-20)

• P(a ≤ z ≤ b)= P(b) – P(a)• P(-a ≤ z ≤ a)= P(-a ≤ z ≤ o)+ P(0 ≤ z ≤ a)• P(-a ≤ z ≤ o) = P(0 ≤ z ≤ a) because of symmetryThey may troll you and have just one restriction • P(z a) => If a > 0: 0.5 - P(a), if a < 0: P(a) + 0.5, Else: 0.5• Z(c) = B, find c = > LOOK AT THE TABLE, Work Backwards

(Example on 5-34 is sufficient for this)

General Procedures

1. Formulate the problem in terms of x values

2. Calculate the corresponding z values, and restate the problem in terms of these z values

3. Find the required areas under the standard normal curve by using the table

Note: It is always useful to DRAW A PICTURE showing the required areas before using the normal table

Example in 5-29 is a good

This is for Cumulative Tables

• P(z ≤ a) => Directly from the Cumulative Table• P(z ≥ a) = 1 - P(z ≥ a) => Slide 5-43 for Table

Quick Check

• 4 Steps determine if binomial • Distribution of the data determines if it is

normal (aka mean = mode = median)

Eg. Rolling a dice is binomialEg. If you roll a dice 200 times and plot the

number of times you got the number, if that plot has mean = mode = median, you have a Normal

Meanception

Meanception

Taking a sample and finding the mean of that specific sample

Eg. Population: AFM StudentsSubject: Marks on the first Stat examMean: Average mark of all AFM students on the

stat examSample: All students in the front rowSample Mean: Mean of marks on the first exam on

all students in the front rowExample in Slide 6-3 is good enough to explain this

Sampling Distribution of the Sample Mean: General Info

Anything with a Bar on top means that it belongs to the sample

Sample Mean: Unbiased Estimator

Sample Deviation: Higher size Lower Variance

Rule:If the population is Normal, then means will be as well To Reduce the Variance (which is SD^2), take more trials!

Example in 6-21

• n = 50, u = 7.6/100, u(bar) = 7.51/100, sd = 0.2

Central Limit Theorem

• The central limit theorem (CLT) states conditions under which the mean of a sufficiently large number of independent random variables, each with finite mean and variance, will be approximately normally distributed (wikipedia)

• As Sample size (n) increases, spread (sd) decreases• An n of usually 30 is sufficient, but if not: