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:
Top Related