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Stat 211 Midterm 2 SOS Session

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)

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Stat 211 Midterm 2 SOS Session

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  1. Stat 211 Midterm 2 SOS Session Ahad Iqbal

  2. 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)

  3. What to memorize Discrete Random Variables Continuous Random Variables Not whole numbers (eg. Speed of your car at specific points in time) • Whole Numbers (eg. Number of people passed the first Stat 211 midterm during different AFM years) • Expected Value = Mean = • Variance =

  4. 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)

  5. The Binomial Distribution #3 L05 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 • What does the equation mean? • The equation for the binomial distribution consists of the product of two factors

  6. Example 4.10 Slide 4-16

  7. Normal Distribution - Memorize The Function: Definition: Mean = Median = Mode Cumulative Normal Curve

  8. 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

  9. 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 symmetry They 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)

  10. 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

  11. 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

  12. 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 binomial Eg. 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

  13. Meanception

  14. Meanception Taking a sample and finding the mean of that specific sample Eg. Population: AFM Students Subject: Marks on the first Stat exam Mean: Average mark of all AFM students on the stat exam Sample: All students in the front row Sample Mean: Mean of marks on the first exam on all students in the front row Example in Slide 6-3 is good enough to explain this

  15. 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!

  16. Example in 6-21 • n = 50, u = 7.6/100, u(bar) = 7.51/100, sd = 0.2

  17. Central Limit Theorem • The central limit theorem (CLT) states conditions under which the mean of a sufficiently large number of independentrandom 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:

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