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Lecture 7

Lecture 7. Dan Piett STAT 211-019 West Virginia University. Last Week. Binomial Distributions 2 Outcomes, n trials, probability of success = p, X = Number of Successes Poisson Distributions Occurrences are measured over some unit of time/space with mean occurrences lambda

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Lecture 7

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  1. Lecture 7 Dan Piett STAT 211-019 West Virginia University

  2. Last Week • Binomial Distributions • 2 Outcomes, n trials, probability of success = p, • X = Number of Successes • Poisson Distributions • Occurrences are measured over some unit of time/space with mean occurrences lambda • X = Number of Occurrences • Finding Probabilities • = • < and ≤ • > and ≥

  3. Overview • Normal Distribution • Empirical Rule • Normal Probabilities • Percentiles

  4. Continuous Distributions • Up until this point we have only talked about discrete random variables. • Binomial • Poisson • Note that in these distributions, X was a countable number. • Number of successes, Number of occurrences. • Now we will be looking at continuous distributions • Ex: height, weight, marathon running time

  5. Continuous Distributions Cont. • Continuous Distributions are generally represented by a curve • Unlike discrete distributions, where the sum of the probabilities equals 1, in the continuous case, the area under the curve is 1. • One additional important difference is that in continuous distributions the P(X=x)=0 • Reason for this has to do with the calculus behind continuous functions. • Because of this ≥ is the same as > • Also, ≤ is the same as < • Therefore, we will only be interested in > or < probabilities.

  6. Normal Distribution • Unlike the Binomial and Poisson distributions that were defined by a set of rigid requirements, the only condition for a normal distribution is that the variable is continuous. • And that the variable follows normal distribution. • MANY variables follow normal distribution. • The normal distribution is one of the most important distribution in statistics. • Normal Distribution is defined the mean and standard deviation • X~N(mu, sigma) • If we are given the variance, we will need to take the square root to get the standard deviation

  7. Normal Distribution Con’t. • Properties: • Mound shaped: bell shaped • Symmetric about µ, population mean • Continuous • Total area beneath Normal curve is 1 • Infinite number of Normal distributions, each with its own mu and sigma

  8. Example: Weight of dogs • Suppose X, the weight of a full-grown dog is normally distributed with a mean of 44 lbs and a standard deviation of 8 pounds X~N(44, 8) 20 28 36 44 52 60 68

  9. The Empirical Rule • The empirical rule states the following: • Approx. 68% of the data falls within 1 stdv of the mean • Approx. 95% of the data falls within 2 stdv of the mean • Approx. 99.7% of the data falls within 3 stdv of the mean

  10. Using the Empirical Rule • Back to the dog weight example, X~N(44,8) • What percent of dogs weigh between 28 and 60 pounds? • 95% by the empirical rule • What percent of dogs weigh more than 60 pounds? • 2.5% by the empirical rule • Why is this?

  11. Finding Normal Probabilities • Like Binomial and Poisson distributions, the cumulative probabilities for the Normal Distribution can be found using tables. • BUT, rather than making tables for different values of mu and sigma, there is only 1 table. • N(0,1) • We will need to convert the normal distribution of our problem to this normal distribution using the formula:

  12. Examples of Finding Z • For X~N(44,8) • Find Z for X = • 52 • 1 • 28 • -2 • 68 • 3 • What do we notice? • Z measures how many standard deviations we are away from the mean

  13. Finding Exact Probabilities • Good news! • For any X, the P(X=x)=0 • We assume it is impossible to get any 1 particular value

  14. Finding Less Than Probabilities • To find less than probabilities. We first convert to our z-score then look up the Z value on the normal table. • Remember, since we are using a continuous distribution, < is the same as <= • For X~N(30, 4), Find • P(X<29) • P(X<40) • P(X≤40)

  15. Greater Than Probabilities • Similar to less than probabilities, first find the z-score, then use the table. Just like Binomial and Poisson we will use 1 – the value in the table. • For X~N(100, 10), Find • P(X>95) • P(X>100) • P(X≥100)

  16. In-Between Probabilities • To find in-between probabilities, you must first find the z-score for both points, call them a and b, and then the probability is just the P(X<b) – P(X<a) • For X~N(18,2), Find • P(14<X<22) • Compare this to the Empirical Rule

  17. Percentiles – Working Backward • Suppose that we want to find what X value corresponds to a percentile of the Normal Distribution • Example: What is the 90th percentile cutoff for SAT Scores? • How to do this • Step 1: Find the z value in the z table that matches closest to .9000. • Step 2: Put this z in the z-score formula • Step 3: Solve for x

  18. Example • Let X be a student’s SAT Math Score with a mean of 500 and a standard deviation of 100. • X~N(500,100) • Find the following percentiles: • 90th • 75th • 50th • Note that these questions could be asked such that: • P(X<C)=.9000. Find C

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