Dan Piett STAT West Virginia University Lecture 7

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 ≥

Overview Normal Distribution Empirical Rule Normal Probabilities Percentiles

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

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.

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

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

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)

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

Using the Empirical Rule Back to the dog weight example, X~N(44,8) 1. What percent of dogs weigh between 28 and 60 pounds? 95% by the empirical rule 2. What percent of dogs weigh more than 60 pounds? 2.5% by the empirical rule Why is this?

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:

Examples of Finding Z For X~N(44,8) Find Z for X = What do we notice? Z measures how many standard deviations we are away from the mean

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

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)

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)

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

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

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: 90 th 75 th 50 th Note that these questions could be asked such that: P(X<C)= Find C