1. Normal distribution and introduction to continuous random variables and continuous probability density functions...

2. Continuous random variable A continuous random variable is a random variable that takes on any value in an interval of numbers. Example: Let X = the weight of a randomly selected Big Mac, 0.20  x  0.30 pound Example: Let Y = the amount of chocolate consumed by a randomly selected student on Valentine’s Day, x  0 pounds

3. Graph: Percent Histogram

4. Histogram (Area of rectangle = probability)

5. Decrease interval size...

6. Decrease interval size more….

7. Continuous probability density functions The curve describes probability of getting any range of values, say P(X > 120), P(X<100), or P(110 < X < 120) Area under the curve = probability. Area under whole curve = 1. Probability of getting specific number is 0, e.g. P(X=120) = 0.

8. Special kind of continuous p.d.f

9. Characteristics of normal distribution Symmetric, bell-shaped curve. Shape of curve depends on population mean  and standard deviation . Center of distribution is the mean . Spread determined by standard deviation . Most values fall around the mean, but some values are smaller and some are larger.

10. Examples of normal random variables testosterone level of male students head circumference of adult females length of middle finger of Stat 250 students

11. What is probability a student gets a grade below 65?  = 70  = 5

12. Probability = Area under curve Calculus?! You’re kidding, right? But somebody did all the hard work for us! We just need a table of probabilities for every possible normal distribution. But there are an infinite number of normal distributions (one for each  and  )!! Solution is to “standardize.”

13. Standardizing Take value X and subtract its mean  from it, and then divide by its standard deviation . Call the resulting value Z. That is, Z = (X-  )/  = number of standard deviations X is above or below the mean. Z is called the standard normal. Its mean  is 0 and standard deviation  is 1. Then, use probability table for Z.

14. Probability given in Standard Normal Z Table

15. How to Read a Standard Normal Z Table Carry out Z calculations to two decimal places, that is X.XX. Round, if necessary. Find the first two digits (X.XX) of Z in column headed by z. Find the third digit of Z (X.XX) in first row. P(Z < z) = probability found at the intersection of the column and row.

16. What is probability student gets grade above 75?  = 70  = 5

17. What is probability student gets grade between 65 and 70?  = 70  = 5

18. Remember! Calculated probabilities are accurate only if the assumptions made are indeed correct! When doing the above calculations, you are assuming that the data are “normally distributed.” When possible, check this assumption! (We’ll learn how later.)