1 ES Chapter 3 ~ Normal Probability Distributions

2 ES Chapter Goals Learn about the normal, bell-shaped, or Gaussian distribution How probabilities are found How probabilities are represented How normal distributions are used in the real world

3 ES Normal Probability Distributions The normal probability distribution is the most important distribution in all of statistics Many continuous random variables have normal or approximately normal distributions Need to learn how to describe a normal probability distribution

4 ES Normal Probability Distribution 1.A continuous random variable 2.Description involves two functions: a.A function to determine the ordinates of the graph picturing the distribution b.A function to determine probabilities 4.The probability that x lies in some interval is the area under the curve 3.Normal probability distribution function: This is the function for the normal (bell-shaped) curve fxe x () ()       1

5 ES The Normal Probability Distribution

6 ES Illustration Probabilities for a Normal Distribution

7 ES Notes The definite integral is a calculus topic We will use a table to find probabilities for normal distributions We will learn how to compute probabilities for one special normal distribution: the standard normal distribution Transform all other normal probability questions to this special distribution Recall the empirical rule: the percentages that lie within certain intervals about the mean come from the normal probability distribution We need to refine the empirical rule to be able to find the percentage that lies between any two numbers

8 ES Percentage, Proportion & Probability Basically the same concepts Percentage (30%) is usually used when talking about a proportion (3/10) of a population Probability is usually used when talking about the chance that the next individual item will possess a certain property Area is the graphic representation of all three when we draw a picture to illustrate the situation

9 ES The Standard Normal Distribution There are infinitely many normal probability distributions They are all related to the standard normal distribution The standard normal distribution is the normal distribution of the standard variable z (the z-score)

10 ES Standard Normal Distribution Properties: The total area under the normal curve is equal to 1 The distribution is mounded and symmetric; it extends indefinitely in both directions, approaching but never touching the horizontal axis The distribution has a mean of 0 and a standard deviation of 1 The mean divides the area in half, 0.50 on each side Nearly all the area is between z = and z = 3.00 Notes: Appendix, Table A lists the probabilities associated with the intervals from the mean (0) to a specific value of z Probabilities of other intervals are found using the table entries, addition, subtraction, and the properties above

11 ES Which Table to Use? An infinite number of normal distributions means an infinite number of tables to look up!

12 ES Solution: The Cumulative Standardized Normal Distribution Z Cumulative Standardized Normal Distribution Table (Portion) Probabilities Shaded Area Exaggerated Only One Table is Needed Z = 0.12

13 ES Standardizing Example Normal Distribution Standardized Normal Distribution Shaded Area Exaggerated

14 ES Example: Normal Distribution Standardized Normal Distribution Shaded Area Exaggerated

15 ES Z Cumulative Standardized Normal Distribution Table (Portion) Shaded Area Exaggerated Z = 0.21 Example: (continued)

16 ES Z Cumulative Standardized Normal Distribution Table (Portion) Shaded Area Exaggerated Z = Example: (continued)

17 ES Notes The symmetry of the normal distribution is a key factor in determining probabilities associated with values below (to the left of) the mean. For example: the area between the mean and z = is exactly the same as the area between the mean and z = When finding normal distribution probabilities, a sketch is always helpful. See course worksheet.

18 ES Example: Normal Distribution Standardized Normal Distribution Shaded Area Exaggerated

19 ES Example: (continued) Z Cumulative Standardized Normal Distribution Table (Portion) Shaded Area Exaggerated Z = 0.30

20 ES Recovering X Values for Known Probabilities Normal Distribution Standardized Normal Distribution

21 ES Assessing Normality Not all continuous random variables are normally distributed It is important to evaluate how well the data set seems to be adequately approximated by a normal distribution

22 ES Assessing Normality Construct charts –For small- or moderate-sized data sets, do stem- and-leaf display and box-and-whisker plot look symmetric? –For large data sets, does the histogram or polygon appear bell-shaped? Compute descriptive summary measures –Do the mean, median and mode have similar values? –Is the interquartile range approximately 1.33  ? –Is the range approximately 6  ? (continued)

23 ES Assessing Normality Observe the distribution of the data set –Do approximately 2/3 of the observations lie between mean 1 standard deviation? –Do approximately 4/5 of the observations lie between mean 1.28 standard deviations? –Do approximately 19/20 of the observations lie between mean 2 standard deviations? (continued)

24 ES With Minitab Assessing Normality

25 ES Applications of Normal Distributions Apply the techniques learned for the z distribution to all normal distributions Start with a probability question in terms of x-values Convert, or transform, the question into an equivalent probability statement involving z-values

26 ES Standardization Suppose x is a normal random variable with mean  and standard deviation  The random variable has a standard normal distribution

27 ES Notes The normal table may be used to answer many kinds of questions involving a normal distribution Example:The waiting time x at a certain bank is approximately normally distributed with a mean of 3.7 minutes and a standard deviation of 1.4 minutes. The bank would like to claim that 95% of all customers are waited on by a teller within c minutes. Find the value of c that makes this statement true. Often we need to find a cutoff point: a value of x such that there is a certain probability in a specified interval defined by x

28 ES Pxc P xc Pz c ()                     Solution

29 ES Notation If x is a normal random variable with mean  and standard deviation , this is often denoted: x ~ N( ,  )  Example: Suppose x is a normal random variable with  = 35 and  = 6. A convenient notation to identify this random variable is: x ~ N(35, 6).

30 ES Notation z-score used throughout statistics in a variety of ways Need convenient notation to indicate the area under the standard normal distribution z (  ) is the algebraic name, for the z-score (point on the z axis) such that there is  of the area (probability) to the right or left of z (  )

31 ES Notes The values of z that will be used regularly come from one of the following situations: 1.The z-score such that there is a specified area in one tail of the normal distribution 2.The z-scores that bound a specified middle proportion of the normal distribution

32 ES Example Example: Find the numerical value of z (0.99) : Because of the symmetrical nature of the normal distribution, z (0.99) = - z (0.01) Using Table A: z (0.01) = z (0.01)

33 ES Example Example: Find the z-scores that bound the middle 0.99 of the normal distribution: Use Table A: z (0.005) = and z (0.995) = z (0.005) z (0.995)

34 ES Chapter Summary Discussed the normal distribution Described the standard normal distribution Evaluated the normality assumption