Normal Distributions

 Symmetric Distribution ◦ Any normal distribution is symmetric Negatively Skewed (Left-skewed) distribution When a majority of the data values fall to the right of the mean Positively Skewed (Right-skewed) distribution when a majority of the data values fall to the left of the mean

Falls to the right of the data (Left-skewed) Falls to the left of the data (Right skewed)  Negatively Skewed  Positively Skewed Mean Med Mode Mean med mode

 The mean, median, and mode are all approximately the same.  Bell-shaped curve  Mean, median, and mode are equal and at the center  Only has one mode  Symmetric  NEVER touches the x- axis  Area under the curve is 100% or 1.00  Fits the Empirical (Normal) Rule

 A normal distribution with a mean of 0 and a standard deviation of 1. ◦ We use this to approximate the area under the curve for any given Normal distribution ◦ Use z-scores to do this (remember these from chapter 3…) ◦ We will also use Table E to help us with the math.

 There are three different possibilities for where the area under the curve that you are looking for is located:  1. To the left of the z-score  2. To the right of the z-score  3. Between any two z-scores

 Look up the z-score using Table E (p. 784)  That is your answer  Ex. Find the area to the left of z =  What does this mean?

 Look up the z-score  SUBTRACT the area from 1  Ex: Find the area to the right of z = -1.19

 Look up both z- scores  SUBTRACT the corresponding area (values)  Ex: Find the area between z = and z = -1.37

 1. Area to the left of z = +.6  2. Area to the right of z =  3. Area to the right of z =  4. Area to the left of z =  5. Area between z = and z =  6. Area between z = and z = -0.23

 Remember that a Normal distribution is a continuous distribution  We can use z-scores to find the probability of choosing any z-value at random  A special notation is used: ◦ If we are finding the probability of any z value between a and b, it is written as: P(a<z<b)

 Find each of the probabilities  1. P(0 < z < 2.32)  2. P(z<1.54)  3. P(z>1.91)  4. P(1.21 < z < 2.34)

 What is the z-value such that the area under the standard normal distribution curve is.2389?

 1. If it is on the negative side of zero, simply find the value.  2. If it is on the positive side of zero, subtract the value from 1 and find the difference value.  3. If it is between 0 and a number larger than zero, add.5 and then find the value.  4. If it is between 0 and a number smaller than zero, subtract.5 and then find the value.

 What is the z-value such that the area under the standard normal distribution curve between 0 and z is.2389?

 P. 311