1111

2222  States that any distribution of sample means from a large population approaches the normal distribution as n increases to infinity ◦ The mean of the population of means is always equal to the mean of the parent population. ◦ The standard deviation of the population of means is always equal to the standard deviation of the parent population divided by the square root of the sample size (n).  If you chart the values, the values will have less variation than the individual measurements  This is true if the sample size is sufficiently large.  What does this mean? VII-5

3333 For almost all populations, the sampling distribution of the mean can be closely approximated by a normal distribution, provided the sample is sufficiently large. Collect many x children, (assumption is infinite number of samples), create histograms. VII-6 Central Limit Theorem Explanation

4444  A number expressing the likelihood that a specific event will occur, expressed as the ratio of the number of actual occurrences to the number of possible occurrences P(n) = probability of n occurrences p= proportion success (what you are looking for) q= proportion failures (what you are not looking for) Example: If a fair coin is tossed, what is the probability of a head occurring? VII-7

5555  You are rolling a fair six-sided die. What are the odds that you will roll a 3? VII-7

6666  You are rolling a fair six-sided die. What are the odds that you will roll a 2 or a 4? A 2 and then a 4? VII-8

7777  You have a standard deck of cards. What are the odds that you draw a 3 or club? VII-11

8888  You have a standard deck of cards. What are the odds that you will draw four aces without replacement? VII-12

9999  Number of ways is a listing of possible successes  Permutation: PN,n, P(n,r), nPr, the number of arrangements when order is a concern – think ‘word’  Combination: CN,n,, nCr, the number of arrangements when order is not a concern VII-13

10  The product of a number and all counting numbers descending from it to 1 6! = 6x5x4x3x2x1=720 Note: 0!=1 VII-13

11  How many 3 letter arrangements can be found from the word C A T? How about 2 letter arrangements?  Three lottery numbers are drawn from a total of 50. How many arrangements can be expected? VII-13

12  How many 3 letter groupings can be found from the word C A T?  Three lottery numbers are drawn from a total of 50. How many combinations can be expected? VII-13

13  A single six-sided die is tossed five times. Find the probability of rolling a four, three times. VII-14

14  Refers to the probability distribution for defect count  Each unit of measure can have 0, 1, or multiple errors, defects, or some other type of measured occurrence.  Consider the following scenarios: ◦ The number of speeding tickets issued in a certain county per week. ◦ The number of calls arriving at an emergency dispatch station per hour. ◦ The number of typos per page in a technical book.  Calculated by: ◦ x = number of occurrences per unit interval (time or space) ◦ λ = average number of occurrences per unit interval VII-16

15  The average number of homes sold by the Acme Realty company is two homes per day. What is the probability that exactly three homes will be sold tomorrow?  μ = 2; since 2 homes are sold per day, on average.  x = 3; since we want to find the likelihood that 3 homes will be sold tomorrow.  e = ; since e is a constant equal to approximately  We plug these values into the Poisson formula as follows:  Solution:  P(x; μ) = (e^-μ) (μ^x) / x!  P(3; 2) = ( ^-2) (2^3) / 3!  P(3; 2) = ( ) (8) / 6  P(3; 2) =  Thus, the probability of selling 3 homes tomorrow is VII-17

16  Symmetrical, Bell-Shaped  Extends from Minus Infinity to Plus Infinity  Two Parameters ◦ Mean or Average ( ) ◦ Standard Deviation ( )  Space under the entire curve is 100% of the data  Mean, median and mode are the same VII-18

17 50% -1  -2  -3  +1  +2  +3  0   ≈ 68% 1   99.73% 3   ≈95% 2 z value = distance from the mean measured in standard deviations LCLUCL See XII-2 VII-18

18  Normal Curve theory tells us that the probability of a defect is smallest if you ◦ stabilize the process (control) ◦ make sigma as small as possible (reduce variation) ◦ get Xbar as close to target as possible (center) So… we first want to stabilize the process, second we will reduce variation and last thing is to center the process. VII-18

19  Specifies the areas under the normal curve  Represents the distance from the center measured in standard deviations  Values found on the normal table Population Sample Remember when we talked about  3  ? The 3 is the z value. See VIII VII-19

20  The known average human height is 5’8” tall with a standard deviation of 5 inches. What are the z values for 6’2” and 4’8”? A positive value indicates a z value to the right of the mean and a negative indicates a z value to the left of the mean. VII-19

21  From our answers from the last exercise, what is the values for: ◦ P(Area > 6’2”)? ◦ P(Area < 4’8”)? ◦ P(4’8”< Area < 6’2”)? ◦ Prove area under the normal curve at  1s,  2s,  3s? VII-19