1. AP Stats Review: Probability Unit Unit #2 – Chapters 6, 7, and Section 8.1 1

2. The Normal Distribution Symmetric, mound-shaped Mean = median Area under the curve = 100% 68-95-99.7% rule - also called the Empirical Rule Crazy formula – no need to know it! 2

3. The 68-95-99.7% Rule The following shows what the 68-95-99.7 Rule tells us: 3

4. For example: 1998 #4 (partial): A company is considering implementing one of two quality control plans – plan P and plan Q – for monitoring the weights of car batteries that it manufactures. If the manufacturing process is working properly, the battery weights are approximately Normally distributed with mean = 2.7 lbs and standard deviation = 0.1 lbs. (a) Quality Control Plan P calls for rejecting a battery as defective if its weight falls more than 1 standard deviation below the mean. If a battery is selected at random, what is the probability that it will be rejected by plan P? 4

5. Solution to 1998 #4 (a): This is the area we’re interested in! 5 So, what’s that area? It’s ½ of the remaining 32%, so it’s 16%!

6. Finding Normal Probabilities When a data value doesn’t fall exactly 1, 2, or 3 standard deviations from the mean, we can use a Z Table to find the area to the left of the value. The calculator can also be used to find areas under a normal curve – it’s easier this way! 6

7. Finding Normal Probabilities (cont.) Table Z is the standard Normal table. We have to convert our values to z-scores before using the table. The figure below shows how to find the area to the left when we have a z-score of 1.80: 7

8. Calculator Use Normalcdf – gives area under a normal curve. You can use z-scores or raw data values: normalcdf (lower, upper, mean, st dev) invNorm – gives z-score or raw data value associated with a percentile: invNorm(area to left as decimal, mean, st dev) Provide complete communication on Free Response questions – use drawings and/or identify the values in your calculator syntax. Don’t just write normalcdf(-1.52,1.28)=0.835 and that’s all; draw a graph, shade in an area, and explain the numbers. Also, make sure that all explanation are in the context of the problem. For example, if the problem is about shirt sizes, then your answers should be about shirt sizes. 8

9. For example, 2003 #3: #3: Men’s shirt sizes are determined by their neck sizes. Suppose that men’s neck sizes are approximately normally distributed with mean 15.7” and standard deviation 0.7”. A retailer sells men’s shirts in sizes S, M, L, and XL, where the shirt sizes are defined in the table below: Shirt SizeNeck Size S14”-15”(b) Using a sketch of a normal M15”-16”curve, illustrate and calculate L16”-17”the proportion of men whose XL17”-18”shirt size is Medium. 9

10. Solution to 2003, #3 (b) (shirt sizes): Normalcdf (15, 16, 15.7, 0.7) = 0.5072 10

11. Question: When do we multiply? Answer: when we want to find the probability of a series of consecutive events Example: We just found that 0.5072 of all men wear size Medium shirts. Suppose we select 3 men at random from a huge population. What is the probability that all three of them wear Medium shirts? Answer: 11

12. What if we are counting how many successes will occur in repeated trials? The BINOMIAL Distribution! 12

13. What does it take to be Binomial? (1) Fixed number of trials (“n”) (2) Fixed probability of success on each trial (“p”) (3) Two outcomes per trials (success or failure) (4) Trials are independent of each other 13

14. Binomial on the Calculator Binomialpdf(n,p,x-value) > only the probability of that specific x-value Binomialcdf (n,p,x-value) > the sum of the probabilities of x=0 up through and including that specific x-value Example: We found that 50.72% of all shirts are mediums. If we select 12 shirts at random, what is the probability that… (i)Exactly 4 of them are Mediums? (ii)No more than 4 of them are Mediums? (iii) At least 4 of them are Mediums? 14

15. Parameters for a Binomially- Distributed Variable X “n” and “p” need to be known > “n” is the number of trials > “p” is the probability of success on any single trial *** 15

16. Question: When do we add? Answer: when the question can be construed as an “OR” situation Example: We found that 0.5072 of all men wear size Medium shirts. Out of a random sample of 12 men, what is the probability that either 3 or 4 (exactly) are Mediums? Pr(x=3 or x=4) = Pr(x=3) + Pr(x=4) Do each of these as a binomialpdf = 0.049 + 0.1139 = 0.1629 16

17. When ADDING, be careful about over-counting! Example: Using the chart above, if I select one bird at random, what is the probability that the bird I select is either a chick or is yellow? Pr(chick or yellow) = Pr(chick) + Pr(yellow) minus Pr(both chick & yellow) = 18/25 + 13/25 – 8/25 = 23/25 17 ChicksHawks Yellow85 Brown102

18. 18 Note: If there hadn’t been any yellow chicks, then there wouldn’t be anything to subtract, because we wouldn’t have overcounted. The characteristics of “yellow” and “chick” would then be called MUTUALLY EXCLUSIVE.

19. Finally, Combining Random Variables Z=X+Y or Z=X-Y Means will be added or subtracted just like the random variables Variances will be added ONLY IF VARIABLES ARE INDEPENDENT! (Remember: Variance = standard deviation squared) 19