W ARM -U P Determine whether the following are linear transformations, combinations or both. Also find the new mean and standard deviation for the following. 1. A = 2.5x 2. B = X + Y 3. C = X – 2y 4. D = -.5y 5. E = xy MeanStandard Deviation X Y204.5

C OUNTING … Find the number of items in the sample space of license plates containing 3 letters and 3 numbers that can be repeated. What if they can’t be repeated?

P ERMUTATIONS An arrangement of objects in a specific order Order Matters and No Repetitions EX: How many ways can you arrange 3 people in a picture?

E XAMPLE 2 Suppose a business owner has a choice of 5 locations. She decides the rank them from best to worst according to certain criteria. How many different ways can she rank them? What if she only wanted to rank the top 3?

P ERMUTATION R ULE ! Where n = total # of objects and r = how many you need.

E XAMPLE 3 A TV news director wishes to use 3 news stories on the evening news. She wants the top 3 out of 8 possible. How many ways can the program be set up?

C OMBINATIONS A selection of “n” objects without regard to order. When different orderings of the same items are not counted separately we have a combination problem. EX: AB is the same as BA When different ordering of the same items are counted separately, we have a permutation. EX: AB is different than BA

C OMBINATION R ULE Example1 : To survey opinions of customers at local malls, a researcher decides to select 5 from 12. How many ways can this be done?

E XAMPLE 2 In a club, there are 7 women and 5 men. A committee of 3 women and 2 men is to be chosen. How many different possibilities are there? What about a committee of 5 with at least 3 women? At most 2 women?

B INOMIAL D ISTRIBUTIONS Each trial has only 2 possible outcomes “success” or “failure” There is a fixed # of trials (n) Trials are independent of each other The probability of a success (p) is constant X ~ B(n, p) q – numerical probability of failure (1 – p) r – number of “successes” in n trials

B INOMIAL D ISTRIBUTION F ORMULA For X ~ B(n, p) then

E XAMPLE 1 A coin is tossed 3 times. Find the probability of getting exactly 2 heads.

E XAMPLE 2 Public Opinion reported that 5% of Americans are afraid of being alone in the house at night. If a random sample of 20 Americans is selected, find the probability that there are exactly 5 people who are afraid of being along in the house at night.

Y OU T RY ! A student takes a random guess at 5 multiple choice questions. Find the probability that the student gets exactly 3 correct. Each question has 4 possible choices.

E XAMPLE 3 X is binomially distributed with 6 trials and a probability of success equal to 1/5 at each. What is the probability of at least one success? Three or fewer successes?

E XAMPLE 2 R EVISITED Public Opinion reported that 5% of Americans are afraid of being alone in the house at night. If a random sample of 20 Americans is selected. Find the probability that at most 3 are afraid. Find the probability that at least 3 are afraid.

Y OU T RY A GAIN ! A student takes a random guess at 5 multiple choice questions. Each question has 4 possible choices. Find the probability that the student gets at most 2 correct. Find the probability that the student gets at least 2 correct.

M EAN & S TANDARD D EVIATION For a binomial distribution: p = probability of success and q = probability of failure μ = p and σ = √(pq)  for 1 trial μ = 2p and σ = √(2pq)  for 2 trials μ = 3p and σ = √(3pq)  for 3 trials In general… μ = np and σ = √(npq)  for n trials

E XAMPLE 1 5% of a batch of batteries are defective. A random sample of 80 batteries is taken with replacement. Find the mean and standard deviation of the number of defective batteries in the sample.

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W ARM U P A biased coin is tossed 6 times. The probability of heads on any toss is 0.3. Let X denote the number of heads that come up. Calculate: P(X = 2) P(X < 3) P(1 < X < 5).

N ORMAL D ISTRIBUTION A normal distribution curve is symmetrical, bell-shaped curve defined by the mean and standard deviation of a data set. The normal curve is a probability distribution with a total area under the curve of 1.

C HARACTERISTICS OF A N ORMAL D ISTRIBUTION What do the 3 curves have in common?

C HARACTERISTICS OF A N ORMAL D ISTRIBUTION The curves may have different mean and/or standard deviations but they all have the same characteristics Bell-shaped curve Symmetrical about the mean Mean, median and mode are the same (not skew!) Area under the curve is always 1 (100%)

S TANDARD N ORMAL D ISTRIBUTION Written as Z ~ N(0, 1) Mean = 0 & Standard Deviation = 1

S TANDARD N ORMAL D ISTRIBUTION Since the total area under the curve is 1, we can consider partial areas to represent probabilities.

Z-S CORES A standard normal distribution is the set of all z-scores. All values can be transformed from a normal distribution to a standard normal by using the z-score. It represents how many standard deviations “x” is always from the mean. The z-score is positive if the data value lies above the mean and negative if the data value lies below the mean.

Z- SCORE E XAMPLES Suppose SAT scores among college students are normally distributed with a mean of 500 and a standard deviation of 100. If a student scores a 700, what would be their z-score?

M ORE Z- SCORE E XAMPLES For which test would a score of 78 have a higher standing? A set of English test scores has a mean of 74 and a standard deviation of 16. A set of math test scores has a mean of 70 and a standard deviation of 8.

E VEN M ORE Z- SCORE E XAMPLES What will be the miles per gallon for a Toyota Camry when the average mpg is 23, it has a z-value of 1.5 and a standard deviation of 5?

A REA WITH A T ABLE Draw the distribution curve Shade the area in which you are interested Use the table to find the areas Might have to add or subtract to get what you want.

E XAMPLES FOR A REA Find the area/probability of the following: Left of z = 1.99 P(z < 1.99) Left of z = 2.55 P(z < 2.55) Right of z = 1.11 P(z > 1.11)

M ORE E XAMPLES FOR A REA Find the area/probability of the following: Left of z = P(z < -2.5) Right of z = P(z > -1.2)

E VEN M ORE E XAMPLES FOR A REA Find the area/probability of the following: P(0 < z < 2.32) P(-1.2 < z < 2.3)

A ND O NE M ORE E XAMPLE FOR A REA Find the area/probability of the following: P(z 2.43)

A PPLICATION 1 A Calculus exam is given to 500 students. The scores have a normal distribution with a mean of 78 and a standard deviation of 5. What percent of the students have scores between 82 and 90? How many students have scores between 82 and 90?

A PPLICATION 2 A Calculus exam is given to 500 students. The scores have a normal distribution with a mean of 78 and a standard deviation of 5. What percent of the students have scores above 70? How many students scored above a 70?

A PPLICATION 3 Find the probability of scoring below a 1400 on the SAT if the scores are normal distributed with a mean of 1500 and a standard deviation of 200.

F INDING Z- SCORES FROM A REA Find the z-score above the mean with an area to the left of z equal to Find the z-score below the mean with an area to the left of z equal to 13.87%

M ORE F INDING Z- SCORES FROM A REA Find the z-score below the mean with an area between 0 and z equal to

E VEN M ORE F INDING Z- SCORES FROM A REA Find the z-score above the mean with an area between 0 and z equal to Find the z to the right of the mean with an area to the right of z equal to

I NVERSE N ORMAL D ISTRIBUTIONS Find k for which P(x < k) = 0.95 given that x is normally distributed with a mean of 70 and a standard deviation of 10.

A PPLICATIONS A professor determines that 80% of this year’s History candidates should pass the final exam. The results are expected to be normally distributed with a mean of 62 and standard deviation of 13. Find the lowest score necessary to pass the exam.

M ORE A PPLICATIONS Researchers want to select people in the middle 60% of the population based on their blood pressure. If the mean is 120 and the S.D. is 8. Find the upper and lower reading that would qualify.

F INDING S TATS B ASED ON P ROBABILITY Sacks of potatoes with a mean weight of 5 kg are packed by an automatic loader. In a test, it was found that 10% of bags were over 5.2 kg. Use this information to find the standard deviation of the process

M ORE F INDING S TATS B ASED ON P ROBABILITY Find the mean and the standard deviation of a normally distributed random variables X, if P(x > 50) = 0.2 and P(x < 20) = 0.3