Warm UpMay 20 th  Please pick up the 11-2 Enrichment sheet from the cart and get started (both sides)

Tougher Probability  Carson is not having much luck lately. His car will only start 80% of the time and his moped will only start 60% of the time. Draw a tree diagram to illustrate the situation. Use the diagram to determine the chance that  Both will start  He has to take his car.  He has to take the bus.

Tougher Probability  A box contains 3 red, 2 blue and 1 yellow marble. Draw a tree diagram to represent drawing 2 marbles.  With replacementWithout replacement  Find the probability of getting two different colors: If replacement occurs If replacement does not occur

Tougher Probability  A bag contains 5 red and 3 blue marbles. Two marbles are drawn simultaneously from the bad. Determine the probability that at least one is red.

Probability Check 1. 13/ / /5 6. 2/5 7. 3/5 8. 4/5 9. 7/ / / / π/5000 = /625 = (144π + 24)/5000 = (144π/5000) = a) 1,12,13,14,1 1,22,23,24,2 1,32,33,34,3 1,42,43,44,4 b) 16 c) 3/16

Probability & Trials  Ch. 19 Dice Simulation

Normal Distribution  Bell-shaped curve defined by the mean and standard deviation of a data set.

Characteristics of a Normal Distribution  What do the 3 curves have in common?

Characteristics of a Normal Distribution  The curves may have different mean and/or standard deviations but they all have the same characteristics Bell-shaped continuous curve Symmetrical about the mean Mean, median and mode are the same and located at the center It approaches, but never touches the x axis Area under the curve is always 1 (100%)

Is it Normal?

Empirical Rule  If data follows a normal distribution… 68% of it will be within 1 standard deviation 95% of it will be within 2 standard deviations 99% of it will be within 3 standard deviations

Empirical Rule

Examples  The heights of the 880 students at East Meck High School are normally distributed with a mean of 67 inches and a standard deviation of 2.5 inches a) Draw and label the normal curve. b) 68% of the students fall between what two heights?

Examples (cont.) c) What percent of the students are between 59.5 and 69.5 inches tall? d) Approximately how many students are more than 72 inches tall?

You Try!  A machine used to fill water bottle dispenses slightly different amounts into each bottle. Suppose the volume of water in 120 bottles is normally distributed with a mean of 1.1 liters and a standard deviation of 0.02 liter. a) Draw and label the normal curve. b) 95% of the water bottles fall between what two volumes? c) What percent of the bottles have between 1.08 and 1.12 liters? d) Approximately how many bottles of water are filled with less than 1.06 liters?

Back to Heights Examples e) If a student is 62 inches tall, how many standard deviations from the mean are they? f) If a student is 71 inches tall, how many standard deviations from the mean are they?

Standard Deviations How to be more specific…  A standard normal distribution is the set of all z-scores (or z-values). It represents how many standard deviations a certain data point is away from the mean. The z-score is positive if the data value lies above the mean and negative if it’s below the mean.

How to find Z-Scores

Examples  Find z if X = 24, µ = 29 and σ = 4.2  You Try! Find Z if X = 19, μ = 22, and σ = 2.6

Back to Heights Examples More Specific e) If a student is 62 inches tall, how many standard deviations from the mean are they? f) If a student is 71 inches tall, how many standard deviations from the mean are they?

You Try!  A machine used to fill water bottle dispenses slightly different amounts into each bottle. Suppose the volume of water in 120 bottles is normally distributed with a mean of 1.1 liters and a standard deviation of 0.02 liter. e) If a water bottle has 1.16 liters, how many standard deviations from the mean is it? f) If a water bottle has 1.07 liters, how many standard deviations away from the mean is it?

Back to Heights Examples  The heights of the 880 students at East Meck High School are normally distributed with a mean of 67 inches and a standard deviation of 2.5 inches g) If you pick a student at random, what is the probability that they will be between 62 and 72 inches tall? h) If you pick a student at random, what is the probability they will be between 65 and 69 inches tall?

Area & Probability  2nd  DISTR (Vars button)  normalcdf(minimum z value, maximum z value) Back to h) If you pick a student at random, what is the probability they will be between 65 and 69 inches tall? (remember mean = 67 and SD = 2.5)

More Examples  The temperatures for one month for a city in California are normally distributed with mean = 81 degrees and sd = 6 degrees. Find each probability and use a graphing calculator to sketch the corresponding area under the curve.  a. P(70 < x < 90)

More Examples  The scores on a standardized test are normally distributed with mean = 72 and sd = 11. Find each probability and use a graphing calculator to sketch the corresponding area under the curve.  Find: P(65 < x < 85)

You Try Again!  A machine used to fill water bottle dispenses slightly different amounts into each bottle. Suppose the volume of water in 120 bottles is normally distributed with a mean of 1.1 liters and a standard deviation of 0.02 liter. g) If you pick a water bottle at random, what is the probability that it will be between 1.06 and 1.12 liters? h) If you pick a water bottle at random, what is the probability it will be between 1.05 and 1.11 liters?

What about these?  The temperatures for one month for a city in California are normally distributed with m = 81 degrees and s = 6 degrees. Find P (x > 95)

What about this one?  The scores on a standardized test are normally distributed with mean = 72 and sd = 11. Find P (x < 89)

You Try!  The heights of the 880 students at East Meck High School are normally distributed with a mean of 67 inches and a standard deviation of 2.5 inches i) What is the probability they will be more than 70 inches tall? j) What is the probability they will be less than 61 inches tall?

You Try Again!  A machine used to fill water bottle dispenses slightly different amounts into each bottle. Suppose the volume of water in 120 bottles is normally distributed with a mean of 1.1 liters and a standard deviation of 0.02 liter. i) What is the probability it will have more than 1.13 liters? j) What is the probability it will have less than 1.04 liters?