U6 - DAY 2 WARM UP! Given the times required for a group of students to complete the physical fitness obstacle course result in a normal curve, and that the mean time 21 minutes and the standard deviation is 4… Use the Empirical Rule to Answer the FOLLOWING! What percent took longer than 29 minutes? What percent took less than 29 minutes? What percent took between 13 and 29 minutes? What percent took between 13 and 25 minutes? What percent took longer than 17 minutes? What time would a student need to have to be in the fastest 2.5% 2.5% 97.5% 95% 81.5% 84% < 13 minutes

Normal Distribution Probabilities with our Calculators!!! If you are looking for values that are not exactly on the standard deviation lines, there are two ways to find the probability – you can use a normal distribution chart OR you can use a calculator. For this course we will ONLY use the calculator!!

Normal Distribution Probability Example #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? DRAW A PICTURE!!!!!

Normal Distribution Probability TI 83/84 directions: Normal Distribution Probability Example #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 is the probability that a student has a score between 82 and 90? Press [2nd][VARS](DISTR) [2] (normalcdf) b. Press [82] [,] [90] [,] [78] [,] [5] [)][Enter] normalcdf(x1, x2, μ, σ) normalcdf(82,90,78,5) = .2036578048 There is a 20.37% probability that a student scored between 82 and 90 on the Calculus exam.

Normal Distribution Probability 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. How many students have scores between 82 and 90? Example #1: Using the probability previously found: 500 * .2037 = 101.85 There are about 102 students who scored between 82 and 90 on the Calculus exam.

Normal Distribution Probability Ex#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 above 60? Hint: Use 1E99 for upper limit (2nd/EE) normalcdf(60,1E99,78,5) = .9998408543 ~ 99.98%

Normal Distribution Probability A PreCalculus exam is given to 500 students. The scores have a normal distribution with a mean of 78 and a standard deviation of 5. How many students have scores above 70? Example #2 500*.9452= 472.6 About 473 students have a score above 70 on the Pre-Calculus exam. 2nd/EE Normalcdf(70,1E99,78,5) = .9452007106 TI 84

Normal Distribution Probability Example #3: Find the probability of scoring below a 1400 on the SAT if the scores are normally distributed with a mean of 1500 and a standard deviation of 200. Hint: Use -1E99 for lower limit

Normal Distribution Probability 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. Example #3 There is a 30.85% probability that a student will score below a 1400 on the SAT. normalcdf(-1E99,1400,1500,200) = .3085375322

Example #4 Kenny and his English Test! Scores of each of the previous English tests were normally distributed with a mean of 76 and standard deviation of 4.5. Kenny will be taking the test tomorrow. What is the probability of Kenny getting at least 70 on the test? (He wants to pass!) What is the probability that he makes between a 93 and a 100? (He really wants and A!) What is the probability that he makes less than a 50? (Oh no…his parents are going to be angry!!) ~ 90.9% ~ .008% ~ .0000004%

Now let’s think Backwards!! Example #5 Scores of each of the previous calculus tests were normally distributed with a mean of 86 and standard deviation of 2.2. How high must a student’s score be to score in the top 20%? Draw a picture…it will help! 2nd /vars/ invNorm (area to the left, μ, σ) invNorm(0.8, 86, 2.2) = ~ 87.9 or 88

PERCENTILES (Use invNorm with percentiles) A percentile is the score at which a specified percentage of scores in a distribution fall below. To say a score of 53 is in the 80th percentile is to say that 80% of all scores are less than 53

More Backwards Questions… Ex #6 Heights are generally found to be normally distributed. Assume that the average adult female height is 65.5 inches with a standard deviation of 3 inches. 2nd /vars/ invNorm (area to the left, μ, σ) What height do 90% of women fall below? What height do 60% of women measure above? What height is the 75th percentile of women? What two heights, symmetric about the mean, contain 80% of all women's heights? 69.3 in. 64.7 in. 67.5 in. 61.7 in. and 69.3 in.

HOMEWORK! Day 2 HW