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Section 3.2, 3.3, 3.4 MATH 166:503 March 3, 2015 Topics from last week: probability using counting techniques, Bernoulli trial, repeated Bernoulli trial, probability of repeated Bernoulli trials, random variable, finite discrete random variables, infinite discrete random variables, continuous random variables, constructing histograms from probability distributions, using histograms to calculate probability, binomial distribution. 3 PROBABILITY DISTRIBUTIONS AND STATISTICS 3.2 Measures of Central Tendency ex. If your homework grades are 100, 86, and 72. What is your homework average? Notes: 1 ex. A cereal company weighs randomly selected 12 oz boxes of cereal A to check for consistency. Below is their data from 100 tests Actual weight (oz) Frequency 11.98 21 11.99 16 12 37 12.01 17 12.02 9 What is the average weight of a 12 oz box of cereal A among those tested? ex. If 100 randomly selected 12 oz boxes of cereal B were also tested with weights Actual weight (oz) Frequency 11.98 6 11.99 15 12 16 12.01 43 12.02 20 What is the average weight of a 12 oz box of cereal B among those tested? ex. If 12 oz boxes of cereal A costs $3.45 and 12 oz boxes of cereal B costs $3.46, which cereal is a better value? Notes: 2 ex. You are playing a game with a friend in which two 6-sided dice are rolled and the top faces are observed. If the sum of the numbers on the dice is a multiple of 3, you pay your friend $5. How much should he pay you when it’s not a multiple of 3 in order to make the game fair? ex. Binomial distribution for n = 15, p = 0.5. Binomial distribution for n = 15, p = 0.8. Binomial distribution for n = 20, p = 0.5. Binomial distribution for n = 20, p = 0.8. Notes: 3 ex. If your test scores were 100, 95, 100, and 0, what is your average test score? What is the median? What is the mode? ex. A cereal company weighs randomly selected 12 oz boxes of cereal A to check for consistency. Below is their data from 100 tests Actual weight (oz) Frequency 11.98 6 11.99 15 What is the median and mode? Notes: 4 12 16 12.01 43 12.02 20 3.3 Measures of Spread ex. Calculate the mean of the random variable corresponding to the following histograms. ex. Mary and Beth participated in a golf tournament. The following are their scores for 4 rounds of the tournament. What is Mary’s average (or mean) score? Mary Beth 70 67 66 71 72 67 70 68 What is Mary’s variance? Her standard deviation? What is Beth’s mean, variance, and standard deviation? Notes: 5 ex. A hardworking grocery store employee decided to check 4-packs of eggs for broken eggs. During one shift, he observed Number broken Frequency 0 12 1 8 2 2 3 4 4 0 ex. Calculate the variance and standard deviation of the random variable corresponding to the following histograms. Notes: 6 ex. Given µ = 2.5, σ = .75, estimate the following a. P (1.75 ≤ X ≤ 3.25) b. P (1 ≤ X ≤ 4) ex. The average American spends 5 hours a day watching television. If we randomly select 15 Americans to be called at a randomly selected time during the day, what is the probability that at least 7 but fewer than 10 of the people polled are watching television when they were called? 3.4 The Normal Distribution ex. If the spinner below has an infinitely fine tip, what is the probabilty that, after flicking the spinner, the spinner will land on 21 ? What is P ( 12 ≤ X ≤ 43 )? Notes: 7 ex. How do we compute P (Z ≤ 2)? Or P (Z ≥ 1)? Or P (−1 ≤ Z ≤ 2)? ex. Compute a. P (Z ≤ −2.31) b. P (Z ≥ .76) c. P (−1 ≤ Z ≤ 1) d. P (−2 ≤ Z ≤ 2) e. P (−3 ≤ Z ≤ 3) Notes: 8 ex. A manufacturer of washing machines found that the time before one of its washers needs its first repair is normally distributed with mean 2 years and standard deviation .608 years. What is the prbability that the machine will need repairs within 4 years? What should the warranty period be so that at least 95% make it through the period without needing repair? Notes: 9