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Probability Distributions Statistical Experiments and Random Variables Statistical Experiments – any process by which measurements are obtained. A quantitative variable x, is a random variable if its value is determined by the outcome of a random experiment. Random variables can be discrete or continuous. Discrete vs. Continuous Discrete random variables – can take on only a countable or finite number of values. (Countable) Continuous random variables – can take on countless values in an interval on the real line. (Measurable) Which measurement involves a discrete random variable? a). Determine the mass of a randomly-selected penny b). Assess customer satisfaction rated from 1 (completely satisfied) to 5 (completely dissatisfied). c). Find the rate of occurrence of a genetic disorder in a given sample of persons. d). Measure the percentage of light bulbs with lifetimes less than 400 hours. Discrete or Continuous? Gas mileage(mpg) of all 2014 Toyota Prius’ Number of songs on a seniors iPod 1 mile time of a high school junior Amount of m&ms in a bag Height of a high school freshman Probability distributions An assignment of probabilities to the specific values or a range of values for a random variable. Discrete Probability Distribution 1) Each value of the random variable has an assigned probability. 2) The sum of all the assigned probabilities must equal 1. Discrete Probability Distribution Rolling a die P(1) = .167, P(2) = .167, P(3) = .167… 0.18 0.16 0.14 1 2 3 4 5 6 0.12 0.1 0.08 0.06 0.04 0.02 0 # on the Die Discrete Probability Distribution Age of AP Statistics Students 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 15 16 17 Age of AP Statistics Students 18 19 Discrete Probability Distribution We asked 200 adults how many cars they own. The results are shown as a probability distribution below. # of cars 0 1 2 3 P(x) .08 .72 .18 .02 1) What is the probability of someone owning 2 cars? 2) What is the probability of someone owning at least 1 car? 3) What is the probability of someone owning no more than 1 car? Probability Distributions for Continuous Random Variables What would a histogram look like if we graphed the weights (to the nearest pound) of 100 newborn babies? What would it look like if we graphed the same 100 newborn babies, however rounded the weights to the nearest tenth? What about hundredth? Probability Distributions for Continuous Random Variables The probability distribution is given as a density curve. The function that defines this curve is called the density function f(x) Probability Distributions for Continuous Random Variables When the density is constant over an interval, the probability distribution is called a uniform distribution. Probability Distribution Features Since a probability distribution can be thought of as a relative-frequency distribution for a very large n, we can find the mean and the standard deviation. When viewing the distribution in terms of the population, use µ for the mean and σ for the standard deviation. Mean and Standard Deviation of Discrete Probability Distribution 𝜇= 𝜎= 𝑥𝑃 𝑥 (Expected Value) (𝑥 − 𝜇)2 𝑃(𝑥) Find the mean and standard deviation of the following… x 0 1 2 3 4 5 6 7 8 9 10 p(x) .002 .001 .002 .005 .02 .04 .17 .38 .25 .12 .01 µ = 7.16, 𝜎 = 1.57 1) What is the probability that a child receives a score within 2 standard deviations of the mean? Sometimes we are interested in the behavior of not just a random variable itself, but a function of the variable. Suppose we know the mean and standard deviation of the number of gallons of propane a random person orders in Old Lyme to be 318 and 42 respectively. A company is considering 2 different pricing models: Model 1: $3 per gallon Model 2: service charge of $50 +$2.80 per gallon Mean and Standard Deviation of Linear Functions Helps us understand the behavior of functions of random variables. Mean of y = a + bx is a + bµ𝑥 Standard deviation of y is ◦ 𝑏 𝜎𝑥 Which pricing model is better? x = # of gallons of propane 𝜎𝑥 = 42 𝜇𝑥 = 318 Model 1= $3.00x Model 2 = $50 + $2.80 per gallon We are interested in y(amount billed) Find the mean and standard deviation of both models and compare them. 𝜎1 = 126, 𝜇1 = 954 𝜎2 = 117.60, 𝜇2 = 940.40 Mean and Standard Deviation of Linear Combinations 𝜇𝑦 = 𝜇𝑥1 +𝜇𝑥2 + ⋯ 𝜎𝑦2 = 𝜎12 + 𝜎22 + ⋯ (must be independent) Mean Standard Deviation Mult. Choice 38 6 Free Response 30 7 If the free response is given twice as much weight, what are the mean and s.d of y? Binomial Experiment (2 outcomes) 1) 2) 3) There are a fixed number of trials. This is denoted by n. The n trials are independent and repeated under identical conditions. Each trial has two outcomes: S = success 4) F = failure For each trial, the probability of success, p, remains the same. Thus, the probability of failure is 1 – p = q. Binomial Probability The central problem is to determine the probability of x successes out of n trials. ◦ Mark is an 80% free throw shooter. What is the probability that he makes exactly 5 out of 10 free throws? ◦ What is the probability he makes exactly 10 out of 10? ◦ What is the probability he makes more than half his free throws? Probability of x successes 𝑃 𝑟 = 𝐶𝑛,𝑟 ∙ 𝑝𝑟 ∙ 𝑞 𝑛−𝑟 𝐶𝑛,𝑟 = 𝑛! 𝑟! 𝑛−𝑟 ! Find the probability of observing 6 successes in 10 trials if the probability of success is p = 0.4. a). 0.111 b). 0.251 c). 0.0002 d). 0.022 Using the Binomial Table 1) Locate the number of trials, n. 2) Locate the number of successes, x. 3) Follow that row to the right to the corresponding p column. Find the probability of observing 3 successes in 5 trials if p = 0.7 Finding the Probability of multiple successes, r’s. Find the probability of observing less than three 3 successes in 5 trials if p = 0.7 P(r<3) = P(0) + P(1) + P(2) Find the probability of observing less than 3 successes in 5 trials if p = 0.7 Graphing a Binomial Distribution Same as a relative frequency histogram with r values on the horizontal axis. Graph n=3, and p=.2 0.6 0.5 0.4 r=0 r=1 r=2 r=3 0.3 0.2 0.1 0 # of Successes Mean and Standard Deviation 𝜇 = 𝑛𝑝 𝜎 = 𝑛𝑝𝑞 Graph n=3, and p=.2 0.6 0.5 0.4 r=0 r=1 r=2 r=3 0.3 0.2 0.1 0 # of Successes 𝜇 = .6 𝜎 = .6928 Think about it! The graph of a distribution with a small probability of success(p) would be skew left, right, symmetrical, uniform or bimodal? How about a probability of success being 0.5? The Normal Curve (Bell Curve) Finding z scores z score gives the number of standard deviations between the x value and the mean µ. z= 𝑥−µ σ Always round to the hundredths. Finding area (probability) using z score and the standard NORMAL DISTRIBUTION Table 2 gives you the area to the left of z. This is the probability of less than z. Area to the right of z Area Between two z values Inverse Normal distribution We have been working with finding an area (probability), given a certain x or z value. We can also find an x or z value given a certain area (probability). Left-tail case: the given area is to the left of z Look up the number A in the body of the table and use the corresponding z value. Right-tail case: the given area is to the right of z Look up the number 1 – A in the body of the table and use the corresponding z value. Center-tail case: the given area is symmetric and centered above z = 0. Look up the number (1 – A)/2 in the body of the table and use the corresponding ±z value. Using Table 3 in the Appendix, find the range of z scores, centered about the mean, that contain 70% of the probability. a). –1.04 to 1.04 b). –2.17 to 2.17 c). –0.30 to 0.30 d). –0.52 to 0.52 Checking for Normality Normal Probability Plot: the scatterplot of (normal score, observed value) If the normal probability plot shows a linear trend, then the data is approximately normal. Using the Correlation Coefficient to Check Normality Correlation Coefficient is obtained using (normal score, observed value) If r is close to 1, then a linear relationship should be represented. How close to 1 is close to 1? If r is less than the critical r for the corresponding n, it is not reasonable to assume a normal distribution If n is between two sample sizes, use the larger n Approximating Discrete/Binomial Distributions using Normal Dist. Approximating Discrete/Binomial Distributions using Normal Dist. How do we tell that a binomial distribution is normal? If np ≥ 10 and nq ≤ 10, then the distribution is approximately normal. Add or subtract .5 to your x-values and compute the probability as you would a normal distribution. A Biologist found that the probability is only 0.65 that a given Arctic Tern will survive the migration from its summer nesting are to its winter feeding groups. A random sample of 500 Artic Terns were branded at their summer nesting area. Find the probability that between 310 and 340 of the branded Artic Terns will survive the migration.