3/30/2009. Probability Distributions. Binomial distribution. TI-83 Binomial Probability

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1 Random variable The outcome of each procedure is determined by chance. Probability Distributions Normal Probability Distribution N Chapter 6 Discrete Random variables takes on a countable number of values (i.e. there are gaps between values). SPECIAL Discrete Random variables Binomial distribution (Sections 5.3, 5.4) Geometric distribution Hypergeometric distribution Poisson distribution (Section 5.5) Continuous Random variables there are an infinite number of values the random variable can take, and they are densely packed together (i.e. there are no gaps between values) SPECIAL Continuous Random variables Normal distribution Eponential distribution Uniform distribution 2 Binomial distribution Fied number of trials There are only two possible outcomes: success or failure The trials are independent The probabilities of success and failure are the remain the same Eample: recording the genders of children in 250 families. µ = np The mean is The standard deviation is = np( 1 p) = npq TI-83 Binomial Probability Press 2nd VARS. Select the option 0:binompdf(. Complete the entry to obtain binompdf(n, p, ), with the appropriate values substituted in. Eample: What is the probability of getting eactly 2 heads when 4 tosses are made? Solution: Using the TI-83 with binompdf(4, 0.5, 2), it follows that the probability for getting 2 heads on 4 throws is Poisson distribution The random variable is the number of occurrences of some events over an interval. Used for describing the behavior of rare events Number of industrial accidents per month in a manufacturing plant. Number of people arriving at a checkout in a day Number of eagles nesting in a region Number of patients arriving at an emergency room The occurrences must be random and independent of each other, and uniformly distributed over the interval. The mean is µ, and the standard deviation is = n Continuous Random Variables Continuous sample spaces contain an infinite number of events. They typically are intervals of possible, continuously-distributed outcomes. E.: Select ANY number between 0 and 1. What is the sample space? S = { all numbers between 0 and 1} E.: Drink ANY volume of water from a 32-ounce bottle. What is the sample space? S = { 0 32 ounce} 5 6 1

2 Continuous Random Variables Special Continuous Probability Distributions A continuous probability distribution function for a random variable X is a continuous function with the property that the area below the graph of the function between any two points a and b equals the probability that a X b. Remember, AREA = PROPORTION = PROBABILITY Uniform distribution Eponential distribution Normal distribution 7 8 Uniform Distribution Eponential Distribution 1. Equally Likely Outcomes 2. Probability Density 1 b a f() 1. Describes Time or Distance Between Events 2. Density Function f(x) λ = 2.0 λ = 0.5 f ( ) = 1 b a 3. Mean & Standard Deviation a+ b b a µ = = 2 12 a Mean Median b f ( ) = 3. Parameters e λ λ 1 1 µ =, = λ λ X Normal Distribution A and B have the same center, but different standard deviations (shape). A and C have the same standard deviations (shape), but different means (shifted). f(x) A B 2 1 ( µ ) f ( ) = ep 2 2π 2 C X Eamples of normal random variables testosterone level of male students head circumference of adult females length of middle finger of Math 225 students test scores in Math 225 height of all kindergarten kids at a school

3 13 Density Bell-shaped curve Mean = 70 SD = 5 Mean = 70 SD = Grades 14 Characteristics of normal distribution Symmetric, bell-shaped curve. Shape of curve depends on population mean µ and standard deviation. Center of distribution is µ. Spread is determined by. Most values fall around the mean, but some values are smaller and some are larger. STANDARD NORMAL DISTRIBUTION: Mean: µ = 0 Standard deviation: =1 Probabilities for Normal Distributions Infinite Number of Tables Probability is area under curve! f() d = c P( c d) f ( ) d? Normal distributions differ by mean & standard deviation. f(x) Each distribution would require its own table. c d X 15 Standardize the Normal Distribution Normal Distribution µ X µ Z = X Standardized Normal Distribution µ = 0 = 1 One table! Z 18 To find probability follow these steps: Draw the normal distribution and shade the area of interest Find the standardized score (z-score) for the given. µ z = Find the probability using the z-table or calculator 3

4 Density TI-83, 84: DISTR 2:normalcdf( upper-tail: normalcdf(z,9999) Probability student scores higher than 75? Grades P(X > 75) lower-tail: normalcdf(-9999,z) Density P(X < 65) Grades Density Between part: normalcdf(z1,z2) Grades P(65 < X < 70) To find from given area follow these steps Draw and shade Find the LOWER tail probability INSIDE the table, and read off the corresponding z- score. OR: use DISTR 3:invNorm( To find use the formula: = z + µ Parameter versus statistic Population: the entire group of individuals in which we are interested but can t usually assess directly. A parameter is a number describing a characteristic of the population. Parameters are usually unknown. Sample: the part of the population we actually eamine and for which we do have data. A statistic is a number describing a characteristic of a sample. We often use a statistic to estimate an unknown population parameter. Eample The Environmental Protection Agency took soil samples at 20 locations near a former industrial waste dump and checked each for evidence of toic chemicals. They found no elevated levels of any harmful substances. Population: ALL the soil near the waste dump Sample: the 20 soil samples Parameter: mean level of toic chemicals in the ground around the waste dump Statistic: the mean level of toic chemicals in the 20 soil samples Notation Variable of interest: Categorical Then we are interested in PROPORTION Notation: Population parameter: p Sample statistic : $p Variable of interest: Quantitative Then we are interested in MEAN Notation: Population parameter: µ Sample statistic: Sampling Variability When we take many samples, the statistics from the samples are usually different from the population figures, and also different from what we got in the first sample. This very intuitive idea, that sample results change from sample to sample, is called sampling variability

5 Comments 1. Parameters are usually unknown, because it is impractical or impossible to know eactly what values a variable takes for every member of the population. 2. Statistics are computed from the sample, and vary from sample to sample due to sampling variability. Sampling Distributions The sampling distribution is a distribution of a sample statistic in infinite number of samples Sampling distribution of the sample mean, Sampling distribution of Histogram of some sample averages OK, we have the sampling distribution of the sample means. Then what? Sampling distributions, like data distributions, are best described by shape, center, and spread Shape, Center, and Spread Shape: Many, but not all, sampling distributions are approimately normal. Center: The mean will be denoted by µ with a subscript to indicate which sampling distribution is being discussed. For eample, the mean of the sampling distribution of the mean is represented by the symbol µ X. (The mean of the sample means.) Spread: the standard deviation of the sampling distribution of the sample means and is X Mean and standard error of the sampling distribution of the sample means Suppose that is the mean of an SRS of size n drawn from a large population with mean μ and standard deviation. Then the sampling distribution of has mean and µ = µ = standard deviation n

6 For any population with mean µ and standard deviation : The mean, or center of the sampling distribution of, is equal to the population mean µ. The standard deviation of the sampling distribution is / n, where n is the sample size. Sampling distribution of / n Mean of a sampling distribution of There is no tendency for a sample mean to fall systematically above or below µ, even if the distribution of the raw data is skewed. Thus, the mean of the sampling distribution of is an unbiased estimator of the population mean μ it will be correct on average in many samples. 31 µ 32 Standard error of a sampling distribution of The standard deviation of the sampling distribution measures how much the sample statistic varies from sample to sample. It is smaller than the standard deviation of the population by a factor of n. Averages are less variable than individual observations. Generating Sampling Distributions 1. Take a random sample of a fied size n from a population. 2. Compute the summary statistics (mean, proportion). 3. Repeat steps 1 and 2 many times. 4. Display the distribution of the summary statistics Eample Etensive studies have found that the DMS odor threshold of adults follows a roughly normal distribution with mean µ =25 micrograms per liter and standard deviation =7 micrograms per liter. With this information, we can simulate many runs of our study with different subjects drawn at random from the population. We take 1000 samples of size 10, find the 1000 sample mean thresholds, and make a histogram of these 1000 values. The results from the 1000 samples 1 st SRS of size 10: 2 nd SRS of size 10: 3 rd SRS of size 10: M 1000 th SRS of size 10: = 36, s = 32. = 22. 8, s = 2. 7 = 30. 4, s = 41. = 28. 9, s =

7 The sampling distribution of the statistic. Frequency C1 Spread: the standard error of the 1000 smaller than the standard deviation = Shape: looks normal. Center: the mean of the 1000 s is µ = The distribution is centered very close to the population mean µ = 25 s is 2.191, notably of the population. For normally distributed populations When a variable in a population is normally distributed, then the sampling distribution of for all possible samples of size n is also normally distributed. If the population is N(µ,), then the sample means distribution is N(µ,/ n ). Sample means Population IQ scores: population vs. sample In a large population of adults, the mean IQ is 112 with standard deviation 16. Suppose 100 adults are randomly selected for a market research campaign. The distribution of the sample mean IQ is A) eactly normal, mean 112, standard deviation 16. B) approimately normal, mean 112, standard deviation 16. C) approimately normal, mean 112, standard deviation 1.6. D) approimately normal, mean 112, standard deviation 4. μ n Application Hypokalemia is diagnosed when blood potassium levels are low, below 3.5mEq/dl. Let s assume that we know a patient whose measured potassium levels vary daily according to a normal distribution N(µ = 3.8, = 0.2). If only one measurement is made, what's the probability that this patient will be misdiagnosed hypokalemic? ( µ ) = = 0.2 z z = 1.5, P(z < 1.5) = % 39 C) approimately normal, mean 112, standard deviation 1.6. Population distribution: N (µ = 112; = 16) Sampling distribution for n = 200 is N (µ = 112; / n = 1.6) If instead measurements are taken on four separate days, what is the probability of such a misdiagnosis? Note: ( µ ) z = = n z = 3, P(z < 1.5) = % Make sure to standardize (z) using the standard deviation for the sampling distribution. 40 But Not all variables are normally distributed. Income is typically strongly skewed for eample. Is still a good estimator of µ then? The Central Limit Theorem will rescue us! The Central Limit Theorem VERY IMPORTANT!!! When randomly sampling from any population with mean µ and standard deviation, when n is large enough, the sampling distribution of approimately normal: N(µ, / n). is

8 Central Limit Theorem The Central Limit Theorem guarantees that a distribution of sample mean to be approimately normal as long as the sample size is large enough. We will depend on the Central Limit Theorem again and again in order to take advantage of normal probability calculations when we use sample mean to draw conclusions about population mean, even if the population distribution is not normal Comments There is no requirement on the shape of the population distribution. This is where the strength of the Central Limit Theorem lies. It tells us that regardless of the shape of the population distribution, averages that are based on a large enough sample will have a normal distribution. The central limit theorem Population with strongly skewed distribution 46 Sampling distribution of for n = 10 observations Sampling distribution of for n = 2 observations Sampling distribution of for n = 25 observations Assessing Normality A normal probability plot is a graph with the original set of data on the -ais, and the corresponding z scores for each data value on the y- ais. If the points appear to lie reasonably close to a straight line and there does not appear to be a systematic pattern that is not a straight line, we can conclude that the data came from a normally distributed population. Normal distribution v Short-tailed distribution right-skewed distribution Long-tailed distribution left-skewed distribution