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Probability and Statistics LECTURE 5 CONTINUOUS PROBABILITY DISTRIBUTIONS AND SAMPLING METHODS Outline 1. Probability density function 2. Normal distribution 3. Sampling methods Adapted from http://www.prenhall.com/mcclave 3-1 3-2 Continuous random variable revisited Questions: What is a continuous random variable? Give some examples. Can we represent probability distributions for continuous random variables the same way as for discrete random variables? 3-3 Probability Density Function 1. f(x) 0 for all values of x 2. The area under the probability density function f(x) over all values of the random variable X within its range, is equal to 1.0 3-4 Probability density function f(x) x Measuring probability for continuous random variables d Probability Is Area Under Curve! P(c x d ) c f ( x ) dx f(x) c 3-5 d • • © 1984-1994 T/Maker Co. Specific distribution having a characteristic bell-shaped form. The most important continuous probability distribution. Why normal distributions are important? 3-7 Suppose X is a continuous random variable, does it make any difference if we write: P(a X b) or P(a X b) or P(a X b) or P(a X b)? X Normal Distribution • Small discussion 3-6 An example (hypothetical) Suppose we gather data about mathscores of all10th graders in country A. Plot a histogram: see next slide. Histogram and smooth curve. 3-8 Histogram 3-9 3 - 10 Normal Distribution • • • • Normal probability density function Many normal distributions Bell-shaped and symmetric Mean = median = mode A normal distribution is completely determined by µ and Notation: 3 - 11 3 - 12 Standardize the Normal Distribution Finding probabilities of normal random variables • • • Probability of a single value Probability of X within an interval What if we try to table the normal distribution probabilities? 3 - 13 Normal Distribution Obtaining the Probability Table 1 in your textbook P(X < 13.3) = P(Z < 0.83) 3 - 15 Notation: Z ~ N(0,1) 3 - 14 Standardizing Example Suppose X is normally distributed with mean = 5, and standard deviation = 10, P(X < 13.3) = ? First convert to Z: Standard Normal Distribution 3 - 16 Solution P(3.8 X 5) Exercise: P(3.8 X 5)=? Normal Distribution Normal Distribution 3 - 17 3 - 18 • 3 - 19 P(Z < 0) – P(Z < -0.12) Shaded area exaggerated Solution P(X 8) Exercise: P(X 8)=? Normal Distribution P(3.8 X 5) = P(-0.12 Z 0) = We have P(X 8) = P(Z 0.30) = 1 P(Z < 0.30) = 1- 0.6179 = 0.3821 3 - 20 Self-study: Finding Z Values for Known Probabilities What is z* given P(Z < z*) = .7704? Self-study: Finding X Values for Known Probabilities Example: part b of the exercise in the next slide What is z* given P(Z > z*) = 0.1935 3 - 21 3 - 22 Solution Problem You work in Quality Control for GE. Light bulb lifetime has a normal distribution with = 2000 hours & = 200 hours. a. What’s the probability that a randomly selected bulb will last between 2000 & 2400 hours? b. (Self-Study) Find the lifetime range of the 5% most durable light bulbs. 3 - 23 3 - 24 Sampling Sampling with/without replacement Some terms: • • Census: Sampling: selecting a sample from population Sampling with replacement: Why sampling is necessary? Cost Practicality Sampling without replacement: Statistical inference will allow us to draw conclusions for population based on sample data 3 - 25 3 - 26 Sampling methods Probability sampling methods Probability sample: a sample selected in such a way that each item or person in the population has a known (nonzero) likelihood of being included in the sample. E.g. Simple random sampling, systematic random sampling, and stratified random sampling. 3 - 27 Sampling methods Nonprobability sampling methods Not all items or people have a chance of being included in the sample. E.g. online survey, surveying customers by going around a shopping mall and asking customers to answer a questionnaire. Results may be biased. 3 - 28 Some Important Probability Sampling Methods Let’s now discuss two of the most important probability sampling methods: simple random sampling and stratified random sampling. 3 - 29 Simple random sampling Example (cont.): we first have to obtain a list of population members. Number the list from 1 to N (the population size). Then, use a computer software to select a random sample. This type of sampling will be demonstrated in R labs. Simple random sampling Selecting a sample in such a way that every possible sample of the same size is equally likely to be chosen. Example: Suppose our lecture class is the population. We are going to select a simple random sample (SRS) of 20 students from the population. 3 - 30 Stratified random sampling To select a stratified random sample, first divide the population into mutually exclusive groups of similar individuals called strata. Then choose a separate SRS from each stratum and combine these SRSs to form the full sample. Example: 3 - 31 3 - 32 Other types of sampling Please refer to your textbook for other type of sampling methods 3 - 33 One student volunteers to summarize the main points of the lecture. 3 - 34 Conclusion 1. Probability density function 2. Normal distribution 3. Sampling methods 3 - 35 Lecture summary