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Probability

Probability. Normal Probability Distributions Any density curve can be used to assign probabilities Normal distribution – probability model Heights of all young women follow a normal distribution with μ of 65.5 inches and σ if 2.5 inches. This is a distribution for a large set of data.

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Probability

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  1. Probability Normal Probability Distributions • Any density curve can be used to assign probabilities • Normal distribution – probability model • Heights of all young women follow a normal distribution with μ of 65.5 inches and σ if 2.5 inches. • This is a distribution for a large set of data. • If you choose any woman C at random, and repeat the randomization many times, the distribution of values of C follow normal distribution.

  2. Probability • Intervals of outcomes • P(0.4 ≤ X ≤ 0.8) =_____________ • P(X ≤ 0.5) = _________________ • P(X > 0.8) = _________________ • P(X ≤ 0.5 or X > 0.8) =_________ 1 A = 1 0 1

  3. Intervals of Outcomes Example 2: • Generate two random numbers between 0 and 1 and take Y to be their sum. • What is the sample space of Y?______ • The density curve of Y can be described as follows: • A: Verify that the area under the curve is 1 • B: What is the probability that Y is less than 1? • C: What is the probability that Y is less than 0.5? 1 0 1 2

  4. Normal Probability Distributions Example: • What is the probability that a randomly chosen young woman had height between 68 and 70 inches? Distribution of Height =N=(64.5, 2.5) Remember – Z table finds probabilities of standardized data.

  5. Probability Example • The random variable X has the standard normal N(0,1) distribution. Find each of the following probabilities: A: P(-1 ≤ X ≤ 1) B: P(1 ≤ X ≤ 2) C: P(0 ≤ X ≤ 2)

  6. SAMPLING DISTRIBUTIONS Statistical estimation and the law of large numbers •  is often an unknown value that we try to estimate using a statistic – Example: Study odor thresholds of sulfur compounds present in wine. Threshold levels vary from person to person. What is the mean threshold of the human population? Computeto estimate  by choosing a random sample of 10 people and measuring their threshold 28, 40, 28, 33, 20, 31, 29, 27, 17, 21 • Calculate = 27.4 to estimate  • How close is to ?

  7. SAMPLING DISTRIBUTIONS Statistical estimation and the law of large numbers • is guaranteed to approach the value of  as sample size approaches the population size. Law of large numbers • If we keep drawing samples of more and more people eventually we will estimate the true mean population odor threshold very accurately. • Draw observations at random from any population with finite mean . As the number of observations drawn increases, the mean of the observed values gets closer and closer to the mean  of the population.

  8. POPULATION DISTRIBUTION Distribution of odor tresholds What can you say about distribution, shape, center and spread Of this distribution?

  9. SAMPLING DISTRIBUTIONS • The law of large numbers assures us that if we measure enough subjects, the statistic will eventually get very close to the unknown parameter  • What happens when we take many samples of odor threshold measurements from the population?

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