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Review A random variable where X can take on a range of values, not just particular ones. Examples: Heights Distance a golfer hits the ball with their driver Time to run 100 meters Electricity usage of a home.
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Review There are two types of continuous distributions we discuss now: uniform and normal distributions. A density curve is the graph of a continuous probability distribution. 1) This curve always positive (or 0) 2) The area under the curve is 1.
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Review For a density curve depicting the probability distribution of a continuous random variable, –the total area under the curve is 1, –there is a direct correspondence between area and probability. –Only the probability of an event occurring in some interval can be evaluated. –The probability that a continuous random variable takes on any particular value is zero.
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Example Find the probability x is at most 5 and at least 2. x P(x) 0 2 4 6 8 10 0.1 0
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Example Find the probability x is at most 5 and at least 2. x P(x) 0 2 4 6 8 10 0.1 0
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Example Find the probability x is at most 5 and at least 2. x P(x) 0 2 4 6 8 10 0.1 0
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Example Find the probability x is at most 5 and at least 2. x P(x) 0 2 4 6 8 10 0.1 0
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Normal Distributions This is the most common observed distribution of continuous random variables. A normal distribution corresponds to bell- shaped curves.
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Normal Distributions Shape of this curve is determined by µ and σ –µ it’s centered, σ is how far it’s spread out.
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Standard Normal Distribution The Standard Normal Distribution is a normal probability distribution that has a mean of 0 and a standard deviation of 1. In this way the formula giving the heights of the normal curve is simplified greatly.
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Z-score We represent a standard normal variable with a z instead of an x. Convert any normal distribution to a standard normal distribution by using the z-score.
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Standard Normal Probabilities P(0 z 1) = 0.3413 This can found in a table in the back of the text (Table IV). The table only gives the areas under the curve to the right between 0 and z. To find other intervals requires some tricks
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Finding Probabilities when given z-scores. For a given z-score, the probability can be found in a table in the back of the text (Table IV), also see inside front cover. Note: The table only gives the areas under the curve to the right between 0 and z. To find other intervals requires some tricks.
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Examples Use the tables in the back of the book to find the following. a) P(0 z 2.43) = 0.4925 b) P(-2.43 z 0) = 0.4925 c) P(1.20 z 2.30) =0.4893 - 0.3849=0.1044 d) P(-1.50 z 2.4) = 0.4918 + 0.4332 = 0.925 e) P( z 1.8) = 0.4641 + 0.5 = 0.9641
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Problems Problems 5.3, 5.4, 5.12 Problems 5.22, 5.26, 5.28, 5.30, 5.36, 5.40, 5.48
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Keys to success Learn the standard normal table and how to use it. We will be using these tables through out the course.
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5.4 How do you know when a data set is normal? 3 methods
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5.4 How do you know when a data set is normal? Method 1: A data set is approximately normal if it is symmetric and bell-shaped.
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5.4 How do you know when a data set is normal? Method 2: A set of data is approximately normal if the data set satisfies the empirical rule: –Within 1 sd: 68% of the data. –Within 2 sd: 95% of the data. –Within 3 sd: 99.7% of the data.
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5.4 How do you know when a data set is normal? Method 3: Find IQR and standard deviation. If the data is approximately normal, then
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Example You are given a data set and determine that: 1.IQR=0.44 and 2.s=0.33 Would you suspect this data is normally distributed?
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Example You are given a data set and determine that: 1.IQR=0.44 and 2.s=0.33 Would you suspect this data is normally distributed?
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Example You are given a data set and determine that: 1.IQR=0.44 and 2.s=0.33 Would you suspect this data is normally distributed? YES
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Problems Problems 5.3, 5.4, 5.12 Problems 5.22, 5.26, 5.28, 5.30, 5.36, 5.40, 5.48 Problem 5.54
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25 Homework Review Chapter 5.1-5.4 Read Chapters 6.1-6.3 for next week Midterm on Thursday, –7:00-8:30 in PS 1072 –Covers chapters 1-4 Quiz during class on Tuesday Next class - optional tutorial
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