Chapter 03 Notes - The Normal Distribution (Online AU 15) - STAT 1450 COURSE NOTES CHAPTER 3 THE NORMAL DISTRIBUTIONS Guided Notes associated with the

Chapter 03 Notes - The Normal Distribution (Online AU 15) -...

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S TAT 1450 C OURSE N OTES – C HAPTER 3 T HE N ORMAL D ISTRIBUTIONS Guided Notes associated with the Lecture Videos for Sections 3.1 & 3.2 Connecting Chapter 3 to our Current Knowledge of Statistics Quick Review: Chapters 1 and 2, we explored distributions for quantitative variables by Chp. 1 - making graphical displays of data. Chp. 1 & 2 - looking for overall patterns and 4 key features Center, spread, shape, and outliers Chps. 2 - computing numerical summaries Describing center, spread, & location Now we are going to extend this by: 1. Approximating the overall pattern of a large sample size with a smooth curve . 2. Using this smooth curve to determine probabilities of events. 3.1 Density Curves A density curves describes the overall pattern of a distribution. The area under the curve for a given range of values along the x-axis is the proportion of the population that falls in that range. The curve has an area of exactly 1 underneath it. Example: Despite any rare arctic blasts, January hi temperatures in Columbus tend to be uniformly distributed between 35 and 40 degrees. a. What would you expect the January hi temperature to be? Taking the average of the endpoints yields 37.5 degrees. Chapter 3, page 1
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b. What proportion of the time is the January hi temperature below 38 degrees? Area= (base)(height) P(X<38)=.60 = (38-35)*(1/5) =3*.20 P(X<38)=.60 As a side note, P(X>38)= .40 3.2 Describing Density Curves The concepts of shape, center, and spread/variability are still used to describe a density curve. Outliers will be less relevant here, as we’ll focus on symmetric distributions. The median of a density curve is the point that divides the area under the curve in half—it is the equal areas point. In Chapter 2, we learned that the mean of a density curve is the balance point. Example: How do the mean and median compare in symmetric and skewed density curves? Density Curve A Density Curve B In a symmetric density curve, the mean and median are the same. We say there is no skew. In a skewed density curve, the mean and median are not the same. This is skewed right, and the mean is pulled to the right. Lower-case greek letters are used to describe density curves. The mean is denoted by μ . The standard deviation is denoted by σ . Chapter 3, page 2
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Notes taken while Completing the Checkpoint for Sections 3.1&3.2 Guided Notes associated with the Lecture Video for Section 3.3 3.3 Normal Distributions Normal distributions are a family of specific density curves that are also good approximations for the results of many kinds of chance outcomes . The statistical inference procedures we will use later in the course can be approximated by Normal distributions. The Normal distribution will be the premier distribution used in this course. Here are some facts about it. All Normal curves have the same overall shape: symmetric, single-peaked, bell-shaped Any specific Normal curve is completely described by giving its mean μ and its standard deviation σ . We use the notation N ( μ , σ ) to abbreviate Changing the mean without changing σ
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