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Chapter 2

Chapter 2 . Density Curves and Normal Distributions. Remember. ALWAYS plot data (usually histogram or stemplot ) Overall Pattern (CUSS) Calculate Numerical Summary NEW – Sometimes overall pattern is so regular it can be described by a smooth curve (called a density curve) fig 2.2 pg78.

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Chapter 2

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  1. Chapter 2 Density Curves and Normal Distributions

  2. Remember • ALWAYS plot data (usually histogram or stemplot) • Overall Pattern (CUSS) • Calculate Numerical Summary • NEW – Sometimes overall pattern is so regular it can be described by a smooth curve (called a density curve) fig 2.2 pg78

  3. Density Curve • Mathematical Model – an idealized description • Always on or above horizontal axis • Has area of exactly 1 underneath it • Areas under the curve represent proportions of the observations

  4. Mean & Median of Density Curves • Median (pg 81) – the point with half the observations on either side or it is the equal areas point. • Mean (pg 82) – point where the curve would balance if made of solid material • Mean of density curve µ ‘mu’ • Standard deviation σ ‘sigma’

  5. The are under the curve is a rectangle with height 1 and width 1. A uniform Distribution Exercise 2.2 20% , the region is a rectangle with height of 1 and base width of 0.2 • Why is the total area under this curve equal to 1? • What percent of the observations lie above 0.8? • What percent of the observations lie below 0.6? • What percent of the observations lie between 0.25 and 0.75? • What is the mean µ of this distribution? 60% 50% Mean = ½ or 0.5, the ‘balance point’ of the density curve

  6. Quartiles • Divides the area under the curve into quarters • ¼ of the area is left of Q1 • ¾ of the area is left of Q3

  7. Symmetric Curve • Median and mean are equal (2.5a, pg 81) • Skewed Curve • Mean pulled toward the long tail of a skewed distribution

  8. HW Monday • 1, 3, 4 • QUIZ BLOCK DAY first thing then 2.2 • This quiz will go in the grade book so be sure to look over your notes and examples in the book.

  9. Normal Curves • Symmetric, single peaked, and bell shaped; describe normal distributions

  10. Normal Distributions • (all have the SAME overall shape) they are described by giving the mean (µ) and the standard deviation (σ) • Notation is N(µ,σ)

  11. Things to Know • Changing µ without changing σ moves the normal curve along the horizontal axis without changing spread. • Standard deviation controls the spread of a normal curve (fig 2.10) • Inflection points are points located at a distance of σ on either side of µ. • Points where there is a change of curvature. • In general µ and σ do not alone specify shape of most distributions, these are special properties of normal curves.

  12. Importance of Normal Distributions • Good descriptions for some distributions of real data • EX – SAT, IQ Tests, Psychological tests, characteristics of biological populations (lengths of cockroaches, yields of corn) • Good approximations to the results of many kinds of chance outcomes • EX – Tossing a coin • Many statistical inference procedures (ch 6-12) based on normal distributions work well with other roughly symmetric distributions.

  13. The Empirical Rule The 68 – 95 – 99.7 Rule

  14. Lets Draw a normal curve with notationN(64.5, 2.5) Find percentiles for the following heights 64.5 inches 59.5 inches 67 inches 72 inches

  15. QUIZ BLOCK DAY first thing, then 2.2 This quiz will go in the grade book so be sure to look over your notes and examples in the book.

  16. 2.2 Standardizing and z-scores • All normal distributions are the same if we measure in units of size σ about the mean µ as center • Changing to these units is called STANDARDIZING

  17. If x is an observation from a distribution that has mean µ and standard deviation σ, the standardized value of x is • z = x - µ σ Standardized value is often called the z-score

  18. Z-score tells you how many standard deviations the original observation (x) falls away from the mean (µ) and in which direction • Lets look back at N(64.5, 2.5) average adult women • Pg. 98 Example 2.7

  19. 54.5 57 59.5 62 64.5 67 69.5 72 74.5 Z = = = 1.4 What proportion of all young women are less that 68 inches tall?

  20. How to use table A to find the proportions of observations from the standard normal distribution that are greater than a specific z-score: z = -2.15 Find -2.15 on the left hand column and .05 in the top row…. The table entry is 0.0158. This means that the area to the left of -2.15 is 0.0158. Since the total area under the curve is equal to 1, the area lying to the right of -2.15 is: 1 - 0.0158 = 0.9842 Figure 2.16 on page 97 illustrates these areas.

  21. Pg. 102 Example 2.10 How to find a value when given the proportion, and . You will need to know how to solve the problem both ways!!!!

  22. Finding Normal Proportions • State problem, draw pic, shade area of interest • Standardize x, draw new pic • Find area under curve with Table A • Write conclusion in CONTEXT of problem Example 2.8 on Pg. 99

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