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Do Now

Do Now.

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Do Now

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  1. Do Now According to the U.S. Census Bureau, the “rough” mean and standard deviation for U.S. household income in 2015 is $50,000 USD and $20,000 USD, respectively.[1] In 1960, these values were $5,700 and $1,100, respectively.[2]* If a person made $75,000 in 2015 and another person made $7,500 in 1960, who was more affluent? Justify your answer with statistical reasoning. Assume factors such as inflation do not affect the standardized scores. *The standard deviation figure was estimated as this information was not available during that year.

  2. CHAPTER 2Modeling Distributions of Data 2.2Density Curves and Normal Distributions

  3. Density Curves and Normal Distributions • ESTIMATE the relative locations of the median and mean on a density curve. • ESTIMATE areas (proportions of values) in a Normal distribution. • FIND the proportion of z-values in a specified interval, or a z-score from a percentile in the standard Normal distribution. • FIND the proportion of values in a specified interval, or the value that corresponds to a given percentile in any Normal distribution. • DETERMINE whether a distribution of data is approximately Normal from graphical and numerical evidence.

  4. Exploring Quantitative Data In Chapter 1, we developed a kit of graphical and numerical tools for describing distributions. Now, we’ll add one more step to the strategy. CUSS! Exploring Quantitative Data • Always plot your data: make a graph, usually a dotplot, stemplot, or histogram. • Look for the overall pattern (shape, center, and spread) and for striking departures such as outliers. • Calculate a numerical summary to briefly describe center and spread. 4. Sometimes the overall pattern of a large number of observations is so regular that we can describe it by a smooth curve.

  5. Density Curves • A density curve is a curve that • is always on or above the horizontal axis, and • has area exactly 1 underneath it. • A density curve describes the overall pattern of a distribution. The area under the curve and above any interval of values on the horizontal axis is the proportion of all observations that fall in that interval. Example The overall pattern of this histogram of the scores of all 947 seventh-grade students in Gary, Indiana, on the vocabulary part of the Iowa Test of Basic Skills (ITBS) can be described by a smooth curve drawn through the tops of the bars.

  6. Describing Density Curves Our measures of center and spread apply to density curves as well as to actual sets of observations. Distinguishing the Median and Mean of a Density Curve The median of a density curve is the equal-areas point, the point that divides the area under the curve in half. The mean of a density curve is the balance point, at which the curve would balance if made of solid material. The median and the mean are the same for a symmetric density curve. They both lie at the center of the curve. The mean of a skewed curve is pulled away from the median in the direction of the long tail.

  7. Describing Density Curves • A density curve is an idealized description of a distribution of data. • We distinguish between the mean and standard deviation of the density curve and the mean and standard deviation computed from the actual observations. • The usual notation for the mean of a density curve is µ (the Greek letter mu). We write the standard deviation of a density curve as σ (the Greek letter sigma).

  8. Normal Distributions One particularly important class of density curves are the Normal curves, which describe Normal distributions. • All Normal curves have the same shape: symmetric, single-peaked, and bell-shaped • Any specific Normal curve is completely described by giving its mean µ and its standard deviation σ.

  9. Normal Distributions • Why are the Normal distributions important in statistics? • Normal distributions are good descriptions for some distributions of real data. • Normal distributions are good approximations of the results of many kinds of chance outcomes. • Many statistical inference procedures are based on Normal distributions. • A Normal distribution is described by a Normal density curve. Any particular Normal distribution is completely specified by two numbers: its mean µ and standard deviation σ. • The mean of a Normal distribution is the center of the symmetric Normal curve. • The standard deviation is the distance from the center to the change-of-curvature points on either side. • We abbreviate the Normal distribution with mean µ and standard deviation σ as N(µ,σ).

  10. Larson/Farber 4th ed Properties of Normal Distributions Normal distribution • A continuous probability distribution for a random variable, x. • The most important continuous probability distribution in statistics. • The graph of a normal distribution is called the normal curve. x

  11. Larson/Farber 4th ed Properties of Normal Distributions • The mean, median, and mode are equal. • The normal curve is bell-shaped and symmetric about the mean. • The total area under the curve is equal to one. • The normal curve approaches, but never touches the x-axis as it extends farther and farther away from the mean. Total area = 1 x μ

  12. Larson/Farber 4th ed Properties of Normal Distributions • Between μ – σ and μ + σ (in the center of the curve), the graph curves downward. The graph curves upward to the left of μ – σ and to the right of μ + σ. The points at which the curve changes from curving upward to curving downward are called the inflection points. Inflection points x μ μ+ σ μ+ 2σ μ+ 3σ μ 3σ μ 2σ μ σ

  13. Larson/Farber 4th ed Means and Standard Deviations • A normal distribution can have any mean and any positive standard deviation. • The mean gives the location of the line of symmetry. • The standard deviation describes the spread of the data. μ = 3.5 σ = 1.5 μ = 3.5 σ = 0.7 μ = 1.5 σ = 0.7

  14. Larson/Farber 4th ed Example: Understanding Mean and Standard Deviation • Which curve has the greater mean? Solution: Curve A has the greater mean (The line of symmetry of curve A occurs at x = 15. The line of symmetry of curve B occurs at x = 12.)

  15. Larson/Farber 4th ed Example: Understanding Mean and Standard Deviation • Which curve has the greater standard deviation? Solution: Curve B has the greater standard deviation (Curve B is more spread out than curve A.)

  16. Larson/Farber 4th ed Example: Interpreting Graphs The heights of fully grown white oak trees are normally distributed. The curve represents the distribution. What is the mean height of a fully grown white oak tree? Estimate the standard deviation. Solution: σ = 3.5 (The inflection points are one standard deviation away from the mean) μ = 90 (A normal curve is symmetric about the mean)

  17. Density Curves and Normal Distributions • ESTIMATE the relative locations of the median and mean on a density curve. • ESTIMATE areas (proportions of values) in a Normal distribution. • FIND the proportion of z-values in a specified interval, or a z-score from a percentile in the standard Normal distribution. • FIND the proportion of values in a specified interval, or the value that corresponds to a given percentile in any Normal distribution. • DETERMINE whether a distribution of data is approximately Normal from graphical and numerical evidence.

  18. Homework • Complete pg. 129 #42-45

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