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## Density Curves

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**Density Curves**Section 2.1**Strategy to explore data on a single variable**• Plot the data (histogram or stemplot) • CUSS • Calculate numerical summary to describe center and spread • Mean and standard deviation • Five number summary**From Histogram to Density Curve**Sometimes the overall pattern of a histogram can be described by a smooth curve.**Density Curve**• Density curve is a mathematical model for the distribution • Idealized description • Scale is adjusted so area under the curve is equal to 1**Density Curve**• A density curve is a curve that • Is always on or above the horizontal axis • Has area exactly 1 underneath it Describes the overall pattern of a distribution The area under the curve and above any range of values is the proportion of observations that fall in that range.**Uniform distribution**• If the total area under the curve is 1, what is the height of the square? • 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 0.40?**1**1 0.5 0 0 0.5 1 1.5 2 2 Area under the curve • Is this a density curve?**Mean and Median of Density Curve**• Mean is the point at which the curve would balance if it were made of solid material. • Median is the equal-areas point, the point that divides the area under the curve in half. • Mean and median are the same for symmetric curve.**Normal Distribution**• Symmetric • Single-peaked • Bell-shaped • Can be defined by mean, m, and standard deviation, s.**68-95-99.7 Rule**68% of the observations fall within 1 standard deviation. 95% of the observations fall within 2 standard deviations. 99.7% of the observations fall within 3 standard deviations.**Example 2.3Young Women’s Heights**• The distribution of heights of young women aged 18 to 24 is approximately normal with mean m = 64.5 inches and standard deviation s = 2.5 inches. • How is the scale on the bottom of the next graph determined from this information?**Example 2.3Young Women’s Heights**What proportion of young women are over 67 inches tall? Between what heights do the middle 68% of all women fall? What is the percentile of women with heights 59.5 inches?**Section 2.2Standard Normal Calculations**• By definition, what is the area under a density curve? • How can we convert a normal distribution curve to a density curve?**Standardizing**• If x is an observation from a distribution that has mean m and standard deviation s, the standardized value of x is A standardized value is often called a z-score. (This is a big deal. Memorize it!)**Using z-scores to compare observations**• While the mean professional baseball batting average has been roughly constant over the decades, the standard deviation has dropped over time. • Using the information on the next slide, determine how far each baseball player stood above his peers.**Normal distribution calculations**• The normal distribution probability density function is given by the formula: Where m is the mean and s is the standard deviation. Not very friendly **Standard Normal Distribution**• In the standardized value, m = 0 and s = 1 • In your book, Table A has a list of the value of p(x) for many z-scores. • If you know the area of the shaded region to the left, can you calculate the area of the unshaded region?**Finding Normal Proportions**• State the problem in terms of the observed variable x. Draw a picture of the distribution and shade the area of interest under the curve. • Calculate the z-score. Draw a picture to show the area of interest under the standard normal curve. • Find the required area under the standard normal curve, using Table A. • Write your conclusion in the context of the problem.**Practice**• Work through example 2.8 • The distribution of heights of adult American men is N(69,2.5). • What percent of men are at least 6 feet tall? • What percent of men are between 5 feet and 6 feet tall? • How tall must a man be to be in the tallest 10% of all adult men?**Using your calculator:**The normalcdf command on the TI-84+ calculator can be used to find the area under a normal distribution and above an interval. Normalcdf(min,max,mean,sd) Normalcdf can be found under DISTR (2nd Vars) Use your calculator to find areas on previous slide and compare with your table results.**Normal Probability Plots**• While many collections of data are normally distributed, we cannot assume that all are. You can examine histograms or box plots to see if data “looks” normal. • An easy way to check the normality is using a normal probability plot. • Put your data in one of the lists in your calculator.**Normal Probability Plot**If the data distribution is close to a normal distribution, the plotted points will lie close to a straight line.