Sec. 2.1 Review 10th, z = -1.26 97th, z = 1.43

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

Density Curves and Normal Distributions ESTIMATE the relative locations of the median and mean on a density curve. USE the 68-95-99.7 Rule to 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.

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No set of real data is exactly described by a density curve No set of real data is exactly described by a density curve. The curve is an approximation that is easy to use and accurate enough for practical use. When you see a density curve, imagine there is a dotplot underneath it. The curve is simply tracing out the distribution of dots.

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).

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.

The 68-95-99.7 Rule Although there are many Normal curves, they all have properties in common. The 68-95-99.7 Rule In the Normal distribution with mean µ and standard deviation σ: Approximately 68% of the observations fall within σ of µ. Approximately 95% of the observations fall within 2σ of µ. Approximately 99.7% of the observations fall within 3σ of µ.

In a survey conducted by the National Center for Health Statistics, the sample mean height of women in the U.S. (ages 20-29) was 64 inches with a standard deviation of 2.75 inches. Estimate the percent of women whose heights are between 64 inches and 69.5 inches. Estimate the percent of the heights that are between 61.25 and 64 inches.

Khan Academy Practice Empirical Rule Questions

The Standard Normal Distribution All Normal distributions are the same if we measure in units of size σ from the mean µ as center. The standard Normal distribution is the Normal distribution with mean 0 and standard deviation 1. If we turn raw values into z-scores then any distribution can be turned into a standard distribution.

Using the Standard Normal Table Using the 68-95-99.7 Rule it is pretty easy to estimate the percent of a population that falls within certain areas of a curve, but that is only if we use nice increments What percent of a normal distribution is to the right of 2 standard deviations? What percent of a distribution is to the left of -1 standard deviation? What percent of a distribution is to the left of -1.5 standard deviations? The Standard Normal Table (pg. T-1 near back of book) or our calculator allows us to find percentages for a wide range of z-values

The Standard Normal Table The standard Normal Table (Table A) is a table of areas under the standard Normal curve. The table entry for each value z is the area under the curve to the left of z. Suppose we want to find the proportion of observations from the standard Normal distribution that are less than 0.81. We can use Table A: P(z < 0.81) = .7910 Z .00 .01 .02 0.7 .7580 .7611 .7642 0.8 .7881 .7910 .7939 0.9 .8159 .8186 .8212

Normal Distribution Calculations We can answer a question about areas in any Normal distribution by standardizing and using Table A or by using technology. How To Find Areas In Any Normal Distribution Step 1: State the distribution and the values of interest. Draw a Normal curve with the area of interest shaded and the mean, standard deviation, and boundary value(s) clearly identified. Step 2: Perform calculations—show your work! Use either the table or a calculator (normalcdf) to find the area of interest. Step 3: Answer the question.

The Standard Normal Distribution Table can be used to easily find the area… to the left of a z-score (find value in table) to the right of a z-score (find value in table and subtract it from 1) between 2 z-scores (find each value and subtract the smaller from the larger)

Find the area under the curve to the left of a z-score of -2.19

Find the area under the curve to the left of a z-score of 2.34

Find the area under the standard normal curve… 0.0051 to the right of 2.57 between -2.16 and -1.35 0.0731

Finding Area on the TI-84 Press 2nd  DISTR  2 Enter values for lower and upper z-scores Use –1E99 for lower limit if you’re finding area to the left of a z-score Use 1E99 for upper limit if you’re finding area to the right of a z-score

Find the area under the standard normal curve… to the left of -2.19 to the left of 2.34 to the right of 2.57 in between -2.16 and -1.35

Sec. 2.2 Assignment (12 points possible) Pg. 129 #39,41,43,47,49,71

Review What percent of a normal distribution is found to the left of 1 standard deviation? What percent of a normal distribution is found to the left of -2 standard deviation? What percent of a normal distribution is found right of 0.67 standard deviations? What percent of a normal distribution is found between 0.79 and 1.61 standard deviations? 84% 2.5% 25% 16%

Density Curves and Normal Distributions FIND a z-score from a percentile in the standard Normal distribution. FIND the value that corresponds to a given percentile in any Normal distribution.

Working Backwards: Normal Distribution Calculations Sometimes, we may want to find the observed value that corresponds to a given percentile. There are again three steps. How To Find Values From Areas In Any Normal Distribution Step 1: State the distribution and the values of interest. Draw a Normal curve with the area of interest shaded and the mean, standard deviation, and unknown boundary value clearly identified. Step 2: Perform calculations—show your work! Use the table or calculator (invNorm) to work backwards and find the required information. Step 3: Answer the question.

Finding a z-score given an area Find the z-score that corresponds to a cumulative area of 0.0384 Table: Find area in table and identify corresponding z-score Calculator: 2ndDISTR3, input area Find the z-score that corresponds to a cumulative area of 0.9901

Finding a z-score Given a Percentile Recall that the area correlates to the percentile so for something at the 20th percentile we are simply looking for the z-score that gives an area of 0.20 Find the z-score for the observation that is greater than 55% of observations.

A slight twist In a recent tournament, tennis player Rafael Nadal averaged 115 mph on his serves. Assuming a normal distribution and a standard deviation of 6 mph, about what percentage of Nadal’s serves would you expect to exceed 120 mph? About 20 % What percent of Nadal’s serves would you expect to be between 100 and 110 mph?

Ch. 2 Review (28 points possible) Pg. 130 #51,53,59,63 a-b Pg. 136 #1,4,5,6,7,8a Pg. 137 #1,7,9,10 Ch. 2 Quiz next time we meet