Mr. Rogers - AP Statistics Objectives

Syllabus 1st Quarter 2nd Quarter 3rd Quarter 4th Quarter 1 Distributions 2 N-Distribution 3 Regression 4 NL Regression 5 Data

Unit Plan Practice Test

Chapter 1: The Normal Distribution AP Statistics Standards I. Exploring Data:  (continued). II. Anticipating Patterns: C. The normal distribution Properties of the normal distribution Using tables of the normal distribution The normal distribution as a model for measurements
Objectives Essential Question: Is the bell-curve real? The Normal Distribution Define density curve. Models a distribution Above horizontal axis Area under it = 1 Area under part of the curve = probability  Describe the normal distribution. Symmetrical Bell Shaped Range negative infinity to positive infinity State where the 2 inflection points fall on a normal distribution. This will help you draw the norm. distr. (± 1 sigma from the mean) State the probability of getting a data point with a specific value. zero. Why? A point has a width of zero, hence, here's zero area under a single point on the normal distribution Correctly use N (mean, sigma) notation. State the effects on a normal distribution of increasing standard deviation. Relevance: The normal distribution is ubiquitous. Discussions about it pop up in all kinds of both technical, news, and popular publications.	Activities   Lesson 1 Key Concept: What is a normal distribution? Purpose: Lay the foundation for using the normal distribution. Interactive Discussion: Objectives. Explain why the 68-95-99.7 Rule is critical for developing "normal distribution intuition". Explain the terms concaved downward, concaved upward. Using a piece of string, demonstrate the effects of changing the standard deviation Normal Distribution, Wikipedia, http://en.wikipedia.org/wiki/Normal_distribution, 9-8-2008
Essential Question: How is the area under the bell-curve related to the standard deviation? Estimating Areas Under Normal Distributions Find probabilities (areas under the curve) using the 68-95-99.7 Rule for normal distributions (sometimes called the empirical rule). Note: in the 1st part of the test for this chapter, you will not be permitted to use calculators or tables. The entire test will be based on the above rule.   Relevance: If you know the empirical rule you can easily double check your answers when solving for areas under the n-distribution. The ability to evaluate answers is a very high level skill.	Lesson 2 Key Concept: Estimating areas under the normal distribution? Purpose: Lay the foundation for using the normal distribution. Interactive Discussion: Objectives. Explain why the 68-95-99.7 Rule is critical for developing "normal distribution intuition". Group Work: Use the small white boards to record the work and the empirical rule to: Find various areas under the n-distribution using problems directly related to the empirical rule. Find areas using the empirical rule to estimate areas when the limits on the areas don't perfectly correspond to the values in the empirical rule.
Essential Question: Does an ogive give the same information as a normal distribution? Convert the N-distribution into an ogive (cumulative frequency plot) using the empirical rule. Estimate the mean, median, sigma, Q1, Q3, and IQR using an ogive for a normal distribution.   Homefun: prob. 2.11, 2.13, 2.15	Lesson 3 Key Concept: The empirical rule and normal distribution can be expressed in ogive form? Purpose: Lay the foundation for using the normal distribution. Interactive Discussion: Objectives. Explain why the 68-95-99.7 Rule is critical for developing "normal distribution intuition". Group Work: Use the small white boards to record the work and the empirical rule to: find various areas under the n-distribution. create an ogive Normal Distribution, Wikipedia, http://en.wikipedia.org/wiki/Normal_distribution, 9-8-2008
Essential Question: How is the area under a non-bell-curve related to standard deviation, median and mean? Describe how the median and mean are related to the area under the curve for any density curve (p.82). Median: the equal area point Mean: the balance point (directly under the center of mass if the distribution were cut out of a piece of plywood) Estimate the median, quartiles, and probabilities for any distribution by estimating the area under the curve: Be as one with the 0-75-89 Rule based on Chebyshev's Theorem, which applies to any distribution. Note: this is the worst case situation for any distr. It gives an indication of how big the error would be in calculating areas if a distribution is incorrectly identified as a normal distribution. Chebyshev's Theorem: p = (1 - 1/k2) Where: p = the lowest possible probability of finding a value between  ±k standard deviations from the mean. Homefun: prob. 2.3, 2.7, 2.9	Lesson 4 Key Concept: Characteristics of "non-normal" distributions Purpose: To relate areas under a non-normal distributions to measures of central tendency and spread. Interactive Discussion: Objectives. Use Chebyshev's Rule rule to help understand the power and limitations of the empirical rule. Group Work: Estimate mean, median, quartiles, and probabilities using the following distributions: uniform triangular skewed
Essential Question: How can the area under the bell-curve be determined more accurately than by estimating? Calculating Areas Under the Normal Distribution Calculate z-scores. z = number of standard deviations from the mean Find areas under the normal distribution using tables. Note, the tables contain a "standardized normal distribution". Here, the mean = zero and the standard deviation = 1. Probability = area under curve Find z-scores corresponding to LL, Q1, median, Q3, UL of a modified box and whiskers plot for normally distributed data. Using a normal distribution, estimate a critical value given the probability of finding a value as extreme or more extreme (a tail area).     Homefun: prob. 2.21, 2.23, 2.25	Lesson 5 Key Concept: Area under the normal distribution and how it relates to a box and whiskers plot.   Purpose: Understand when a data set is normally distributed. Interactive Discussion: Objectives 2-person teams: Derive an equation for the z-score Seat Work: Find the area under n-distribution using tables and the TI-83 calculator. 2-person teams: Derive the z-scores of Q1, Q2, LL, UL Draw a box and whiskers diagram of an n-distribution How to Calculate the Normal Distribution pdf = 1/[σ(2π)] • exp[ - (x - μ)2 / (2σ2)] Where: pdf = probability density function (height or y-axis coordinate of the N-distribution) x = horizontal axis coordinate μ = mean σ = the standard deviation Note: determining the shape of the normal distribution requires a knowledge of both the mean and standard deviations
Essential Question: How can we tell if a distribution is normal and why would we care? Judge the normality of a distribution by examining histograms, stem plots, dot plots, ogives, or box and whiskers plots.   Bell shaped Sym-metrical Follows 68-95-99.7 Rule whisker ≈ 1.5 x ( box width) histograms, stem plots, dot plots x x x   ogives   x x   box & whiskers plots.   x   x   Judge the normality of a distribution using a normal quantile plot on a TI-83. n-distributions look like straight lines on normal quantile plots.   Essential Question: What is the best way to find probabilities associated with normal distributions? Work quality control reject rate problems. Note: rejects occur in the tail areas. Find areas under the normal distribution using tables using a TI -83 calculator. normalcdf (L,U,M,S) Have you taken your LUMS today? Lower value Upper value Mean Standard deviation Given a tail area, find the critical value (see objective 15) using a TI-83 calculator. invNorm (A,M,S) I think therefore I AMS. Area (must be a decimal fraction) Mean Standard deviation Remember, in tables for finding standard normal distribution probabilities, the mean = zero and the standard deviation = 1. If these values are input into invNorm along with an area, the result will be a z-score. Homefun: prob. 2.27   Essential Question: How can I make an "A" on the test? Normal Distributions Review Work the practice test. Review the objectives. Look over free response problems from previous years. Master the vocabulary (see example below). Summative Assessment: Test--Objectives 1-20	Lesson 6 Key Concept: Determining normality.   Purpose: Understand when a data set is normally distributed. Interactive Discussion: Objectives Seat Work: evaluate whether various data sets are normally distributed by using the TI-83 calculator.