University:
California State University, Northridge
Course:
MATH 140 | Introductory Statistics
Academic year:

2010
Views:

438

Pages:

16
Author:

skinnystatusvp2

Math 140 Introductory Statistics Chapter 2 Part 3 Section 2.4 Recentering and Rescaling Recentering a data set (adding the same number c to all the values in the set) Shape or spread do not change. It slides the entire distribution by the amount c, adding c to the median and the mean. Rescaling a data set (multiplying all the values in the set by the same positive number d) Basic shape doesn’t change. It stretches or shrinks the distribution, multiplying the spread (IQR or standard deviation) by d and multiplying the center (median or mean) by d. Example City Country Temperature (oF) Addis Ababa Ethiopia 32 Algiers Algeria 32 Bangkok Thailand 50 Madrid Spain 14 Nairobi Kenya 41 Brazilia Brazil 32 Warsaw Poland -22 The Influence of Outliers A summary statistic is resistant to outliers if it does not change very much when an outlier is removed. sensitive to outliers if the summary statistic is greatly affected by the removal of outliers. Example Percentiles and CRF plots You are responsible to read through this and understand the concepts of percentile, and cumulative relative frequency plot. 2.5 The Normal Distribution Shape SD Center: Mean SD Mean x sum of values ∑ x= = n number of values Spread: Standard Deviation σ n −1 = 2 ( x − x ) ∑ n −1 Applications of the Normal Distribution The normal distribution tells us how: Variability in measures behaves. Variability in population behaves. Averages and some other summary statistics behave when you repeat a random process. Nice property: A normal distribution is determined by its mean and standard deviation! (If you know mean and SD you know everything) The Two Main Problems. The distribution of the SAT scores for the University of Washington was roughly normal in shape, with mean 1055 and standard deviation 200. 1. What percentage of scores were 920 or below? (Unknown percentage problem) 2. What SAT score separates the lowest 25% of the SAT scores from the rest? (Unknown value problem) Unknown percentage problem. The distribution of the SAT scores for the University of Washington was roughly normal in shape, with mean 1055 and standard deviation 200. 1. What percentage of scores were 920 or below? Unknown value problem. The distribution of the SAT scores for the University of Washington was roughly normal in shape, with mean 1055 and standard deviation 200. 2. What SAT score separates the lowest 25% of the SAT scores from the rest? Which one is it? 1. Unknown percentage problem. Given x, Find P. 2. Unknown value problem. Given P, Find x. The Standard Normal Distribution. It is the normal distribution with Mean = 0, and standard deviation = 1. The area under the curve equals 1 (or 100%) SD SD -1 0 1 The Standard Normal Distribution. It is the normal distribution with Mean = 0, and standard deviation = 1. The area under the curve equals 1 (or 100%) The Standard Normal Distribution is important because any normal distribution can be recentered and/or rescaled to the standard normal distribution. This process is called standarizing or converting to standard units. Also, the two main problems can be easily solved in the Standard Normal Distribution with the help of tables or a calculator. The Two Main Problems in the Standard Normal Distribution. Unknown Percentage. (Given z, find P ) With Table A (end of the textbook) Use the units and the first decimal to locate the row and the closest hundredths digits to locate the column. The number found is the percentage of the number of values below z. With Calculator Enter normalcdf(-99999, z) to get the percentage of the number of values below z. Example (given z find P) Calculator P = normalcdf(-99999,1.23) = .8906513833 ~ 89.07% Table A Look for -row labeled 1.2 -column labeled .03 The intersection shows P =.8907 = 89.07%