Concepts in Statistics

Transcription

1 Concepts in Statistics -- A Theoretical and Hands-on Approach Statistics The Art of Distinguishing Luck from Chance Statistics originally meant the collection of population and economic information vital to the state, and it was considered to belong more to the arts than to the mathematics. The notes are designed in such a way that some fundamental concepts in statistics are introduced at many levels from simple verbal explanations to relatively rigorous mathematical deductions, and then the contents are further explained by many AP statistics problems. All example problems and exercises have answers and many have detailed solutions. So, the notes can be viewed as the self-study materials. 0. Introduction 1. Sampling. Design of Studies 3. Exploring Data 4. Regressions 5. Probability 6. Distributions 7. Confidence Interval 8. Hypothesis Tests 9. R Projects Contents 1

2 N. INTRODUCTION In this chapter many concepts in statistics are briefly introduced through examples. The Probability [MATH] A proposition is a declarative sentence that can be either true or false. It must be one or the other, and it cannot be both. An atomic proposition is the smallest statement or assertion that must be true or false. A set is a collection of objects. Each object in a set is an element of that set. For any given experiment, the set of all possible outcomes is called the sample space or probability space. An event is a subset of the sample space. Probability is a measure of the likelihood that an event will occur. Example [MC1434] A sample of 94 homeowners are classified, in the two-way frequency table below, by the number of credit cards they have and the number of years they have owned their current homes. Of the homeowners in the sample who have four or more credit cards, what proportion have owned their current homes for at least one year? Another way to state the problem is that a homeowner, known to have four or more credit cards, is selected at random, what is the probability that this homeowner has owned his/her home for at least one year?

3 Solution: This is a conditional posterior probability problem. Assume that the events A and B are: A: number of people who have owned their current homes for at least 1 year. B: number of people who have four or more credit cards. Then, if we know that a homeowner has four or more credit cards, the probability that the homeowner has owned his/her current home for at least one year is P ( A B) = =» Example [MC1404] The distribution of colors of candies in a bag is as follows: If two candies are randomly drawn from the bag with replacement, what is the probability that they are the same color? (This is a classic probability problem.) Solution: assume that each drawing is independent. Let the events S: The two picked candies have the same color B: The two picked candies have the brown color R: The two picked candies have the red color Y: The two picked candies have the yellow color G: The two picked candies have the green color O: The two picked candies have the orange color Then Pr( S) = Pr( B) + Pr( R) + Pr( Y ) + Pr( G) + Pr( O) = 0.3 =

4 Population and Sample Distributions A method for collecting data is to sample from the relevant population. The graph below illustrates the distributions of a survey result from a population of 5000 individuals and some samples with different sampling sizes. The Bell Curve is referred as to as the normal distribution which is described by the probability density function : p(x) 1 p ( x) = e s p -1æ x-mö ç è s ø where the meanm is the center of the curve, and standard deviation s is often viewed as a measure of the spread of the data from the mean, or as the variability of the data. 4

5 The amazing fact is that is only dependent on the mean and standard deviation. The area under is the probability. Some examples are p(x) p(x) Calculation of Standard Deviation Pr ( x -m < s) = AREA ( m-s < x< m+ s )» ( x -m < s) = AREA ( m- s < x< m+ s ) ( x -m < 3s) = AREA ( m- 3s < x< m+ 3s ) Pr» Pr» For the population data points ( x1, x, L, xn ), the standard deviation of the equally likely points can be calculated as s = n å i= 1 ( x - x) i n 5

6 x1 + x + L+ xn where x = is the mean or the arithmetic average of the data points and n is the data size. The standard deviation can be viewed as the normalized distance from the data point to its mean x : n Example Calculate the standard deviation for each of the following cases: a.) x, x ) (1, 9), and b.) x, x ) (6, 4). ( 1 = ( 1 = Solution: In both cases, the mean is the same as x = = = 5. (1-5) + (9-5) a.) d = (1-5) + (9-5) = 4, s = = 4 (6-5) + (4-5) b.) d = (6-5) + (4-5) =, s = = 1 This example illustrates that with the same mean, the further apart the data is, the larger the standard deviation is. Or in another view, the closer the data point to the line x = x1, the smaller the standard deviation is. Example [MC1414] Which of the following statements must be true about the data sets A and B displayed in the histograms below? 6

7 (A) The mean of data set A is equal to the mean of data set B. (B) The median of data set A is equal to the median of data set B. (C) The range of data set A is equal to the range of data set B. (D) The standard deviation of data set A is less than the standard deviation of data set B. (E) The standard deviation of data set A is greater than the standard deviation of data set B. Solution: The answer is E. (A) When the distribution is symmetrical, the mean is located at the center. But those are discrete distributions, so the means and can only be estimated between 30 and 40. m A (B) The median divides the distribution into two congruent areas. When the distribution is symmetrical, the median is located at the center. The same reason as (A) that two medians can only be known for between 30 and 40. (C) Similar reason for ranges. (D) The more data are clustered around mean, the smaller the variability or the standard deviation. (E) The standard deviation measures the spread of the data from the mean. There are more data away from the center in Data Set A than in Data Set B. So, the choice is (E). m B Example The probability Pr(n ) is a function that describes the likelihood of event occurs. The posterior probability can be approximated by finding the area under the frequency function constructed from the observed data. n Given the above frequency function constructed from an observed data set, find 7

8 Pr( n = 5) = Pr( 3 n 5) = Pr( n - 4 ³ 3) = Solution: 4.5 Pr( n = 5) =» Pr( 3 n 5) = =» Pr( n - 4 ³ 3) = P( n 1 & n³ 7) = =»

9 Quick-Check Introduction QC [MC1434M] A sample of 94 homeowners are classified, in the two-way frequency table below, by the number of credit cards they have and the number of years they have owned their current homes. Of the homeowners in the sample who have at most three credit cards, what proportion have owned their current homes for no more than three years? QC [MC1404M] The distribution of colors of candies in a bag is as follows. If two candies are randomly drawn from the bag with replacement, what is the probability that they are not the same color? 9

10 QC [MC139] For a roll of a fair die, each of the outcomes 1,, 3, 4, 5, or 6 is equally likely. A red die and a green die are rolled simultaneously, and the difference of the outcomes (red green) is computed. This is repeated for a total of 500 rolls of the pair of dice. Which of the following graphs best represents the most reasonable distribution of the differences? QC For the given normal distribution below: a.) Identify the mean, median and mode of the distribution. 10

11 b.) By varying the standard deviation s, the shape of distribution changes. Explain how the shape of distribution would change by varying the standard deviation. QC You generated 10,000 digits of 1,, 3,, 9 with a computer program. The results were formulated as a histogram given below. You believed that a sample with a size of was large enough, so you intended to believe that the results were close to the true distribution of this digit-generating program. All the possible outcomes are the non-zero digits denoted as W = {1,,3,4,5,6,7,8,9 }. Let H 0 = {,3, 4,5, 6} and H a = {1, 7,8,9}, find the probability Pr ( xî H a ). (The notation xî means that the digit x belongs to set H ) H a a QC [Extra Credit] Assume the above distribution is approximately normal distributed. Estimate the mean and standard deviation. 11

12 Answers QC C: number of people who have owned their current homes for no more than 3 years. D: number of people who have at most three credit cards P ( C D) = =» QC Pr( Different Color) = 1- Pr( Same Color) = 1-0.= QC The answer is C. Since both of them are fair dice, the means or expected values should be the same. That is, the mean of the difference of the outcomes is zero. Most of the outcomes of the difference for the 500 rolls of the pair of dice should follow a distribution that are clustered around zero with rapid decay when the numbers are farther away from zero (the distribution curve is normal distribution that is concave down near the mean and then concave up towards the tails. The difference of two normal distributions is also a normal distribution. QC a.) Since the curve is symmetrical and peaked at center, the mean, median and mode are all equal tom. b.) As thes reduces, the curve becomes narrower. The data is more clustered around the mean, and hence the spread or variability is minimized Pr Î a = = QC ( x H ).95% QC Since the mode is at 4 and 99% of digits lie within 1 to 7 inclusive, then the 7 mean and standard deviation can be approximated by m = 4, 6s = 7Þs =, 6 1