University:
California State University, Northridge
Course:
MATH 340 | Introduction to Probability and Statistics
Academic year:

2024
Views:

88

Pages:

21
Author:

Mckenna Welch

Basic Probability Distributions Part 1 How can it be that mathematics, being after all a product of human thought independent of experience, is so admirably adapted to the objects of reality Albert Einstein Random Variables A random variable is a numerical description of the outcome of an experiment. A discrete random variable may assume either a finite number of values or an infinite sequence of values. Count it. x= number of students showing up in class, 11, 12,13, ….. x= male and junior (0), female and junior (1), male and senior (3), female and senior (4). The range can be finite or infinite A continuous random variable may assume any numerical value in an interval or collection of intervals. Measure it. x= length of the lecture in minutes, 45, 45.000001, 45,000002, .. 3 Chapter 7 Sampling and Sampling Distributions  Selecting a Sample  Point Estimation  Introduction to Sampling Distributions  Sampling Distribution of x  Sampling Distribution of p  Other Sampling Methods 4 Introduction An element is the entity on which data are collected A population is a collection of all the elements of interest.. A sample is a subset of the population. The sampled population is the population from which the sample is drawn. A frame is a list of the elements that the sample will be selected from. 5 Introduction The reason we select a sample is to collect data to answer a research question about a population. The sample results provide only estimates of the values of the population characteristics. The reason is simply that the sample contains only a portion of the population. With proper sampling methods, the sample results can provide “good” estimates of the population characteristics. 6 Selecting a Sample   Sampling from a Finite Population Sampling from an Infinite Population 7 Sampling from a Finite Population  Finite populations are often defined by lists such as:     Organization membership roster Credit card account numbers Inventory product numbers A simple random sample of size n from a finite population of size N is a sample selected such that each possible sample of size n has the same probability of being selected. 8 Sampling from a Finite Population  Replacing each sampled element before selecting subsequent elements is called sampling with replacement.  Sampling without replacement is the procedure used most often.  In large sampling projects, computer-generated random numbers are often used to automate the sample selection process. 9 Sampling from a Finite Population  Example: St. Andrew’s College St. Andrew’s College received 900 applications for admission in the upcoming year from prospective students. The applicants were numbered, from 1 to 900, as their applications arrived. The Director of Admissions would like to select a simple random sample of 30 applicants. 10 Sampling from a Finite Population  Example: St. Andrew’s College Step 1: Assign a random number to each of the 900 applicants. The random numbers generated by Excel’s RAND function follow a uniform probability distribution between 0 and 1. Step 2: Select the 30 applicants corresponding to the 30 smallest random numbers. 11 Sampling from a Finite Population Using Excel  Excel Formula Worksheet 1 2 3 4 5 6 7 8 9 A Applicant Number 1 2 3 4 5 6 7 8 B Random Number =RAND() =RAND() =RAND() =RAND() =RAND() =RAND() =RAND() =RAND() Note: Rows 10-901 are not shown. 12 Sampling from a Finite Population Using Excel  Excel Value Worksheet 1 2 3 4 5 6 7 8 9 A B Applicant Random Number Number 0.61021 1 0.83762 2 3 0.58935 0.19934 4 0.86658 5 0.60579 6 7 0.80960 8 0.33224 Note: Rows 10-901 are not shown. 13 Sampling from a Finite Population Using Excel  Put Random Numbers in Ascending Order Step 1 Select any cell in the range B2:B901 Step 2 Click the Home tab on the Ribbon Step 3 In the Editing group, click Sort & Filter Step 4 Choose Sort Smallest to Largest 14 Sampling from a Finite Population Using Excel  Excel Value Worksheet (Sorted) 1 2 3 4 5 6 7 8 9 A B Applicant Random Number Number 12 0.00027 773 0.00192 408 0.00303 58 0.00481 116 0.00538 185 0.00583 510 0.00649 394 0.00667 Note: Rows 10-901 are not shown. 15 Sampling from an Infinite Population  Sometimes we want to select a sample, but find it is not possible to obtain a list of all elements in the population.  As a result, we cannot construct a frame for the population.  Hence, we cannot use the random number selection procedure.  Most often this situation occurs in infinite population cases. 16 Sampling from an Infinite Population  Populations are often generated by an ongoing process where there is no upper limit on the number of units that can be generated.  Some examples of on-going processes, with infinite populations, are: • parts being manufactured on a production line • transactions occurring at a bank • telephone calls arriving at a technical help desk • customers entering a store 17 Sampling from an Infinite Population  In the case of an infinite population, we must select a random sample in order to make valid statistical inferences about the population from which the sample is taken.  A random sample from an infinite population is a sample selected such that the following conditions are satisfied. • Each element selected comes from the population of interest. • Each element is selected independently. 18 Point Estimation Point estimation is a form of statistical inference. In point estimation we use the data from the sample to compute a value of a sample statistic that serves as an estimate of a population parameter. We refer to mean µ. as the point estimator of the population s is the point estimator of the population standard deviation σ. is the point estimator of the population proportion p. 19 Point Estimation  Example: St. Andrew’s College Recall that St. Andrew’s College received 900 applications from prospective students. The application form contains a variety of information including the individual’s Scholastic Aptitude Test (SAT) score and whether or not the individual desires on-campus housing. At a meeting in a few hours, the Director of Admissions would like to announce the average SAT score and the proportion of applicants that want to live on campus, for the population of 900 applicants. 20 Point Estimation  Example: St. Andrew’s College However, the necessary data on the applicants have not yet been entered in the college’s computerized database. So, the Director decides to estimate the values of the population parameters of interest based on sample statistics. The sample of 30 applicants is selected using computer-generated random numbers. 21 Point Estimation Using Excel  Excel Value Worksheet (Sorted) 1 2 3 4 5 6 7 8 9 A Applicant Number 12 773 408 58 116 185 510 394 B Random Number 0.00027 0.00192 0.00303 0.00481 0.00538 0.00583 0.00649 0.00667 C SAT Score 1207 1143 1091 1108 1227 982 1363 1108 D On-Campus Housing No Yes Yes No Yes Yes Yes No Note: Rows 10-31 are not shown. 22

Basic Probability Distributions, Part 1

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