Statistical Methods STAT 302 Chapter 5 Fundamentals of Statistical Inference Copyright 2025 by Aburweis All rights reserved No part of this work may be reproduced in any form without written permission STATISTICS Descriptive Statistics Inferential Statistics Estimation Hypothesis Testing Point Interval Estimation Estimation 2 Chapter 5 Learning Objectives 1 Identify appropriate point estimates for various population parameters 2 Explain the concept of a sampling distribution and how sample statistics can vary across 3 Apply the Central Limit Theorem CLT to explain the behavior and characteristics of 4 Use the CLT to estimate the mean and standard deviation of sampling distributions for both sample means and sample proportions 5 Use the CLT to calculate probabilities associated with sample means and sample 6 Construct and interpret confidence intervals to estimate unknown population parameters 7 Perform hypothesis tests using appropriate test statistics and interpret p values to draw different samples sampling distributions proportions conclusions 3 Fundamentals of Statistical Inference Point Estimates and Sampling Variability Sampling Distributions Central Limit Theorem CLT Confidence Intervals Hypothesis Testing 4 Introduction This chapter introduces the fundamental principles of statistical inference which involve analyzing data from a sample to draw conclusions and make generalizations about unknown population parameters Through the study of sample data we can obtain estimates of these parameters accompanied by a quantifiable degree of uncertainty The chapter emphasizes methods for estimating key population characteristics including means and proportions 5 Fundamental Terms Population Proportion or The fraction or percentage of a population that satisfies a specific condition Sample Proportion or A point estimate of the unknown population proportion calculated from sample data Population Mean The average of all observations in a population Sample Mean or The average of observations in a sample serving as a point estimate for the unknown population mean 6 Point Estimate A point estimate is a single numerical value calculated from a random sample data that serves as an approximation of an unknown population parameter Examples proportion The sample proportion is a point estimate for the population The sample mean is a point estimate for the population mean The sample variance 2 is a point estimate for the population variance 2 7 Point Estimate Cont The sample proportion is the most commonly used point estimate of the population proportion It is defined as where denotes the number of observed successes in the sample and represents the total sample size 8 Example 1 Facebook In a survey of 1550 U S adults 1 054 reported using the social media website Facebook Calculate the point estimate for the population proportion of U S adults who use Facebook Solution The number of successes is the number of adults who use Facebook so 1054 The sample size 1550 so the sample proportion is 0 68 1054 1550 or 9 Your Turn 1 In a survey of 1 000 U S adults 662 indicated that they believe it is acceptable to check personal email while at work Calculate the point estimate for the population proportion of U S adults who believe it is acceptable to check personal email while at work 10 Your Turn 1 Solution 11 Sampling Variability Sample statistics can vary from one sample to another Quantifying this variability helps estimate the margin of error associated with our point estimate Before we get to quantifying the variability among samples let s try to understand how and why point estimates vary from sample to sample 12 Sampling Variability Cont Suppose we randomly sample 1 000 adults from each state in the United States Would you expect the sample means of their heights to be the same somewhat different or substantially different Not the same but only somewhat different 13 Activity 1 Using the dataset from the student information survey conducted on the first day of class weights 1 Randomly select five students from the class 2 Ask each selected student to collect a random sample of five other students 3 Have each student calculate the average mean weight of their sample 4 Compare the mean weights obtained by the five students and discuss any differences or similarities observed among the samples 14 Activity 1 Solution Sample 1 Sample 2 Sample 3 Sample 4 Sample 5 Mean 15 Sampling Variability Cont Suppose the proportion of American adults who support the expansion of solar energy is which is our parameter of interest Is a randomly selected American adult more or less likely to support the expansion of solar energy More likely 16 Sampling Variability Cont In most real world situations we do not have access to the entire population of American adults To estimate the proportion of adults who support the expansion of solar power we can draw a sample from the population and use the sample proportion as our best estimate of the unknown population proportion Activity 1 Draw a random sample of 1 000 American adults without replacement from the population and record whether each individual supports or does not support the expansion of solar power 2 Compute the sample proportion of individuals in favor of solar power expansion 3 Compile and plot the distribution of the sample proportions obtained by all members of the class 17 Sampling Variability Simulation 1 Create a set of 250 million entries where 88 of them are support and 12 are not pop size 250000000 possible entries c rep support 0 88 pop size rep not 0 12 pop size 2 Sample 1000 entries without replacement sampled entries sample possible entries size 1000 3 Compute p hat count the number that are support then divide by the sample size sum sampled entries support 1000 18 Sampling Distribution The sampling distribution of a statistic such as a sample proportion or sample mean represents the distribution of all possible values of the statistic when multiple random samples of the same size are drawn from the same population Elementary statistics Mario F Triola Elementary statistics Mario F Triola 19 Sampling Distribution Cont Sampling Distribution Distribution of a statistics such as the sample mean or sample proportion obtained from many samples drawn from the same population Population Distribution Probability Parameter or Parent Distribution The distribution of values for all individuals in the entire population Data Distribution The distribution of
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