ENVE-GEOE 224 - 18 - Experimental Design - Handout

pdf

School

University of Waterloo *

*We aren’t endorsed by this school

Course

224

Subject

Civil Engineering

Date

Oct 30, 2023

Type

pdf

Pages

Uploaded by deraokonkwo

2021-03-01 1 ENVE/GEOE 224: Probability & Statistics Experimental Design Prof. Philip J. Schmidt Department of Civil & Environmental Engineering, University of Waterloo Winter 2021 Week 7: Sampling This week includes three modules focused upon sampling and how this sampling effects our estimates of the mean and standard deviation. We’ll learn about the central limit theorem and two new distributions (and corresponding tables)—the 𝑡 and 𝜒 ଶ distributions. This theory is foundational to the confidence intervals and hypothesis tests that we are going to be doing throughout the rest of the course! This material will be revisited in online lectures in the next two weeks as we make our way through material that applies it. 2 of 11 ENVE/GEOE 224 (W2021) – P. Schmidt 1 2

2021-03-01 2 Learning Objectives Learn about the importance of experimental design → Know what a simple random sample is → Know what it means for a sampling method to be unbiased 3 of 11 ENVE/GEOE 224 (W2021) – P. Schmidt Transitioning from Probability to Statistics We have spent weeks 2-6 learning the building blocks of probability theory so that we can say something about the probabilities associated with various outcomes of clearly defined random experiments If we know enough details to mathematically specify the problem, what is the probability of some outcome of interest? Statistics is concerned with the collection and analysis of data from random experiments to draw inferences and aid decision-making 4 of 11 ENVE/GEOE 224 (W2021) – P. Schmidt 3 4

2021-03-01 3 Sampling It is almost never practical (or even possible) to collect data until we understand a random process with absolute certainty—so we sample 5 of 11 ENVE/GEOE 224 (W2021) – P. Schmidt Consider the Election Poll… Which party is likely to win the most seats in the next election? → We can’t ask everybody who they plan to vote for and in which riding, so we take a sample (e.g., 2000 people) → Is it possible to reach a particular person, will they answer, and will they be honest? → Is the sample representative (e.g., covering all ridings and other variables such as age, gender, ethnicity, socio-economic status, …) → Are today’s responses representative of election day? 6 of 11 ENVE/GEOE 224 (W2021) – P. Schmidt 5 6

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

2021-03-01 4 Experimental Design Collecting data is often experimental, but what is experimental design? Design implies: → A well-defined process being analyzed → Thinking about questions to be answered before collecting the data → Thinking about data analysis before collecting the data → A purposeful approach to collecting data 7 of 11 ENVE/GEOE 224 (W2021) – P. Schmidt Sampling Design The way to get statistically dependable results is to choose the sample at RANDOM. Suppose that we have a large population of 𝑁 objects and a procedure for selecting 𝑛 of them. If the procedure ensures that ALL POSSIBLE SAMPLES OF 𝑛 OBJECTS ARE EQUALLY LIKELY, then we call the procedure a SIMPLE RANDOM SAMPLE . 8 of 11 ENVE/GEOE 224 (W2021) – P. Schmidt 7 8

2021-03-01 5 Sampling Design The simple random sample has two properties that make it the standard against which we measure all other sampling methods: Unbiased each object has the same chance of being chosen Independent selection of one object has no influence on the selection of other objects In the REAL WORLD, unfortunately, completely unbiased, independent samples are hard to find. For instance, surveying voters by randomly dialling telephone numbers is biased: it ignores voters without a telephone and oversamples people with more than one phone number. 9 of 11 ENVE/GEOE 224 (W2021) – P. Schmidt Sampling with Bernoulli Trials Consider a series of 𝑛 Bernoulli trials with probability of success 𝑝 . The probability 𝑝 governs the outcomes we see in the so-called “real world” We may not know what 𝑝 is, but we’d like to find out. So, we take a random sample of size 𝑛 and find the number of successes 𝑥 . Remember, 𝑥 is the observed value of the number of successes for the binomial random variable 𝑋 . 10 of 11 ENVE/GEOE 224 (W2021) – P. Schmidt 9 10

2021-03-01 6 Sampling with Bernoulli Trials The proportion of successes in the sample should be somewhere around 𝑝 (the true probability of success) so we call this estimate 𝑝̂ 𝑝̂ = ௫ ௡ , 𝑃 ෠ = ௑ ௡ The mean of 𝑃 ෠ is 𝐸 𝑃 ෠ = 𝑝 𝑃 ෠ is an unbiased estimator of 𝑝 ! 𝐸 𝑃 ෠ = 𝐸 ௑ ௡ = ଵ ௡ 𝐸 𝑋 = ଵ ௡ 𝑛𝑝 = 𝑝 The standard deviation of 𝑃 ෠ is 𝜎 𝑃 ෠ = ௣ ଵି௣ ௡ Note the 1 𝑛 ⁄ ! 𝑉𝑎𝑟 𝑃 ෠ = 𝑉𝑎𝑟 ௑ ௡ = ଵ ௡ మ 𝑉𝑎𝑟 𝑋 = ଵ ௡ మ 𝑛𝑝 1 − 𝑝 = ௣ ଵି௣ ௡ For large 𝑛 , 𝑃 ෠ is approximately normally distributed 11 of 11 ENVE/GEOE 224 (W2021) – P. Schmidt 11

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version