MAS 261 - Lecture 9
Central Limit Theorem
Housekeeping
Today’s plan
Review Question about the sampling distribution of the sample mean.
A few EXTRA minutes for R Questions 🪄
- Demo videos for installation
Reveiw of the the sampling distribution of the sample mean.
Introduction to the Central Limit Theorem
Introduction to Practice Questions for Quiz 1
Time for questions about HW 4, Practice Questions, Installing R
Reminders
HW 4 is due 9/25.
Quiz 1 is on October 1st.
There is no class on October 3rd.
R and RStudio
In this course we will use R and RStudio to understand statistical concepts.
You will access R and RStudio through Posit Cloud.
- Sign up for a Free Posit Cloud Account
I will post R/RStudio files on Posit Cloud that you can access in provided links.
I will also provide demo videos that show how to access files and complete exercises.
NOTE: The free Posit Cloud account is limited to 25 hours per month.
I will demo how to download completed work so that you can use this allotment efficiently.
For those who want to go further with R/RStudio:
- After Test 1, I will provide videos on how to download the software (R/RStudio/Quarto) and lecture files to your computer.
Lecture 9 In-class Exercises - Q1-Q2
This is related to HW 4 - Questions 7 and 8.
Alto Cinco, a local Mexican restaurant on Westcott Street, is known for its great burritos. On weekend evenings during the school year, they receive an average (\(\mu\)) of 41 burrito take-out/delivery orders with a standard deviation (\(\sigma\)) of 4.
In the next couple of months (n=16 weekend nights), the sales manager notices that sales are staying very high. If the average number of sales based on these 16 weekend nights is 45 burritos (\(\overline{X} = 45\)), how do you interpret this sales number?
Q1. How many standard deviations is the observed mean sales (\(\overline{X}\)) for 16 nights from the the population mean, \(\mu\)?
Q2 What are the most likely reasons for this Z value?
Comparison of Distributions of \(X\) and \(\overline{X}\)
Recall the quality control example of a soda factory from Lecture 8.
X is 1 measurement from 1 can from a normal distribution
\(X\sim N(12,0.4)\)
\(\overline{X}\) is the sample mean from 4 \((n=4)\) can measurements.
\(\overline{X}\sim N(12,\frac{0.4}{\sqrt{4}})\)
Sampling Distribution of the Sample Mean
The sample mean is the average of multiple measurements or observations which provides more information.
This increase in information translates to a more precise and more narrow normal distribution
- The size of the sample used to create the mean effects how precise the distribution is.
X is an observation from a normal distribution with mean, \(\mu\), and standard deviation sigma, \(\sigma\). X is normally distributed.
- \(X\sim N(\mu,\sigma)\)
\(\overline{X}\) is also normally distributed with mean, \(\mu\), standard deviation sigma divided by the square root of the sample size, \(\sigma/\sqrt{n}\)
- \(\overline{X}\sim N(\mu,\frac{\sigma}{\sqrt{n}})\)
Adjusting for this sample size adjustment is straightforward in the R commands we have covered.
What is the Central Limit Theorem (CLT)?
The CLT is comprised of a few related concepts (see video)
For MAS 261 there is one primary concept you need to know:
Often times we are sampling from a population that does not have a normal (bell-shaped) distribution.
We also may have no way of knowing how the population is distributed.
The Central Limit Theorem states that:
If our sample size is large enough, then the sampling distribution of the sample mean is NORMAL, even if the population distribution is not normal or is unknown.
A sample size of 30 more is sufficient no matter how the original population is distributed.
Central Limit Theorem Visual Demo
Central Limit Theorem Visual Demo
Central Limit Theorem Visual Demo
Central Limit Theorem Visual Demo
Central Limit Theorem Visual Demo
Central Limit Theorem Visual Demo
Central Limit Theorem Visual Demo
A Short (6 min.) Good Video About the CLT
I reviewed quite a few online resources to help explain the CLT.
This video by Dr. Nic of the Statistics Learning Center is excellent and she describes the concepts using DRAGONS.
It also reviews some of the previous concepts we have discussed.
Lecture 9 In-class Exercises - Q3
Today we discussed that even if our population distribution is non-normal, or unknown, our sample mean IS normally distributed if our sample size is at least 30
What if the population IS Normal? What is the minimum sample size to ensure that our sample mean is ALSO normally distributed?
A. 30
B. 20
C. 10
D. 5
E. 1
Lecture 9 In-class Exercises - Q4
The name of the helpful concept that we learned today is
A. The central theorem
B. The center limiting theorem
C. The central limit theorem
D. The center limit theory
E. The center limit thingy
F. All of the above are correct.
Introduction of practice Questions
Test 1 is on October 1st and it will be 50 minutes long.
I have posted a new R Project with two sets of practice questions.
Part 2includes multi-part questions similar to what you will see on Test 1.We will use these questions to guide and augment our review on Thursday.
If there is material we don’t cover in lecture, I create videos to assist with the questions we don’t get to, that will be posted by Saturday at the latest.
Key Points from Today
Central Limit Theorem
- Regardless of the shape of population distribution, the sampling distribution of the sample mean will be normal if the sample size is at least 30 (\(n \geq 30\)).
Sampling distribution of the sample mean, \(\overline{X}\)
Although we only have one sample mean, there many possible sample means, and all of those possible means form a distribution.
The mean, \(\mu\), is the same as the mean of the population.
The standard deviation, \(\sigma_{\overline{X}}\), is the population standard deviation divided by the square root of the sample size, \(\sigma_{\overline{X}} = \frac{\sigma}{\sqrt{n}}\).
To submit an Engagement Question or Comment about material from Lecture 9: Submit it by midnight today (day of lecture).