1 / 25

Using Simulations to understand the Central Limit Theorem

Using Simulations to understand the Central Limit Theorem. Parameter : A number describing a characteristic of the population ( usually unknown ). The mean gas price of regular gasoline for all gas stations in Maryland . The mean gas price in Maryland is $______.

Download Presentation

Using Simulations to understand the Central Limit Theorem

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Using Simulations to understand the Central Limit Theorem

  2. Parameter: A number describing a characteristic of the population (usually unknown) The mean gas price of regular gasoline for all gas stations in Maryland

  3. The mean gas price in Maryland is $______ Statistic: A number describing a characteristic of a sample.

  4. In Inferential Statisticswe use the value of a sample statistic to estimate a parameter value.

  5. We want to estimate the mean height of MC students. The mean height of MC students is 64 inches

  6. Will x-bar be equal to mu? What if we get another sample, will x-bar be the same? How much does x-bar vary from sample to sample? By how much will x-bar differ from mu? How do we investigate the behavior of x-bar?

  7. What does the x-bar distribution look like?

  8. Graph the x-bar distribution, describe the shape and find the mean and standard deviation

  9. Simulation Rolling a fair die and recording the outcome randInt(1,6) Press MATH Go to PRB Select 5: randInt(1,6)

  10. Rolling a die n times and finding the mean of the outcomes. Let n = 2 and think on the range of the x-bar distribution What if n is 10? Think on the range Mean(randInt(1,6,10) Press 2nd STAT[list] Right to MATH Select 3:mean( Press MATH Right to PRB 5:randInt(

  11. Rolling a die n times and finding the mean of the outcomes. The Central Limit Theorem in action

  12. The Central Limit Theorem in action

  13. For the larger sample sizes, most of the x-bar values are quite close • to the mean of the parent population mu. • (Theoretical distribution in this case) • This is the effect of averaging • When n is small, a single unusual x value can result in an x-bar value far from the center • With a larger sample size, any unusual x values, when averaged • with the other sample values, still tend to yield an x-bar value close to mu. • AGAIN, an x-bar based on a large will tends to be closer to mu than will an x-bar based on a small sample. This is why the shape of the x-bar distribution becomes more bell shaped as the sample size gets larger.

  14. Normal Distributions

  15. The Central Limit Theorem in action Closing stock prices ($) Variability of sample means for samples of size 64 26 – 2.526 + 2.5 26 + 2*2.5 __|________|________|________X________|________|________|__ 18.5 21 23.5 26 28.5 31 33.5

  16. Closing stock prices ($) Variability of sample means for samples of size 64 2.5% | 95% | 2.5% 26 – 2.5 26 + 2.5 26 + 2*2.5 __|________|________|________X________|________|________|__ 18.5 21 23.5 26 28.5 31 33.5 About 95% of samples of 64 closing stock prices have means that are within $5 of the population mean mu About 99.7% of samples of 64 closing stock prices have means that are within $7.50 of the population mean mu

  17. We want to estimate the mean closing price of stocks by using a SRS of 64 stocks. Assume the standard deviation σ = $20. X ~Right Skewed (μ = ?, σ = 20) __|________|________|________X________|________|________|__ μ-7.5 μ-5 μ-2.5 μμ+2.5 μ+5 μ+7.5 We’ll be 95% confident that our estimate is within $5 from the population mean mu We’ll be 99.7% confident that our estimate is within $7.50 from the population mean mu

  18. Simulation Roll a die 5 times and record the number of ONES obtained: randInt(1,6,5) Press MATH Go to PRB Select 5: randInt(1,6,5)

  19. Roll a die 5 times, record the number of ONES obtained. Do the process n times and find the mean number of ONES obtained. The Central Limit Theorem in action

  20. Use website APPLETS to simulate proportion problems

More Related