class: middle background-image: url(data:image/png;base64,#LTU_logo.jpg) background-position: top left background-size: 30% # STM1001 [Topic 4](https://bookdown.org/content/88ef9b7c-5833-4a70-84f2-93470957d1f9/) Workshop ## Sampling Distributions ### La Trobe University This workshop complements the [Topic 4 readings](https://bookdown.org/content/88ef9b7c-5833-4a70-84f2-93470957d1f9/) --- # Topic 4: Sampling Distributions ## In this week's readings: We will not have time to cover every concept, so please make sure you read this topic's readings thoroughly. <iframe src="https://bookdown.org/content/88ef9b7c-5833-4a70-84f2-93470957d1f9/" width="100%" height="400px" data-external="1"></iframe> --- # Today's workshop Today, we will introduce the following topics: * Sampling * The sample mean * The Central Limit Theorem * The distribution of the sample mean We will learn how the distribution of an underlying population relates to the distribution of the sample mean --- name: stat class: middle background-image: url(data:image/png;base64,#slide_1.png) background-size: 110% --- name: stat class: middle background-image: url(data:image/png;base64,#slide_6.png) background-size: 100% --- name: stat class: middle background-image: url(data:image/png;base64,#slide_7.png) background-size: 100% --- # Group activity 1 As a group, and using a whiteboard, answer the following questions: 1. Make a list of each of your height in cm. If you do not know your exact height, an estimate will be fine 1. What is the average height of your group? (calculate the sample mean) 1. What is the sample standard deviation of heights in your group? (you can use [this calculator](https://www.gigacalculator.com/calculators/standard-deviation-calculator.php?embed=true) to calculate the sample standard deviation) 1. Draw a histogram of the heights represented by your group --- # Collating the results * We will now draw a histogram (or dot plot) of the sample means provided by each group * Comparing the histogram of sample means compared with your group's histogram of individual heights, which one contains more variability? --- # The Central Limit Theorem * It turns out that the distribution of the sample ***means*** is **less variable** than the distribution of the values for each ***individuals***. -- * If we know that the underlying distribution is Normal, we can assume that `\(\overline{X}\sim N\left(\mu,\frac{\sigma^2}{n}\right)\)` -- * Otherwise, the Central Limit Theorem may apply... --- # The Central Limit Theorem .content-box-blue[ .center[ **The Central Limit Theorem (CLT)** ] Let `\(X_1, \ldots, X_n\)` be a random sample from a distribution with finite mean `\(\mu\)` and finite variance `\(\sigma^2\)`. For `\(\overline{X}\)` denoting the sample mean, if the sample size `\(n\)` is sufficiently large then `$$\overline{X}\stackrel{\tiny \text{approx.}}\sim N\left(\mu,\frac{\sigma^2}{n}\right)$$` where `\(\stackrel{\tiny \text{approx.}}\sim\)` denotes 'approximately distributed as'. ] -- * For example, if the **variance** `\(\sigma^2\)` of the population was `\(9.85^2\)`cm, then the **variance** of the sample mean would be `\(\displaystyle \frac{9.85^2}{n}\)` * Equivalently, if the **standard deviation** `\(\sigma\)` of the population was `\(9.85\)`cm, then the **standard deviation** of the sample mean would be `\(\displaystyle \frac{9.85}{\sqrt{n}}\)` --- #Group activity 2 For this activity, assume that the underlying distribution of heights is Normal. 1. Using the sample standard deviation for your group, what is the estimated standard deviation of the distribution of the mean (also known as the standard error)? 2. You can check your answer by looking again at the results from the Standard Deviation Calculator used in the previous exercise. The answer will be displayed in the SEM row. Is the answer your group calculated in (1) the same as the SEM provided in the calculator's results? --- # Examples: Central Limit Theorem in practice .pull-left[ <img src="data:image/png;base64,#Topic_4_Workshop_files/figure-html/unnamed-chunk-2-1.svg" width="100%" style="display: block; margin: auto;" /> ] .pull-right[ * We can think of the green histogram as the histogram of heights you created for your group in the first activity {{content}} ] -- * We can think of the red histogram(s) as the histogram of means we collated together on the whiteboard {{content}} -- * The higher the sample size, the less spread out the distribution of the sample mean {{content}} -- * However, the mean itself always stays in the same place --- # Examples: Central Limit Theorem in practice .pull-left[ <img src="data:image/png;base64,#Topic_4_Workshop_files/figure-html/unnamed-chunk-3-1.svg" width="100%" style="display: block; margin: auto;" /> ] .pull-right[ * Another amazing fact: even if the underlying distribution is highly skewed, as long as the sample size is large (normally 30 or greater), the distribution of the means will still be approximately normal! ] --- # More details in the readings * For more detail, see [this week's readings](https://bookdown.org/content/88ef9b7c-5833-4a70-84f2-93470957d1f9/) * There are also important guidelines for determining the distribution of the sample mean based on three scenarios: 1. It is known that the underlying population distribution is normal 1. The underlying population distribution is not known, and the sample size is 30 or more 1. The underlying population distribution is not known, and the sample size is less than 30 * We also see how we can use the distribution of the sample mean to answer questions: a very important concept leading into Topic 5 --- background-image: url(data:image/png;base64,#computerlab.jpg) background-position: bottom background-size: 75% class: center # See you in the computer labs! Continue with this topic's readings: [Topic 4 Readings](https://bookdown.org/content/88ef9b7c-5833-4a70-84f2-93470957d1f9/) --- class: middle <font color = "grey"> These notes have been prepared by Amanda Shaker. The copyright for the material in these notes resides with the authors named above, with the Department of Mathematics and Statistics and with La Trobe University. Copyright in this work is vested in La Trobe University including all La Trobe University branding and naming. Unless otherwise stated, material within this work is licensed under a Creative Commons Attribution-Non Commercial-Non Derivatives License <a href = "https://creativecommons.org/licenses/by-nc-nd/4.0/" target="_blank"> BY-NC-ND. </a> </font>