Statistical Reasoning for everyday life Intro to Probability and Statistics Mr. Spering – Room 117

5.3 The Central Limit Theorem Is it Normal?  Times in the 100-meter dash…  YES--NORMAL  The weights of new Casino Dice…  NO--UNIFORM  Number of candy bars in a Twix package…  NO--UNIFORM  Heights of Sequoia trees…  YES--NORMAL  Weights of 5000 randomly selected Boston Terriers…  YES--NORMAL

5.3 The Central Limit Theorem  THE CENTRAL LIMIT THEOREM When we take many random samples of size n for a variable with any distribution and record the distribution of the means of each sample. Then, 1. The distribution of means will be approximately a normal distribution for large sample sizes. 2. The mean of the distribution of means approaches the population mean, μ, for large sample sizes. 3. The standard deviation of the distribution of means approaches σ/√n for large sample sizes, where σ is the standard deviation of the population. practical purposes--the distribution will need more than 30 valesccccccp… Ooops, I think I feel asleep…> 30 values. Recall, If normal then the rule holds true.

5.3 The Central Limit Theorem  Summary:  Within the central limit theorem, we always start with a particular variable, such as the outcomes of a die roll or weights of people, that varies randomly over a population. The variable has a certain mean, μ, and standard deviation, σ, which we may or may not know. This variable can have any sort of distribution. When we take many samples of that variable, with n items on each sample, and make a histogram of the means from the many samples, we will see a distribution that is close to “normal”. The larger the sample size, n, the more closely the distribution of means approximates a normal distribution. Check out this website on rolling dice & the C.L.T

5.3 The Central Limit Theorem VISUAL EXAMPLE: Pictures of a distribution being "smoothed out" by summation (showing original density of distribution and three subsequent summations, obtained by convolution “averaging” of density functions)summationdensity of distributionconvolution Works Cited: ntral_limit_theorem More visuals: Page 218 TheCentralLimitTheorem.nbp

5.3 The Central Limit Theorem EXAMPLE:  Suppose you are a principal of a middle school and your 100 eighth-graders are about to take a national standardized test. (sound familiar?) The test is designed so that the mean score is μ = 400 with a standard of deviation σ = 70. Assume the scores are normally distributed.  A) What is the likelihood that one of your eighth-graders, selected at random, will score below 380 on the exam? How do we start? Percentage? Z-scores correspond to a percentage from the table 5.1 on page 232. Therefore, the standard score of 380 is is approximately 38 th percentile, or 38% of all students (about 4 out of 10).

5.3 The Central Limit Theorem EXAMPLE: B) Your performance as a principal depends on how well your entire group of eighth-graders scores on the exam. What is the likelihood that your group of 100 eighth-graders will have a mean score below 380? How do we start? Central Limit Theorem? According to the C.L.T. if we take random groups of say 100 students and study their means, then the means distribution will approach normal. Hence, the μ = 400 and its standard of deviation is according to the C.L.T. Therefore, the z-score for a mean of 380 is The percentile for the z-score on page 232 is 0.19 th percentile or , which is very small, about 1 in 500. EMPHASIS POINT: THERE IS MORE VARIATION IN THE SCORES OF INDIVIDUALS THAN IN THE MEANS OF GROUPS OF INDIVIDUALS

5.3 The Central Limit Theorem HOMEWORK: Pg 220 # 1-20 all Z-ScorePercentile ≈ -1.2 and th ≈ -0.7 and th ≈ and th ≈ 0.0 and th ≈ 0.30 and nd ≈ 0.65 and th ≈ 1.1 and th ≈ 3.5 and above99.98 th When listening to corn pop, you are hearing normal distribution and the C.L.T William A. Massey {Princeton}