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DatStat UNIT 4-B Schedule (Topics 16-18) Day 1: ________ Journal U4.B1 Topic 16: Sampling Distributions: Proportions SWBAT: Understand how Sampling Distributions of Sample Proportions( pˆ ), behave. Define the Central Limit Theorem for pˆ . Activity: Reese’s Pieces HW 3: p. 358 #16 – 6 , 7, 9 (a,b,c only) Day 2: ________ Journal U4.B2 Topic 17: Sampling Distributions: Means SWBAT: Understand how Sampling Distributions of Sample Means ( x ), behave. Define the Central Limit Theorem for x . Activity: Pennies HW 4: p. 382 # 17-9,10 Day 4: ________ Journal U4.B3 Topic 18: The Central Limit Theorem SWBAT: define the CLT and decide when the CLT applies to Sampling Distributions of Sample Means and Sample Proportions. HW 5: p 396 # 18-6, 7, 11 Day 6: Journal U4.B4 REVIEW DAY Day 7: QUIZ UNIT 4B (Topics 16-18) _______ 1 Topic 16: The Sampling Distribution of Proportions Sampling Distribution: _____________________________________________(DUH!) Value: Population = Parameter Sample = Statistic Mean Standard Deviation Percent (Proportion) Size Activity 16-2: I will give you a Sample of 25 Reese’s Pieces candies from a Population of the candies in a bag. I want you to make a count and proportion of each color. Only then may you eat them! Orange Yellow Brown Count Proportion(%) a) Is the proportion of Orange candies in your sample a parameter or a statistic? _______ What symbol is used to denote it?______ b) Is the proportion of Orange candies in the teacher’s bag a parameter or a statistic? ___ What symbol is used to denote it?______ c) Do you know what the population proportion of Orange candies is from the bag? ____ d) How would you estimate that proportion? e) We will build a distribution of the samples of the proportion of Orange candies. 0 .04 .08 .12 .16 .20 .24 .28 .32 .36 .40 .44 .48 .52 .56 .60 .64 .68 .72 .76 .80 f) How would you describe this SAMPLING DISTRIBUTION (this class SUCS)? g) Between what two values do the middle 95% of all the samples fall? 2 Sample Proportions: 1. Log onto the website: http://www.rossmanchance.com/applets/ Select “Reeses Pieces” 2. Choose “Sample Size” to be 10. 3. Hit the “Draw Samples” button. How many candies were selected (n) ? ________ What portion was orange ( )? __________. Repeat at least 5 times. 4. Undo the “Animate” box. Choose “Number of Samples” to be 10. Click “Draw Samples” at least 20 times. 5. Describe what you see in the dot plot. Shape:________________________________________ Center:_______________________________________ Spread:_______________________________________ Unusual Points: _________________________________ 6. Choose “Sample Size” to be 25. Select the “Animate” box. Choose “Number of Samples” to be 1. 7. Hit the “Draw Samples” button. How many candies were selected (n) ? ________ What portion was orange ( )? __________. Repeat at least 5 times. 8. Undo the “Animate” box. Choose “Number of Samples” to be 10. Click “Draw Samples” at least 20 times. 9. Describe what you see in the dot plot. Shape:________________________________________ Center:_______________________________________ Spread:_______________________________________ Unusual Points: _________________________________ 10. What did we learn about the sampling distributions of sample proportions as n gets bigger? Talk about how shape, center, and spread changes. _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ ____________________________________________________ The Central Limit Theorem: If n is large (usually above 30), the sampling distributions of the sample means and proportions are distributed Normally. They are centered at µ and p respectively. 3 CENTRAL LIMIT THEOREM (CLT) for SAMPLE PROPORTIONS: If we take a SRS of size n from a large population, then the sampling distribution of the pˆ 's is ___________________, with pˆ p When does this happen? pˆ and np ≥ 10 and p(1 p) n n(1 – p ) ≥ 10 Ex1) Find the probability of getting a sample proportion of Orange candies of size 25 that is greater than 50%. Ex2) Find the probability of getting a sample proportion of Orange candies of size 100 that is greater than 50%. Ex3) Bonus Challenge (for the rest of the bag!): Without your calculators, predict the standard deviation of the sampling distribution of the sample props pˆ , for n = 400 . 4 Today, your goal is to learn about how SAMPLE MEANS behave. So what is a sampling distribution? It’s a distribution of samples Under score ( “Shift” then “ – “ Sample Means: ) 1) Log onto the website: http://onlinestatbook.com/stat_sim/sampling_dist/index.html 2) On the left hand side, click the “Begin” button. 3) What do you see? Note below: n = 5 . Describe the shape of the distribution:______________________________ What is the mean of the “Parent Population”? ___________ What statistical symbol should be used for the mean?_____________ What is the Standard Deviation? ___________________ What statistical symbol should be used for the std. deviation?_____________ 4) Now hit the “Animated” button. What happened? Repeat at least 5 times. 5) What does the little blue “chunks” represent? __________________________________ 6) Now click the “5” button. You are taking 5 samples at a time. Repeat at least 10 times and watch it grow. 7) Now click the “1,000” button. You just took 1,000 samples from the population. 8) Compare the Blue Sampling Distribution with the Black Parent Population. What happened to Spread? __________________________________________ What happened to the mean? ________________________________________ 9) In the drop-down menu at the top right (it started with Normal), Choose “Skewed”. Choose n=25 below for the “Distribution of Means”. 10) Now hit the “Animated” button. What happened? Repeat at least 3 times. 11) Now click the “5” button. You are taking 5 samples at a time. Repeat at least 10 times and watch it grow. 12) Now click the “1,000” button. You just took 1,000 samples from the population. 13) Compare the Blue Sampling Distribution with the Black Parent Population. What happened to Spread? __________________________________________ What happened to the mean? ________________________________________ 14) Try clicking the “Fit Normal” box. What happened? Do you think sample means behave normally? 15) In the drop-down menu at the top right (it started with Normal), Choose “Custom”. Click and drag your cursor over the histogram and create your own distribution. Repeat the steps above. 16) What can we conclude about the distribution of Sample Means? Talk about shape, center, and spread in relation to the parent population. ______________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ 5 Topic 17: Sample Means Activity 17-1: PENNIES In this activity, you will be charting the age of pennies. For example, how old is a penny from 2001? We will chart the ages from the entire population. Then, I will ask you to start sampling from the population. We will look at the sampling distribution of sample means ( x ) of size n = 10 . Make a rough sketch of each distribution with a smooth curve Population: sampling distribution of ( x ) of size n = 10 a) Describe the shape of the population: ______________________________________ b) Guesstimate µ, the population mean: ______________________ c) Describe the shape of the SamplingDistribution: _____________________________ d) Now guesstimate µ, the population mean: ______________________ Which guess, b or d, do you feel more confident in?____ e) Describe how the distribution changed (SUCS) from n = 1(the population) to n = 10 (the distribution of x ) . Population (n = 1) Sampling Dist. (n = 10) Shape Unusual Points Center Spread CENTRAL LIMIT THEOREM (CLT) for SAMPLE MEANS: If we take a SRS of size n from a large population, then the sampling distribution of the x ’s is ___________________, with x When does this happen? x n and When n ≥ 30 6 Ex 1) Starbucks claims that a pour of a “Venti” coffee drink is Normally distributed and will average 20 oz. They admit that there is some variation in pours and that the .5oz for their population of “Venti” coffees. What is the probability that my next coffee is less than 19.3 oz.? Ex2) If I take and measure 15 coffees, what is the probability that the sample mean of these coffees is less than 19.3 oz.? Ex 3) If I take and measure 30 coffees, what is the probability that the sample mean of these coffees is less than 19.3 oz.? Bonus: What statistical principal is being displayed by examples 1,2,3 (What happens to the probabilities as n increases). Describe in your own words what the CLT says about sample means and proportions (Hint: Talk about how they are distributed and what happens to SPREAD as n increases). 7 Topic 18: The CLT Activity 18-1 (p. 387): The CDC (Center for Disease Control and Prevention) reported that 22.9% of all American adults smoke regularly in 1998. a) Is .229 a parameter or a statistic? What symbol should be used to denote it? ________ b) We want to see if that value has changed. Suppose you take a SRS of 100 American adults and ask whether they smoke. What do we know about the distribution of this sample proportion? (Talk about Shape, Center and Spread) c) We found that 19 of our 100 adults were actually smokers. What is the probability that we find a sample as low or lower than the one we obtained? d) Interpret your answer from part (c). Does this constitute strong evidence that the proportion of smokers is becoming smaller? e) The CDC is encouraged by the results from ( c and d ). So now they hire you to perform an extensive survey to determine if the proportion of smokers is actually becoming smaller. They pay you to randomly sample 2500 American adults. You found that 470 indicated that they smoke regularly. What is the probability that we find a sample as low or lower than the one we obtained? f) They now want you to estimate the true proportion of American adult smokers. You want to be 95% sure that you create a window (min to a max) that will catch the true proportion. What is your minimum and maximum estimates? (Hint: 68-95-99.7% Rule) 8 Activity 18-3 (p.391) BigBar Candy Bar Company advertises that their famous BigBar weighs 2.13 oz. Their website claims that the distribution of BigBar weights is Normal with a mean of 2.2 oz and a standard deviation of .04 oz. a) What is the probability that you purchase a BigBar and get cheated (less than the advertised weight)?! b) A box of BigBars contains 50 BigBars. What can we say about the distribution of the sample mean weight of 50 BigBars? c) What is the probability that you purchase a box of 50 BigBars and the average weight is less than the advertised weight of 2.13? d) Now draw the two distributions (1 single BigBar and the average of 50 BigBars) on top of eachother. Shade the proportion that gets “cheated”. e) What does your drawing indicate about the probability of a value getting far away from the mean as sample size (n) increases? f) Suppose your box of 50 BigBars had an average weight of 2.12 oz. What would you say about the true weight of the BigBar? 9