Chapter 18: Sampling Distribution Models

Size: px

Start display at page:

Download "Chapter 18: Sampling Distribution Models"

Prudence King
5 years ago
Views:

1 Chater 18: Samlig Distributio Models This is the last bit of theory before we get back to real-world methods. Samlig Distributios: The Big Idea Take a samle ad summarize it with a statistic. Now take aother samle will you get the same value for the statistic? Of course ot! Every samle is differet. Thus, every time we take a samle, we get a differet (radom) value of the statistic. Thus, you ca thik of a statistic as a radom variable! This is a icredibly imortat idea. Of course, oce you thik of a statistic as a radom variable, there is a logical et questio: what do we kow about the distributio (ceter, sread, shae) of that variable? The distributio of a statistic is called a samlig distributio (because the value of the statistic comes from samlig). I this chater, we ll derive the distributios of the two rimary statistics roortios, ad meas. Proortios Lookig at the Biomial A few chaters ago, I metioed that the shae of the biomial distributio deeded o two thigs ow it s time to actually do it. There are two arameters i a biomial model ad. I m goig to geerate histograms of biomial radom variables for lots of differet values of ad first u, a fied value of ad various values of. The Effect of I ll set the samle size at 10. First u: 0.5. Biomial Distributio (=10,=0.5) What does that look like? It s defiitely symmetric Now let 0.4. HOLLOMAN S AP STATISTICS BVD CHAPTER 18, PAGE 1 OF 15

2 Biomial Distributio (=10,=0.4) Let s try 0.2. Biomial Distributio (=10,=0.2) How about 0.1? HOLLOMAN S AP STATISTICS BVD CHAPTER 18, PAGE 2 OF 15

3 Biomial Distributio (=10,=0.1) Let s go the other way: 0.6. Biomial Distributio (=10,=0.6) Now 0.8. HOLLOMAN S AP STATISTICS BVD CHAPTER 18, PAGE 3 OF 15

4 Biomial Distributio (=10,=0.8) Fially, 0.9. Biomial Distributio (=10,=0.9) For a fied samle size, the closer the arameter is to 0.5, the more symmetric the biomial distributio will be. The farther the arameter is from 0.5, the more skewed the biomial distributio will be. The Effect of Now let s set the arameter at a fairly etreme value say, 0.1 ad see what haes as the samle size icreases. We ll start with 20. HOLLOMAN S AP STATISTICS BVD CHAPTER 18, PAGE 4 OF 15

5 Biomial Distributio (=20,=0.1) How about 40? Biomial Distributio (=40,=0.1) Let s look at 60. HOLLOMAN S AP STATISTICS BVD CHAPTER 18, PAGE 5 OF 15

6 Biomial Distributio (=60,=0.1) Oe more 80. Biomial Distributio (=80,=0.1) The effect seems fairly clear the larger the samle size, the closer the biomial (ad the samlig) distributio gets to ormal. Is t that iterestig! The Bottom Lie So the larger is, the more ormal the biomial distributio will be (regardless of ), ad the closer is to 0.5, the more symmetric (ad maybe eve ormal) the biomial distributio will be. Here s how we re goig to ut that ito a ice rule: the biomial distributio will be aroimately ormal whe 10 ad These rules force to be large eough ad to be close eough to 0.5 to make the biomial sufficietly close to the ormal. HOLLOMAN S AP STATISTICS BVD CHAPTER 18, PAGE 6 OF 15

7 Chagig Couts to Proortios First of all, we eed to get our heads straight about where roortios come from. Let s say that we re lookig at blue M&M s i a bag. How do you determie the roortio of blue M&M s? That s right cout the umber of blue, ad divide by the umber i the bag. Divide a cout by the total ad you get a roortio. I symbols: ˆ. The is the biomial cout, ad the is the fied umber of trials (or samle size). Now let s do the same thig to the ceter ad sread of a biomial to fid the ceter ad sread for roortios! The Samlig Distributio of Ceter Recall that b a biomial radom variable? 1. That is biomial what s the mea of b. Thus, 1 Yes! If is biomial the Sread Recall that b 1., so b. Thus,. The variace of a biomial radom This is the variace; we really wat stadard deviatio, so take the square root! 2 2 variable is 1. That makes 1 1. Proortios: A Summary The distributio of the samle roortio ( ) will have mea, stadard deviatio 1, ad a aroimately ormal shae (rovided that 10 ad 1 10). So What? Now that we have some iformatio about the distributio of, we ca do robability roblems ivolvig samle roortios which is what I ll do with the followig eamles. HOLLOMAN S AP STATISTICS BVD CHAPTER 18, PAGE 7 OF 15

Distribution of Sample Proportions

Distribution of Sample Proportions Distributio of Samle Proortios Probability ad statistics Aswers & Teacher Notes TI-Nsire Ivestigatio Studet 90 mi 7 8 9 10 11 12 Itroductio From revious activity: This activity assumes kowledge of the