Sampling Distribution of Differences

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Jeffry Ford
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1 . Chapter 0 Samplig Distributio of Differeces Topic covers the distributio of a differece betwee two idepedet sample proportios or two idepedet sample meas. Topic Samplig Distributio of a Differece Betwee Two Idepedet Sample Proportios or Two Idepedet Sample Meas (Simulatios) Samplig Distributio of a Differece Betwee Two Idepedet Sample Proportios This simulatio will reiforce the mathematical reality that the mea of the distributio of p p is p p (the differece of the two populatio proportios) ad the variace p p p = + p = p ( p ) p( p) + (the sum of the two populatio variaces) ad thus p( p) p( p) ^ p ^ p = +. ^ ^ Note: I later topics it will be hypothesized that p = p (or p - p = 0), so you will look at that case here. The differece eed ot be zero, however, ad i the ext sectio you will observe such a example. 00 TEXAS INSTRUMENTS INCORPORATED

2 30 ADVANCED PLACEMENT STATISTICS WITH THE TI-89 Example: = 50, p = 0.33, ad = 35, p = Oe hudred samples from populatio oe were geerated i Topic 9, screes 5, 6, ad 7 ad saved i list. Chage to folder RACE, sice you will be usig some data from Topic 9.. From the Home scree, set RadSeed Calculate tistat.radbi(50,.33,00)/50!list. 3. Press to display scree. 4. Use ½ to calculate mea(list) mea =.39 p = Use ½ to calculate stddev(list) stddev = s x = p( p) (. 33)(. 67) ^ p = = = () 6. Press ± Z ad the press to display the third lie i scree. 7. Use ½ to calculate variace(list). (See scree.) p( p) variace = s x.004 ^ p = = Compare with Topic 9, scree 7. Simulate 00 samples from populatio two ad store i list, similar to Topic 9 ad show i screes 3 ad 4.. Set RadSeed Calculate tistat.radbi(35,.33,00)/35!list. 3. Press to display scree 3. () 4. Use ½ to calculate mea(list) mea = p = Use ½ to calculate stddev(list) p ( p ) (.33)(.67) stddev = p m = = = (3) 00 TEXAS INSTRUMENTS INCORPORATED

CHAPTER 0: SAMPLING DISTRIBUTION OF DIFFERENCES 3 6. Use ½ to calculate variace(list) p( p) (.33)(.67) variace =.00639 m = = =.00637 p 35 (See scree 4.) (4) 7. Store the differece usig list list!

3 CHAPTER 0: SAMPLING DISTRIBUTION OF DIFFERENCES 3 6. Use ½ to calculate variace(list) p( p) (.33)(.67) variace = m = = = p 35 (See scree 4.) (4) 7. Store the differece usig list list!list3 ad fid the mea, stadard deviatio, ad variace of this distributio of differeces (scree 5). Note that list3[] = list[] list[] = =.0857 from screes, 3, ad 5. (5) 8. Calculate the mea, stddev, ad variace of list3. mea = µ ^ p ^ p =. 00 = p p = p( p) p( p) (. 33)(. 67) (. 33)(. 67) stddev = ^ p ^ p =. 04 = + = p( p) p( p) (. 33)(. 67) (. 33)(. 67) variace =.0090 ^ p ^ p =. 007 = + = (See scree 5). It is importat to ote that it is ot true that ^ ^ = ^ + ^. = =. 5 = + ^ ^ ^ ^ p p p p O the other had it is true that = +. ^ ^ ^ ^ p p p p ^ ^ =. 0 = = ^ + ^ p p p p p p p p 9. From the Stats/List Editor, set up ad defie Plot, Plot, ad Plot 3 as modified boxplots with Mark: Square, usig list, list, ad list3, respectively. 0. From the Plot Setup scree, press ZoomData ad Trace, which graphically shows the distributio of p s at the top, p s i the middle with the same mea as p s but with greater spread, ad the the distributio of p p s at the bottom cetered at zero with still greater spread (scree 6). (6) 00 TEXAS INSTRUMENTS INCORPORATED

4 3 ADVANCED PLACEMENT STATISTICS WITH THE TI-89. From the Stats/List Editor, press Plots ad select 3:Plotsoff to tur off the plots.. Press Plots ad :Norm Prob Plot to defie Plot 4 usig list3 with Mark: Dot. 3. Press to retur to the Stats/List Editor. 4. Select Plots, :Plot Setup ad Zoom Data to get a probability plot o the distributio of the differeces i list3 (scree 7). Observe that the distributio of the differeces also is ormally distributed because the sample size is large eough. (7) Samplig Distributio of a Differece Betwee Two Idepedet Sample Meas The simulatio will reiforce the fact that µ x x = µ µ ad the variace x x = x + x = +, ad the Stadard Deviatio x x = +. Simulate 00 samples of size 9 from a ormal populatio with µ = 85 ad = 5 ad store i list (similar to Topic 0, scree 7, but i that case it was a uiform distributio). For this sectio, chage to folder BLDTALL.. From the Home scree, set RadSeed 3.. Calculate seq(mea(tistat.radorm(85,5,9)),x,,00)!list (secod lie of scree 8). (8) Simulate 00 samples of size 6 from a ormal populatio with µ = 60 ad = 0 ad store i list.. Calculate seq(mea(tistat.radorm(60,0,6)),x,,00)!list (third lie of scree 8). 00 TEXAS INSTRUMENTS INCORPORATED

CHAPTER 0: SAMPLING DISTRIBUTION OF DIFFERENCES 33. Store the differeces usig list list!list3 (top of scree 9). (9) 3.

85 = µ = µ x mea(list) = 60.003 60 = µ = µ x mea(list3) = 4.97 5 = µ µ = µ x x (0) Note: stddev(list3) does ot equal + = 5+ 5=0.

= x = + x variace(list) = 5.5659 5 = 5 = = 9 x variace(list) = 8.9005 5 = 0 = = x 6 () x x x x variace(list3) = 55.

5 CHAPTER 0: SAMPLING DISTRIBUTION OF DIFFERENCES 33. Store the differeces usig list list!list3 (top of scree 9). (9) 3. Fid the mea, stadard deviatio, ad variace of list, list, ad list3 (screes 9, 0, ad ). Note that: mea(list) = = µ = µ x mea(list) = = µ = µ x mea(list3) = = µ µ = µ x x (0) Note: stddev(list3) does ot equal + = 5+ 5=0. stddev(list) = = stddev(list) = = 5 = = 9 x 0 = = 6 x stddev(list3) = = = x = + x variace(list) = = 5 = = 9 x variace(list) = = 0 = = x 6 () x x x x variace(list3) = = + = + = 4. From the Stats/List Editor, set up ad defie Plot, Plot, ad Plot 3 as modified boxplots with Mark: Square, usig list, list, ad list3. 5. Deselect all other plots. 00 TEXAS INSTRUMENTS INCORPORATED

6 34 ADVANCED PLACEMENT STATISTICS WITH THE TI From the Plot Setup scree, press ZoomData to display scree with list at the top, list i the middle, ad the distributio of the differece with the greater spread at the bottom. 7. For aother look at the results, set up ad defie Plot, Plot, ad Plot 3 as histograms with x: list, list, ad list3, ad Hist. Bucket Width: 4. () 8. Set up the widow usig $ with the followig etries: xmi = 3 xmax = 99 xscl = 4 (3) ymi = -5 ymax = 45 yscl = 0 xres = (See scree 3.) 9. Press % (scree 4). (4) 00 TEXAS INSTRUMENTS INCORPORATED

Parameter, Statistic and Random Samples

Parameter, Statistic and Random Samples Parameter, Statistic ad Radom Samples A parameter is a umber that describes the populatio. It is a fixed umber, but i practice we do ot kow its value. A statistic is a fuctio of the sample data, i.e.,