Kelly's math stuff
Correlation Basics + CalculatorCalculator (to find a, b, r, r2):StatEditEnter x’s in L1 and y’s in L22nd QuitStat>Calc4:linregL1, L2Use Zoom 9 to view a scatter diagramRemember:y = all blankstatplots 1 (on) 2 & 3 (off)Output:r = linear correlation coefficientr2 = coefficient of determination (AKA proportion of variance)a = slope of regression lineb = intercept of regression lineTo turn “diagnostics” on and be able to see r and r22nd 0 (zero)D (the x-1 button)Diagnostics onEnter 2xPearson Correlation Critical Values16383006159500Proportion in ONE Tail.25.10.05.025.01.005Proportion in TWO TailsDF.50.20.10.05.02.011.7071.9511.9877.9969.9995.99992.5000.8000.9000.9500.9800.99003.4040.6870.8054.8783.9343.95874.3473.6084.7293.8114.8822.91725.3091.5509.6694.7545.8329.87456.2811.5067.6215.7067.7887.83437.2596.4716.5822.6664.7498.79778.2423.4428.5494.6319.7155.76469.2281.4187.5214.6021.6851.734810.2161.3981.4973.5760.6581.707911.2058.3802.4762.5529.6339.683512.1968.3646.4575.5324.6120.661413.1890.3507.4409.5140.5923.641114.1820.3383.4259.4973.5742.622615.1757.3271.4124.4821.5577.605516.1700.3170.4000.4683.5425.589717.1649.3077.3887.4555.5285.575118.1602.2992.3783.4438.5155.561419.1558.2914.3687.4329.5034.548720.1518.2841.3598.4227.4921.536821.1481.2774.3515.4132.4815.525622.1447.2711.3438.4044.4716.515123.1415.2653.3365.3961.4622.505224.1384.2598.3297.3882.4534.495825.1356.2546.3233.3809.4451.486926.1330.2497.3172.3739.4372.478527.1305.2451.3115.3673.4297.470528.1281.2407.3061.3610.4226.462929.1258.2366.3009.3550.4158.455630.1237.2327.2960.3494.4093.448731.1217.2289.2913.3440.4032.442132.1197.2254.2869.3388.3972.435733.1179.2220.2826.3338.3916.429634.1161.2187.2785.3291.3862.423835.1144.2156.2746.3246.3810.418236.1128.2126.2709.3202.3760.412837.1113.2097.2673.3160.3712.407638.1098.2070.2638.3120.3665.402639.1084.2043.2605.3081.3621.397840.1070.2018.2573.3044.3578.393241.1057.1993.2542.3008.3536.388742.1044.1970.2512.2973.3496.384343.1032.1947.2483.2940.3457.380144.1020.1925.2455.2907.3420.376145.1008.1903.2429.2876.3384.372146.0997.1883.2403.2845.3348.368347.0987.1863.2377.2816.3314.364648.0976.1843.2353.2787.3281.361049.0966.1825.2329.2759.3249.3575-762001231900050.0956.1806.2306.2732.3218.3542-7620012509500Proportion in ONE Tail.25.10.05.025.01.005Proportion in TWO TailsDF.50.20.10.05.02.0151.0947.1789.2284.2706.3188.350952.0938.1772.2262.2681.3158.347753.0929.1755.2241.2656.3129.344554.0920.1739.2221.2632.3102.341555.0912.1723.2201.2609.3074.338556.0904.1708.2181.2586.3048.335757.0896.1693.2162.2564.3022.332858.0888.1678.2144.2542.2997.330159.0880.1664.2126.2521.2972.327460.0873.1650.2108.2500.2948.324861.0866.1636.2091.2480.2925.322362.0858.1623.2075.2461.2902.319863.0852.1610.2058.2441.2880.317364.0845.1598.2042.2423.2858.315065.0838.1586.2027.2404.2837.312666.0832.1574.2012.2387.2816.310467.0826.1562.1997.2369.2796.308168.0820.1550.1982.2352.2776.306069.0814.1539.1968.2335.2756.303870.0808.1528.1954.2319.2737.301771.0802.1517.1940.2303.2718.299772.0796.1507.1927.2287.2700.297773.0791.1497.1914.2272.2682.295774.0786.1486.1901.2257.2664.293875.0780.1477.1888.2242.2647.291976.0775.1467.1876.2227.2630.290077.0770.1457.1864.2213.2613.288278.0765.1448.1852.2199.2597.286479.0760.1439.1841.2185.2581.284780.0755.1430.1829.2172.2565.283081.0751.1421.1818.2159.2550.281382.0746.1412.1807.2146.2535.279683.0742.1404.1796.2133.2520.278084.0737.1396.1786.2120.2505.276485.0733.1387.1775.2108.2491.274886.0728.1379.1765.2096.2477.273287.0724.1371.1755.2084.2463.271788.0720.1364.1745.2072.2449.270289.0716.1356.1735.2061.2435.268790.0712.1348.1726.2050.2422.267391.0708.1341.1716.2039.2409.265992.0704.1334.1707.2028.2396.264593.0700.1327.1698.2017.2384.263194.0697.1320.1689.2006.2371.261795.0693.1313.1680.1996.2359.260496.0689.1306.1671.1986.2347.259197.0686.1299.1663.1975.2335.257898.0682.1292.1654.1966.2324.256599.0679.1286.1646.1956.2312.2552100.0675.1279.1638.1946.2301.2540733361624688800072866252554605714375025546057143750270700570104002707005701040028473406867525290703068199003059430555307530568906048375255460555054502907030563880029521155941060280225555530752802255576262529070305638800280225558102502754630563880026498555941060264985557626252599690589597524688804286250255460503924300296418045053253056890404114030594304448175284734044005502554605428625030118054152900255460539693852847340428625028022553924300259969031337252952115306705030568902886075305689030194252907030276225029070300028860752906396003019425274510600282892527546300027146252799715271462525996901638300301180518478503164205172402530118051581150290449015240002754630139065027070051390650255460586201252553971084124812752090831532528594058208010296417908010525316420540005031642052476503011805952502859405-571502707005-209550255460568192651783080Linear relationship020000Linear relationship857241849755Linear relationship020000Linear relationship31337251783080NOT a linear relationship020000NOT a linear relationship55530754221480Critical Value = 0.4438020000Critical Value = 0.443810191754135755Critical Value = –0.4438020000Critical Value = –0.4438-114300220980Suppose n = 20, DF = 18The chart value is 0.443800Suppose n = 20, DF = 18The chart value is 0.44386410325185864520383501783079794385032689806667500326898053244753259455382905032689802533650326898012096753259455120967523831552533650238315538290502383155532447523641056667500236410579438502383155-2762253259455-2762252364105-27622533547050-2762263507105r = – 1r = – 0.7r = – 0.3r = 0r = 0.3r = 0.7r = 1020000r = – 1r = – 0.7r = – 0.3r = 0r = 0.3r = 0.7r = 1Example One: Perfect Positive Linear Relationship321310087630Fahrenheit020000Fahrenheit 35941001473200Celsius025303750100Fahrenheit32778698.6122212 5511800347980Celsius020000Celsiusa =360680096520b =r2 =r =Example Two: Perfect Negative Linear Relationship10 question quiz321310087630Wrong020000Wrong 35941001473200Right01256810Wrong10985420 5511800347980Right020000Righta =360680096520b =r2 =r =Example 3: Not a “perfect” linear relationshipName of CerealSugar (g)Calories /servingCocoa Puffs10103Apple Jacks10100Pops9120Fruity Pebbles9110Raisin Bran Crunch19190Honey Nut Crunch9105Lucky Charms10103Marshmallow Alpha-Bits10115Kix3107Honeycomb10127Fruitloops12109Special K9111Frosted Flakes11111Cheerios1106Rice Krispies3126Cap’n Crunch w/ Crunch Berries11103Frosted Mini Wheats Big Bites12203Corn Flakes2100Cinnamon Jacks 9111Cinnamon Toast Crunch 9127What is the linear correlation between grams of sugar and calories/serving?Are grams of sugar per serving and calories per serving linearly related?What proportion of variance is explained between these variables?What is the predicted amount of calories for a cereal with 10 grams of sugar per serving?Example 4: Not a “perfect” linear relationshipData were collected for 17 countries on two variables, deaths from coronary heart disease per 100,000 and annual per capita cigarette consumption. Per CapitaDeaths fromCigaretteCoronary Heart DiseaseCountryConsumption(per 100,000) United States3900259.9Canada3350211.6Australia3220238.1New Zealand3220211.8United Kingdom2790194.1Ireland2770187.3Iceland2290110.5Finland2160233.1West Germany1890150.3Netherlands1810124.7Austria1770182.1Belgium1700118.1Mexico1680 31.9Italy1510114.3Sweden1270126.9Spain1200 43.9Norway1090136.3What is the linear correlation between these two variables? Does this indicate a significant linear relationship ( = .05)What does the coefficient of determination indicate about the relationship? Use the data above to predict the number of deaths (per 100,000) in a country with a per capita consumption of cigarettes of 2000.Example 5: Not LinearUse the data below to determine whether a person’s body mass index islinearly related to their weight. BMI2628222725242321Weight150140125189200165120105What is the linear correlation?Does this indicate a linear relationship?What does the coefficient of determination say about the variance explained? Predict the weight of a person whose BMI is 22.What if the weight was measured in ounces? How does it affect the outputs?How would changing the weight measurement to ounces affect the correlation (higher, lower, same)?Switch which variable is x and which is y. How does that change the calculator outputs?Correlation: Conceptual ReviewWhich of the following correlations would indicate the strongest linear relationship?r = 0.001b. r = -0.953 c. r = 0.555d. r = 0.876Match the variables to the most likely correlation value:x = Weight in poundsy = Weight in ouncesr = -0.777x = High school GPAr = 1.0y = College GPAx = number of late paymentsr = 0.653y = credit scoreTom and Mary each did a correlation project relating weight and blood pressure. Tom measured weight in pounds and Mary measured weight in ounces. How would their correlation coefficients relate?Tom’s would be higherTom’s would be lowerThey would both be about the sameCan’t be determined, sometimes higher, sometimes lowerIn the case when two variables x and y are linearly related, what is the best way to predict an outcome for the variable y knowing x?In the case when two variables x and y are NOT linearly related, what is the best way to predict an outcome for the variable y knowing x?What if the effect of switching which variable is x and which is y?a, b, r, and r2 all switch signsa, b, r, and r2 all remain the samea, b, remain the same and r, and r2 changea, b, change and r and r2 remain the sameCorrelation PracticeA group of husbands and wives were asked to rank the taste of 10 Lean Cuisine meals. Their rankings are as follows:PlayerABCDEFGHIJHus.11029384756Wife29183105647What is the linear relationship between the ratings?At the level, are the rankings of husbands and wives linearly related?What is the proportion of variance explained?Predict the wife’s ranking for a husband’s ranking of 5.Use the data below to determine whether a person’s body mass is significantly related to their neck size. Neck12.51211.21313.512.711.511Weight150140125189200165120105What is the linear relationship between neck size and weight?At the level, are they linearly related?What is the proportion of variance explained?Predict the weight for a neck size of 12.Project One (10 points)Use a sample size of about 20Use a significance level of 0.05Task: Collect real data for about 20 sample items. Collect information on two variables of your choosing. Determine if there is a linear relationship between your variables.Report out:List of datarr2 as a percentIndication of whether the relationship is linear (showing CV)Sample variable pairs (not required to use these, they’re just examples):Cereal: Sugar in grams, CaloriesMcDonald’s Sandwiches: Fat in grams, CaloriesMLB Pitchers: ERA, WinsNCAA BB: Seed, RPIStates: Median income, SAT scoresPatients: Age, Resting Pulse ................
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