Ejercicio Regresión Lineal Simple
knitr::opts_chunk$set(echo = FALSE)
library(ggplot2)Estima el modelo:
\[ Testscore = \beta_0 + \beta_1StudentTeacherRatio +u_i \]
- Leer el archivo:
Extraemos los datos que nos interesan:
Para realizar los cálculos necesitamos la media de \(X\) y de \(Y\) así que las calculamos:
## [1] 24.02282## [1] 753.8542- Ahora podemos estimar \(\hat{\beta}_1\) por medio de la fórmula:
\[ \hat{\beta}_1 = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sum (X_i - \bar{X})^2} \]
Primero la parte de arriba
## [1] -6007.325Ahora la parte de abajo:
## [1] 6322.869Calculamos \(\hat{\beta}_1\):
## [1] -0.9500948- Para calcular \(\hat{\beta}_0\) usamos la fórmula:
\[ \hat{\beta}_0 = \bar{Y} - \hat{\beta}_1 \bar{X} \]
## [1] 776.6782## [1] 776.6782- Por último calculamos los residuos con la fórmula:
\[\hat{u}_i = Y_i - \hat{Y}_i\]
## [1] -23.38811591 3.50175773 -47.24100846 -66.81782067 -18.94089816
## [6] 51.74030706 -19.04800143 -62.12939330 -74.16865652 108.82230250
## [11] -56.33027287 -28.93194682 -21.25574448 34.09141855 37.35303562
## [16] 61.90995385 116.58622810 -66.66668057 65.84213852 -64.38067625
## [21] -12.19920358 6.08137369 77.91654551 -63.39271056 14.98910657
## [26] -4.07566644 -28.37354911 75.52367562 -62.84393588 136.73739222
## [31] -51.12565033 228.10949344 -4.51297611 -51.95590616 -43.71813482
## [36] 12.08989879 -47.39702806 60.14621530 167.92018339 2.14277048
## [41] 145.22756965 31.83359432 -4.52987857 -78.62540638 -81.72418525
## [46] -30.93277978 111.80328240 51.17417413 -40.40154017 -28.35076265
## [51] -32.46913302 7.17201953 52.26102615 103.09527995 -87.89931219
## [56] -5.82301742 -3.97570665 -17.60520976 -60.33479021 9.85749106
## [61] 187.25988365 9.69651187 46.17041791 17.38148692 81.31776617
## [66] -16.48324168 56.60957457 31.36906300 24.65958992 -108.97455855
## [71] 10.36775231 -20.85593005 1.72683286 57.61814545 24.10227968
## [76] 13.53885872 2.31441475 32.88637957 23.36621252 -47.45380790
## [81] 16.04367827 -38.85996271 119.38318228 -3.51696495 35.36404438
## [86] -53.03127416 -13.17581847 -15.95378777 -107.65531998 9.77333930
## [91] -8.16063565 -109.12543801 54.21546031 -8.24476872 120.26686823
## [96] 46.25703235 29.12793552 -1.93187148 39.53736548 59.42590087
## [101] 66.37573586 -1.78967069 -81.52290434 23.61473960 -53.87852778
## [106] 104.18834340 25.55898885 9.54757805 11.53958785 6.55439756
## [111] -39.74585393 6.48469132 20.75590915 16.96752199 91.42095225
## [116] -3.53082332 -57.91427663 6.29342162 -37.01719058 -106.09544345
## [121] -80.27424840 -22.47074519 68.69413445 28.10701695 12.81571377
## [126] -8.07566644 47.66555694 -35.83641217 51.38978469 -59.07743928
## [131] -82.18200102 -11.49102559 -15.51678745 128.44282748 -51.32487989
## [136] 28.63628695 -16.07559383 37.90146999 -97.54696984 26.22394806
## [141] 113.88747222 -84.77327748 21.41823093 -55.38792883 74.91950558
## [146] -50.46250204 -66.40613810 -42.08269882 -29.73893143 70.09453640
## [151] -159.94192169 44.83028602 -55.03720816 -9.63494325 2.74344294
## [156] 8.40739624 53.38179529 21.42455561 96.85023405 -18.26585137
## [161] -6.04928220 14.66804280 -15.02598751 -35.45553877 -72.22590708
## [166] -25.30040499 -73.95881452 81.61154831 -57.61867091 0.96761992
## [171] -124.42897765 -37.17928182 19.23666403 12.14092193 88.16896519
## [176] 90.57841870 77.37417932 -10.30101871 -15.53435821 186.87426314
## [181] 14.52966640 -24.69999983 -80.14237905 77.97832990 17.68769286
## [186] 42.53427277 -50.51297611 28.14814848 120.34017907 52.42201830
## [191] 86.65653978 73.65083064 -0.77941395 14.56078425 30.20150044
## [196] 36.16498849 85.96651711 14.03124670 -31.43594342 47.54778810
## [201] -64.29513130 -22.57541739 -45.43638777 15.49422557 -17.41983541
## [206] -63.42825799 -41.29380768 34.76486039 -78.78285693 -34.04384328
## [211] -113.09392958 -3.34837166 -24.70570152 14.22424874 36.79209121
## [216] 153.31014531 -4.83279080 -75.19571440 -0.50784511 -57.03432368
## [221] 75.16539170 -97.58767863 21.31891801 29.42102415 -25.91251558
## [226] -1.38670884 70.72386425 -122.65292537 11.95120992 -71.87832397
## [231] 91.81702069 20.20228669 -13.40331358 41.65281451 109.86178338
## [236] 56.70251172 -18.67475505 -40.70734874 49.70557721 -24.98014628
## [241] -81.19696826 61.82420433 -27.13761377 -41.38107909 -3.51284790
## [246] 13.27852410 -0.35131938 15.20056305 -42.53648895 125.20259147
## [251] -30.42802158 -120.28550592 -113.56069548 79.77838568 -94.63152791
## [256] -25.00290337 39.23129085 -40.51880841 -7.06837598 -71.12004253
## [261] -9.81958454 124.72898409 45.42894108 34.14788872 -12.92450760
## [266] 24.03238478 -55.26818736 -61.54038734 -80.33435197 42.12618568
## [271] -72.14871955 -55.67572586 -11.67344846 -20.61264635 4.19417677
## [276] 17.62810076 78.05453432 -40.34430908 -88.64303980 -57.92583548
## [281] -0.86645558 -24.89458421 78.32760152 -18.89795541 21.56188166
## [286] -53.88936035 -77.25507244 -81.77710360 46.57582237 -122.16203487
## [291] -9.88389575 -14.39002352 -19.20947365 -29.82628297 -13.74352408
## [296] -27.53231277 41.00537969 -2.46272127 3.88659581 -62.83411305
## [301] 22.95469403 18.11884128 76.08364063 -35.93482048 47.36588701
## [306] 33.14190178 92.99279875 -61.57197488 134.58917441 -107.42617715
## [311] -70.57545839 -50.34037866 -81.67219581 -73.73911334 48.59038516
## [316] 2.54703842 -8.57802419 -75.76932717 -58.14697245 -41.31033322
## [321] 64.42254753 17.11968947 -77.38001598 -18.19484495 44.13781507
## [326] -110.27601332 28.36283871 126.97435417 -1.89373307 -96.62721051
## [331] -86.15780592 39.46382843 66.16561748 -51.87410202 -52.08953778
## [336] 15.86822366 64.60721004 -67.48495190 -17.96616205 53.52397867
## [341] 141.23054869 -68.02121064 -50.34714390 -24.21617155 -3.66554529
## [346] -16.17529575 33.22448660 6.55753706 -82.94173512 13.69645084
## [351] 26.65692213 96.24094285 -67.93209772 -28.50140169 -14.91961933
## [356] -1.11831607 -59.01447685 -81.34271954 -117.44146268 -19.03067655
## [361] 115.27721303 14.11565176 20.84337300 -40.80685857 -94.71753853
## [366] -59.77700368 133.98661419 -28.97861010 -63.10787710 -42.98053368
## [371] 43.77435109 -35.56933486 -46.16611910 -49.19608136 -11.57380950
## [376] 47.65905374 -36.46600959 -89.04227811 42.28279853 50.46990922
## [381] 17.83394345 16.78918661 -76.31650284 -40.87779961 45.87812806
## [386] -51.87574687 47.91575071 78.09837544 33.09160512 -18.70713954
## [391] -42.93405482 28.69216709 -102.86496010 0.05216615 25.87558393
## [396] -46.89744585 31.26822544 -41.61751745 77.20568326 -57.79049265
## [401] 4.66914254 -60.99202691 -39.82641660 6.11211732 -17.59386159
## [406] -15.99138377 108.76300670 28.96223324 47.74927395 -26.12369273
## [411] -62.30694799 -67.15199785 -121.45649359 73.93330696 -61.39308598
## [416] -32.10429284 -12.38912484 109.15324884 -26.68473785 13.46018252
## [421] 10.38535051 -27.74646965 136.52384506 -2.76784751 -79.05248713
## [426] 2.81005435 -100.86276685 20.36409291 -104.03896165 16.54209142
## [431] -71.89800736 51.98630449 -25.18886101 36.24271104 16.94477821
## [436] -43.56744904 -34.34004802 -69.29682647 1.13700098 -5.17362787
## [441] 42.30547649 55.54282814 26.65259259 -13.02117875 30.53713214
## [446] 89.23739531 -33.13272013 6.94225432 3.84817475 41.22608957
## [451] -49.87850886 54.74408670 -10.93008323 -43.22555841 41.65475600
## [456] 111.95136763 -54.26075395 44.70812598 82.01031516 29.22024053
## [461] -58.81308105 -2.23976868 35.46297956 47.16488479 -45.23133224
## [466] 19.75136297 -1.20889714 44.71274305 -32.32021430 42.99301433
## [471] 38.82413172 -14.57608492 22.55563561 32.32191534 20.59060161
## [476] 136.64254039 16.07305974 -93.86565651 49.29441048 60.08324915
## [481] -10.43721507 -128.72097770 7.65725313 -20.16090542 96.97446960
## [486] 108.11252487 -61.25586545 -14.17609453 -84.54010738 62.74763506
## [491] 53.79035344 -17.62921152 -48.68128924 37.06782176 49.88532131
## [496] -73.98468973 -0.26878976 2.27345057 106.31631484 -11.94097297## [1] -3.365952e-14- Para validar nuestros resultados, podemos comparar con lo que nos da R al ajustar el modelo de regresión lineal simple:
##
## Call:
## lm(formula = testscore ~ str_s, data = datos)
##
## Residuals:
## Min 1Q Median 3Q Max
## -159.942 -43.042 -2.086 37.490 228.109
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 776.6782 18.4064 42.196 <2e-16 ***
## str_s -0.9501 0.7579 -1.254 0.211
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 60.27 on 498 degrees of freedom
## Multiple R-squared: 0.003145, Adjusted R-squared: 0.001144
## F-statistic: 1.571 on 1 and 498 DF, p-value: 0.2106- Ejercicio:
- Calcula el error estándar.
## [1] 60.26901- Calcula la \(R^2\).
## [1] 0.003145303- Grafica los datos con la pendiente estimada
## Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
## ℹ Please use `linewidth` instead.
## This warning is displayed once per session.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.