{"nbformat":4,"nbformat_minor":0,"metadata":{"colab":{"provenance":[],"authorship_tag":"ABX9TyNsZf5zi2FkUdTOxM8NxKLX"},"kernelspec":{"name":"ir","display_name":"R"},"language_info":{"name":"R"}},"cells":[{"cell_type":"markdown","source":["# **TD3**"],"metadata":{"id":"mbqa-f_OY_fD"}},{"cell_type":"markdown","source":["## **Exercice 1**"],"metadata":{"id":"CrZgyN--ZFWg"}},{"cell_type":"markdown","source":["### 1.Create a Data Frame"],"metadata":{"id":"XB5D_xX7bodO"}},{"cell_type":"code","source":["ourdata<- data.frame( x1 = c(1100, 1200, 1430, 1500, 1520, 1620, 1800, 1820, 1800),\n","                   x2 = c(300, 400, 420, 400, 510, 590, 600, 630, 610),\n","                   y = c(60, 120, 190, 250, 300, 360, 380, 430, 440)\n","                 )\n","ourdata"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":380},"id":"O4LpQj3kZOEF","executionInfo":{"status":"ok","timestamp":1705841228873,"user_tz":-120,"elapsed":366,"user":{"displayName":"Data Science","userId":"07742026815209387612"}},"outputId":"fd5b3fbd-cf9e-4d62-f114-c049f3d2acac"},"execution_count":23,"outputs":[{"output_type":"display_data","data":{"text/html":["<table class=\"dataframe\">\n","<caption>A data.frame: 9 × 3</caption>\n","<thead>\n","\t<tr><th scope=col>x1</th><th scope=col>x2</th><th scope=col>y</th></tr>\n","\t<tr><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th></tr>\n","</thead>\n","<tbody>\n","\t<tr><td>1100</td><td>300</td><td> 60</td></tr>\n","\t<tr><td>1200</td><td>400</td><td>120</td></tr>\n","\t<tr><td>1430</td><td>420</td><td>190</td></tr>\n","\t<tr><td>1500</td><td>400</td><td>250</td></tr>\n","\t<tr><td>1520</td><td>510</td><td>300</td></tr>\n","\t<tr><td>1620</td><td>590</td><td>360</td></tr>\n","\t<tr><td>1800</td><td>600</td><td>380</td></tr>\n","\t<tr><td>1820</td><td>630</td><td>430</td></tr>\n","\t<tr><td>1800</td><td>610</td><td>440</td></tr>\n","</tbody>\n","</table>\n"],"text/markdown":"\nA data.frame: 9 × 3\n\n| x1 &lt;dbl&gt; | x2 &lt;dbl&gt; | y &lt;dbl&gt; |\n|---|---|---|\n| 1100 | 300 |  60 |\n| 1200 | 400 | 120 |\n| 1430 | 420 | 190 |\n| 1500 | 400 | 250 |\n| 1520 | 510 | 300 |\n| 1620 | 590 | 360 |\n| 1800 | 600 | 380 |\n| 1820 | 630 | 430 |\n| 1800 | 610 | 440 |\n\n","text/latex":"A data.frame: 9 × 3\n\\begin{tabular}{lll}\n x1 & x2 & y\\\\\n <dbl> & <dbl> & <dbl>\\\\\n\\hline\n\t 1100 & 300 &  60\\\\\n\t 1200 & 400 & 120\\\\\n\t 1430 & 420 & 190\\\\\n\t 1500 & 400 & 250\\\\\n\t 1520 & 510 & 300\\\\\n\t 1620 & 590 & 360\\\\\n\t 1800 & 600 & 380\\\\\n\t 1820 & 630 & 430\\\\\n\t 1800 & 610 & 440\\\\\n\\end{tabular}\n","text/plain":["  x1   x2  y  \n","1 1100 300  60\n","2 1200 400 120\n","3 1430 420 190\n","4 1500 400 250\n","5 1520 510 300\n","6 1620 590 360\n","7 1800 600 380\n","8 1820 630 430\n","9 1800 610 440"]},"metadata":{}}]},{"cell_type":"markdown","source":["### 2.Plot the Data"],"metadata":{"id":"1Iioi1jDcmHL"}},{"cell_type":"code","source":["par(mfrow = c(1, 2))\n","\n","plot(ourdata$x1, ourdata$y, main = \"Scatter Plot: Working Hours\",\n","     xlab = \"Working Hours\", ylab = \"Production\",  col = \"blue\")\n","\n","\n","plot(ourdata$x2 , ourdata$y , main = \"Scatter Plot: Capital Employed\",\n","     xlab = \"Capital Employed\", ylab = \"Production\", col = \"green\")\n"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":437},"id":"yfml9u_4crVz","executionInfo":{"status":"ok","timestamp":1705841229287,"user_tz":-120,"elapsed":28,"user":{"displayName":"Data Science","userId":"07742026815209387612"}},"outputId":"88d6dc83-b2d8-404a-e8b7-311b6488c36d"},"execution_count":24,"outputs":[{"output_type":"display_data","data":{"text/plain":["Plot with title “Scatter Plot: Capital Employed”"],"image/png":"iVBORw0KGgoAAAANSUhEUgAAA0gAAANICAIAAAByhViMAAAACXBIWXMAABJ0AAASdAHeZh94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\nnDlzRh3UjqUZMmSIO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ABkGwAwAAMAiC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\nAFAEwQ4AAEARBDsAAABF/C8x/hfn0ptNqQAAAABJRU5ErkJggg=="},"metadata":{"image/png":{"width":420,"height":420}}}]},{"cell_type":"markdown","source":["### 3.Fit a Linear Model"],"metadata":{"id":"V4gU-TKcfOgz"}},{"cell_type":"code","source":["model <- lm(y ~ x1 + x2 , data = ourdata)\n"," summary(model)"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":347},"id":"ZxSRDAipfbl0","executionInfo":{"status":"ok","timestamp":1705841229288,"user_tz":-120,"elapsed":23,"user":{"displayName":"Data Science","userId":"07742026815209387612"}},"outputId":"79f4116d-c9fa-4d33-923f-a9fcac97e40d"},"execution_count":25,"outputs":[{"output_type":"display_data","data":{"text/plain":["\n","Call:\n","lm(formula = y ~ x1 + x2, data = ourdata)\n","\n","Residuals:\n","    Min      1Q  Median      3Q     Max \n","-34.050 -10.129   4.526  17.080  21.850 \n","\n","Coefficients:\n","              Estimate Std. Error t value Pr(>|t|)    \n","(Intercept) -437.71363   57.93097  -7.556 0.000279 ***\n","x1             0.33653    0.08966   3.753 0.009474 ** \n","x2             0.41002    0.19614   2.090 0.081555 .  \n","---\n","Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1\n","\n","Residual standard error: 23.07 on 6 degrees of freedom\n","Multiple R-squared:  0.9784,\tAdjusted R-squared:  0.9712 \n","F-statistic: 135.9 on 2 and 6 DF,  p-value: 1.007e-05\n"]},"metadata":{}}]},{"cell_type":"markdown","source":["### 4.Test if the 𝜀𝑖 are normally distributed.\n"],"metadata":{"id":"Eks0wBhkgBjA"}},{"cell_type":"code","source":["residuals <- residuals(model)\n","shapiro.test(residuals)"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":104},"id":"5I4ef9vXgQZf","executionInfo":{"status":"ok","timestamp":1705841229289,"user_tz":-120,"elapsed":22,"user":{"displayName":"Data Science","userId":"07742026815209387612"}},"outputId":"569a9c27-4c4b-4d77-8a16-e3242d29f7e0"},"execution_count":26,"outputs":[{"output_type":"display_data","data":{"text/plain":["\n","\tShapiro-Wilk normality test\n","\n","data:  residuals\n","W = 0.91569, p-value = 0.3578\n"]},"metadata":{}}]},{"cell_type":"markdown","source":[" Interpretation: W = 0.91569 very close to 1 → normal distribution\n"],"metadata":{"id":"xlVTeav5gYw8"}},{"cell_type":"markdown","source":["5.Get the Coefficients"],"metadata":{"id":"ZVgvR2Ykgl2S"}},{"cell_type":"code","source":["coef(model)"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":34},"id":"tyuIQYBkhbcI","executionInfo":{"status":"ok","timestamp":1705841229699,"user_tz":-120,"elapsed":430,"user":{"displayName":"Data Science","userId":"07742026815209387612"}},"outputId":"3d94b3b8-01a4-433d-a0b6-41e5a4249b79"},"execution_count":27,"outputs":[{"output_type":"display_data","data":{"text/html":["<style>\n",".dl-inline {width: auto; margin:0; padding: 0}\n",".dl-inline>dt, .dl-inline>dd {float: none; width: auto; display: inline-block}\n",".dl-inline>dt::after {content: \":\\0020\"; padding-right: .5ex}\n",".dl-inline>dt:not(:first-of-type) {padding-left: .5ex}\n","</style><dl class=dl-inline><dt>(Intercept)</dt><dd>-437.71363282332</dd><dt>x1</dt><dd>0.336530289507513</dd><dt>x2</dt><dd>0.410015695762618</dd></dl>\n"],"text/markdown":"(Intercept)\n:   -437.71363282332x1\n:   0.336530289507513x2\n:   0.410015695762618\n\n","text/latex":"\\begin{description*}\n\\item[(Intercept)] -437.71363282332\n\\item[x1] 0.336530289507513\n\\item[x2] 0.410015695762618\n\\end{description*}\n","text/plain":[" (Intercept)           x1           x2 \n","-437.7136328    0.3365303    0.4100157 "]},"metadata":{}}]},{"cell_type":"markdown","source":["### 5.Construct a 95% confidence interval for parameters 𝛽0, 𝛽1 and 𝛽2"],"metadata":{"id":"tUnpD7TwhgBu"}},{"cell_type":"code","source":["confint(model, level = 0.95)"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":161},"id":"aypPPjzEh4sj","executionInfo":{"status":"ok","timestamp":1705841229700,"user_tz":-120,"elapsed":113,"user":{"displayName":"Data Science","userId":"07742026815209387612"}},"outputId":"e2f26008-3728-40ca-9ffc-a605bb17177b"},"execution_count":28,"outputs":[{"output_type":"display_data","data":{"text/html":["<table class=\"dataframe\">\n","<caption>A matrix: 3 × 2 of type dbl</caption>\n","<thead>\n","\t<tr><th></th><th scope=col>2.5 %</th><th scope=col>97.5 %</th></tr>\n","</thead>\n","<tbody>\n","\t<tr><th scope=row>(Intercept)</th><td>-579.46562195</td><td>-295.9616437</td></tr>\n","\t<tr><th scope=row>x1</th><td>   0.11712905</td><td>   0.5559315</td></tr>\n","\t<tr><th scope=row>x2</th><td>  -0.06993004</td><td>   0.8899614</td></tr>\n","</tbody>\n","</table>\n"],"text/markdown":"\nA matrix: 3 × 2 of type dbl\n\n| <!--/--> | 2.5 % | 97.5 % |\n|---|---|---|\n| (Intercept) | -579.46562195 | -295.9616437 |\n| x1 |    0.11712905 |    0.5559315 |\n| x2 |   -0.06993004 |    0.8899614 |\n\n","text/latex":"A matrix: 3 × 2 of type dbl\n\\begin{tabular}{r|ll}\n  & 2.5 \\% & 97.5 \\%\\\\\n\\hline\n\t(Intercept) & -579.46562195 & -295.9616437\\\\\n\tx1 &    0.11712905 &    0.5559315\\\\\n\tx2 &   -0.06993004 &    0.8899614\\\\\n\\end{tabular}\n","text/plain":["            2.5 %         97.5 %      \n","(Intercept) -579.46562195 -295.9616437\n","x1             0.11712905    0.5559315\n","x2            -0.06993004    0.8899614"]},"metadata":{}}]},{"cell_type":"markdown","source":["### 6.Anova"],"metadata":{"id":"zjl5Jyv3iT3h"}},{"cell_type":"code","source":[" # Compute anova table\n","  tab <- anova(model)\n","\n","  # Obtain number of predictors\n","  p <- nrow(tab) - 1\n","\n","  # Add predictors row\n","  predictorsRow <- colSums(tab[1:p, 1:2])\n","  predictorsRow <- c(predictorsRow, predictorsRow[2] / predictorsRow[1])\n","\n","  # F-quantities\n","  Fval <- predictorsRow[3] / tab[p + 1, 3]\n","  pval <- pf(Fval, df1 = p, df2 = tab$Df[p + 1], lower.tail = FALSE)\n","  predictorsRow <- c(predictorsRow, Fval, pval)\n","\n","  # Simplified table\n","  tab <- rbind(predictorsRow, tab[p + 1, ])\n","  row.names(tab)[1] <- \"Predictors\"\n","  return(tab)\n","\n"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":161},"id":"249Y5NI6jZyR","executionInfo":{"status":"ok","timestamp":1705841229702,"user_tz":-120,"elapsed":109,"user":{"displayName":"Data Science","userId":"07742026815209387612"}},"outputId":"c26d4578-f3d7-441c-976b-e839bb1dac5f"},"execution_count":29,"outputs":[{"output_type":"display_data","data":{"text/html":["<table class=\"dataframe\">\n","<caption>A anova: 2 × 5</caption>\n","<thead>\n","\t<tr><th></th><th scope=col>Df</th><th scope=col>Sum Sq</th><th scope=col>Mean Sq</th><th scope=col>F value</th><th scope=col>Pr(&gt;F)</th></tr>\n","\t<tr><th></th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th></tr>\n","</thead>\n","<tbody>\n","\t<tr><th scope=row>Predictors</th><td>2</td><td>144695.122</td><td>72347.5611</td><td>135.9164</td><td>1.00717e-05</td></tr>\n","\t<tr><th scope=row>Residuals</th><td>6</td><td>  3193.767</td><td>  532.2944</td><td>      NA</td><td>         NA</td></tr>\n","</tbody>\n","</table>\n"],"text/markdown":"\nA anova: 2 × 5\n\n| <!--/--> | Df &lt;dbl&gt; | Sum Sq &lt;dbl&gt; | Mean Sq &lt;dbl&gt; | F value &lt;dbl&gt; | Pr(&gt;F) &lt;dbl&gt; |\n|---|---|---|---|---|---|\n| Predictors | 2 | 144695.122 | 72347.5611 | 135.9164 | 1.00717e-05 |\n| Residuals | 6 |   3193.767 |   532.2944 |       NA |          NA |\n\n","text/latex":"A anova: 2 × 5\n\\begin{tabular}{r|lllll}\n  & Df & Sum Sq & Mean Sq & F value & Pr(>F)\\\\\n  & <dbl> & <dbl> & <dbl> & <dbl> & <dbl>\\\\\n\\hline\n\tPredictors & 2 & 144695.122 & 72347.5611 & 135.9164 & 1.00717e-05\\\\\n\tResiduals & 6 &   3193.767 &   532.2944 &       NA &          NA\\\\\n\\end{tabular}\n","text/plain":["           Df Sum Sq     Mean Sq    F value  Pr(>F)     \n","Predictors 2  144695.122 72347.5611 135.9164 1.00717e-05\n","Residuals  6    3193.767   532.2944       NA          NA"]},"metadata":{}}]},{"cell_type":"markdown","source":["### 7.Hypothesis Testing for 𝛽1 & 𝛽2"],"metadata":{"id":"qFX4BjFQkXoh"}},{"cell_type":"markdown","source":["**Step1: Set the Hypothesis**\n","\n","---\n","H0: 𝛽1 = 0\n","\n","H1: 𝛽1 &ne; 0\n"],"metadata":{"id":"FoaN5osdlHAC"}},{"cell_type":"markdown","source":["**Step2: Test-Statistic**\n","\n","---\n","T-test for significance\n","\n","\n","\n"],"metadata":{"id":"fUoVcv1xlHAE"}},{"cell_type":"code","source":["summary <- summary(model)\n","summary$coefficients"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":161},"id":"yrNhYyNplHAF","executionInfo":{"status":"ok","timestamp":1705841229703,"user_tz":-120,"elapsed":102,"user":{"displayName":"Data Science","userId":"07742026815209387612"}},"outputId":"6d519e3e-d7b6-48e4-cb3c-ee10363e31ec"},"execution_count":30,"outputs":[{"output_type":"display_data","data":{"text/html":["<table class=\"dataframe\">\n","<caption>A matrix: 3 × 4 of type dbl</caption>\n","<thead>\n","\t<tr><th></th><th scope=col>Estimate</th><th scope=col>Std. Error</th><th scope=col>t value</th><th scope=col>Pr(&gt;|t|)</th></tr>\n","</thead>\n","<tbody>\n","\t<tr><th scope=row>(Intercept)</th><td>-437.7136328</td><td>57.93097494</td><td>-7.555779</td><td>0.0002789705</td></tr>\n","\t<tr><th scope=row>x1</th><td>   0.3365303</td><td> 0.08966455</td><td> 3.753215</td><td>0.0094738487</td></tr>\n","\t<tr><th scope=row>x2</th><td>   0.4100157</td><td> 0.19614345</td><td> 2.090387</td><td>0.0815546320</td></tr>\n","</tbody>\n","</table>\n"],"text/markdown":"\nA matrix: 3 × 4 of type dbl\n\n| <!--/--> | Estimate | Std. Error | t value | Pr(&gt;|t|) |\n|---|---|---|---|---|\n| (Intercept) | -437.7136328 | 57.93097494 | -7.555779 | 0.0002789705 |\n| x1 |    0.3365303 |  0.08966455 |  3.753215 | 0.0094738487 |\n| x2 |    0.4100157 |  0.19614345 |  2.090387 | 0.0815546320 |\n\n","text/latex":"A matrix: 3 × 4 of type dbl\n\\begin{tabular}{r|llll}\n  & Estimate & Std. Error & t value & Pr(>\\textbar{}t\\textbar{})\\\\\n\\hline\n\t(Intercept) & -437.7136328 & 57.93097494 & -7.555779 & 0.0002789705\\\\\n\tx1 &    0.3365303 &  0.08966455 &  3.753215 & 0.0094738487\\\\\n\tx2 &    0.4100157 &  0.19614345 &  2.090387 & 0.0815546320\\\\\n\\end{tabular}\n","text/plain":["            Estimate     Std. Error  t value   Pr(>|t|)    \n","(Intercept) -437.7136328 57.93097494 -7.555779 0.0002789705\n","x1             0.3365303  0.08966455  3.753215 0.0094738487\n","x2             0.4100157  0.19614345  2.090387 0.0815546320"]},"metadata":{}}]},{"cell_type":"markdown","source":["t-value for 𝛽1 = 3.753215"],"metadata":{"id":"yosZpO5XzoKe"}},{"cell_type":"markdown","source":["**Step3: T-Critical**"],"metadata":{"id":"jtUSFiO3lHAH"}},{"cell_type":"code","source":["# find t-critical\n","t_critical <- qt(df= 9 -2, p=0.025)\n","\n","abs(t_critical)"],"metadata":{"id":"WKlv0jFNlHAI","executionInfo":{"status":"ok","timestamp":1705841229703,"user_tz":-120,"elapsed":98,"user":{"displayName":"Data Science","userId":"07742026815209387612"}},"outputId":"2a409585-7d0a-4380-b779-dc1d789ff4e6","colab":{"base_uri":"https://localhost:8080/","height":34}},"execution_count":31,"outputs":[{"output_type":"display_data","data":{"text/html":["2.36462425159278"],"text/markdown":"2.36462425159278","text/latex":"2.36462425159278","text/plain":["[1] 2.364624"]},"metadata":{}}]},{"cell_type":"markdown","source":["## **Exercice 2**"],"metadata":{"id":"PSADXgZYIX10"}},{"cell_type":"markdown","source":["### 1) Create a Data Frame"],"metadata":{"id":"BEv6ol1EIX11"}},{"cell_type":"code","source":["ourdata<- data.frame( x1 = c(124, 49, 181, 4, 22, 152, 75, 54, 43, 41, 17, 22, 16, 10, 63, 170, 125, 12, 221, 171, 97, 254),\n","                      x2 = c(33, 31, 38, 17, 20, 39, 30, 29, 35, 31, 23, 21, 8, 23, 37, 40, 38, 25, 39, 33, 38, 39),\n","                      x3 = c(8, 6, 8, 2, 4, 6, 7, 7, 6, 5, 4, 3, 3, 3, 6, 8, 6, 4, 7, 7, 6, 8),\n","                      y  = c(81, 55, 80, 24, 78, 52, 88, 45, 50, 69, 66, 45, 24, 43, 38, 72, 41, 38, 52, 52, 66, 89)\n","                    )\n","ourdata"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":788},"executionInfo":{"status":"ok","timestamp":1705841229704,"user_tz":-120,"elapsed":92,"user":{"displayName":"Data Science","userId":"07742026815209387612"}},"outputId":"15af12ba-e4d4-41ee-e8e1-0d2cf83c0c00","id":"FAbFTL05IX11"},"execution_count":32,"outputs":[{"output_type":"display_data","data":{"text/html":["<table class=\"dataframe\">\n","<caption>A data.frame: 22 × 4</caption>\n","<thead>\n","\t<tr><th scope=col>x1</th><th scope=col>x2</th><th scope=col>x3</th><th scope=col>y</th></tr>\n","\t<tr><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th></tr>\n","</thead>\n","<tbody>\n","\t<tr><td>124</td><td>33</td><td>8</td><td>81</td></tr>\n","\t<tr><td> 49</td><td>31</td><td>6</td><td>55</td></tr>\n","\t<tr><td>181</td><td>38</td><td>8</td><td>80</td></tr>\n","\t<tr><td>  4</td><td>17</td><td>2</td><td>24</td></tr>\n","\t<tr><td> 22</td><td>20</td><td>4</td><td>78</td></tr>\n","\t<tr><td>152</td><td>39</td><td>6</td><td>52</td></tr>\n","\t<tr><td> 75</td><td>30</td><td>7</td><td>88</td></tr>\n","\t<tr><td> 54</td><td>29</td><td>7</td><td>45</td></tr>\n","\t<tr><td> 43</td><td>35</td><td>6</td><td>50</td></tr>\n","\t<tr><td> 41</td><td>31</td><td>5</td><td>69</td></tr>\n","\t<tr><td> 17</td><td>23</td><td>4</td><td>66</td></tr>\n","\t<tr><td> 22</td><td>21</td><td>3</td><td>45</td></tr>\n","\t<tr><td> 16</td><td> 8</td><td>3</td><td>24</td></tr>\n","\t<tr><td> 10</td><td>23</td><td>3</td><td>43</td></tr>\n","\t<tr><td> 63</td><td>37</td><td>6</td><td>38</td></tr>\n","\t<tr><td>170</td><td>40</td><td>8</td><td>72</td></tr>\n","\t<tr><td>125</td><td>38</td><td>6</td><td>41</td></tr>\n","\t<tr><td> 12</td><td>25</td><td>4</td><td>38</td></tr>\n","\t<tr><td>221</td><td>39</td><td>7</td><td>52</td></tr>\n","\t<tr><td>171</td><td>33</td><td>7</td><td>52</td></tr>\n","\t<tr><td> 97</td><td>38</td><td>6</td><td>66</td></tr>\n","\t<tr><td>254</td><td>39</td><td>8</td><td>89</td></tr>\n","</tbody>\n","</table>\n"],"text/markdown":"\nA data.frame: 22 × 4\n\n| x1 &lt;dbl&gt; | x2 &lt;dbl&gt; | x3 &lt;dbl&gt; | y &lt;dbl&gt; |\n|---|---|---|---|\n| 124 | 33 | 8 | 81 |\n|  49 | 31 | 6 | 55 |\n| 181 | 38 | 8 | 80 |\n|   4 | 17 | 2 | 24 |\n|  22 | 20 | 4 | 78 |\n| 152 | 39 | 6 | 52 |\n|  75 | 30 | 7 | 88 |\n|  54 | 29 | 7 | 45 |\n|  43 | 35 | 6 | 50 |\n|  41 | 31 | 5 | 69 |\n|  17 | 23 | 4 | 66 |\n|  22 | 21 | 3 | 45 |\n|  16 |  8 | 3 | 24 |\n|  10 | 23 | 3 | 43 |\n|  63 | 37 | 6 | 38 |\n| 170 | 40 | 8 | 72 |\n| 125 | 38 | 6 | 41 |\n|  12 | 25 | 4 | 38 |\n| 221 | 39 | 7 | 52 |\n| 171 | 33 | 7 | 52 |\n|  97 | 38 | 6 | 66 |\n| 254 | 39 | 8 | 89 |\n\n","text/latex":"A data.frame: 22 × 4\n\\begin{tabular}{llll}\n x1 & x2 & x3 & y\\\\\n <dbl> & <dbl> & <dbl> & <dbl>\\\\\n\\hline\n\t 124 & 33 & 8 & 81\\\\\n\t  49 & 31 & 6 & 55\\\\\n\t 181 & 38 & 8 & 80\\\\\n\t   4 & 17 & 2 & 24\\\\\n\t  22 & 20 & 4 & 78\\\\\n\t 152 & 39 & 6 & 52\\\\\n\t  75 & 30 & 7 & 88\\\\\n\t  54 & 29 & 7 & 45\\\\\n\t  43 & 35 & 6 & 50\\\\\n\t  41 & 31 & 5 & 69\\\\\n\t  17 & 23 & 4 & 66\\\\\n\t  22 & 21 & 3 & 45\\\\\n\t  16 &  8 & 3 & 24\\\\\n\t  10 & 23 & 3 & 43\\\\\n\t  63 & 37 & 6 & 38\\\\\n\t 170 & 40 & 8 & 72\\\\\n\t 125 & 38 & 6 & 41\\\\\n\t  12 & 25 & 4 & 38\\\\\n\t 221 & 39 & 7 & 52\\\\\n\t 171 & 33 & 7 & 52\\\\\n\t  97 & 38 & 6 & 66\\\\\n\t 254 & 39 & 8 & 89\\\\\n\\end{tabular}\n","text/plain":["   x1  x2 x3 y \n","1  124 33 8  81\n","2   49 31 6  55\n","3  181 38 8  80\n","4    4 17 2  24\n","5   22 20 4  78\n","6  152 39 6  52\n","7   75 30 7  88\n","8   54 29 7  45\n","9   43 35 6  50\n","10  41 31 5  69\n","11  17 23 4  66\n","12  22 21 3  45\n","13  16  8 3  24\n","14  10 23 3  43\n","15  63 37 6  38\n","16 170 40 8  72\n","17 125 38 6  41\n","18  12 25 4  38\n","19 221 39 7  52\n","20 171 33 7  52\n","21  97 38 6  66\n","22 254 39 8  89"]},"metadata":{}}]},{"cell_type":"markdown","source":["### 2) Simple Linear Regression (x1 and y)"],"metadata":{"id":"4CIIs_8RLAvk"}},{"cell_type":"code","source":["#Fit a Linear model\n","model1 <- lm(y~x1 , data = ourdata)\n","\n","#Get the coefficients\n","model1$coefficients"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":34},"id":"PjnqlkRhLKNQ","executionInfo":{"status":"ok","timestamp":1705841229705,"user_tz":-120,"elapsed":88,"user":{"displayName":"Data Science","userId":"07742026815209387612"}},"outputId":"c15d1d41-87e6-417b-f039-49987d434e5d"},"execution_count":33,"outputs":[{"output_type":"display_data","data":{"text/html":["<style>\n",".dl-inline {width: auto; margin:0; padding: 0}\n",".dl-inline>dt, .dl-inline>dd {float: none; width: auto; display: inline-block}\n",".dl-inline>dt::after {content: \":\\0020\"; padding-right: .5ex}\n",".dl-inline>dt:not(:first-of-type) {padding-left: .5ex}\n","</style><dl class=dl-inline><dt>(Intercept)</dt><dd>46.2363327564604</dd><dt>x1</dt><dd>0.120021154112257</dd></dl>\n"],"text/markdown":"(Intercept)\n:   46.2363327564604x1\n:   0.120021154112257\n\n","text/latex":"\\begin{description*}\n\\item[(Intercept)] 46.2363327564604\n\\item[x1] 0.120021154112257\n\\end{description*}\n","text/plain":["(Intercept)          x1 \n"," 46.2363328   0.1200212 "]},"metadata":{}}]},{"cell_type":"markdown","source":["So the linear regression equation for this model is:\n","\n","y_hat = 46.23633 +  0.12002 xi\n","\n","\n"],"metadata":{"id":"nA8DWfwDL0Bl"}},{"cell_type":"markdown","source":[" **Test the significance of this simple linear\n","regression:**"],"metadata":{"id":"dNhoFw4NMUJ0"}},{"cell_type":"code","source":["summary(model1)"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":329},"id":"AZFXHtQZMzWl","executionInfo":{"status":"ok","timestamp":1705841229706,"user_tz":-120,"elapsed":84,"user":{"displayName":"Data Science","userId":"07742026815209387612"}},"outputId":"ffc550ac-1a1d-4c7f-e397-31df59df949b"},"execution_count":34,"outputs":[{"output_type":"display_data","data":{"text/plain":["\n","Call:\n","lm(formula = y ~ x1, data = ourdata)\n","\n","Residuals:\n","    Min      1Q  Median      3Q     Max \n","-24.157 -14.190  -2.637  12.219  32.762 \n","\n","Coefficients:\n","            Estimate Std. Error t value Pr(>|t|)    \n","(Intercept) 46.23633    5.77111   8.012 1.14e-07 ***\n","x1           0.12002    0.05044   2.379   0.0274 *  \n","---\n","Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1\n","\n","Residual standard error: 17.47 on 20 degrees of freedom\n","Multiple R-squared:  0.2206,\tAdjusted R-squared:  0.1817 \n","F-statistic: 5.662 on 1 and 20 DF,  p-value: 0.02741\n"]},"metadata":{}}]},{"cell_type":"markdown","source":["We can see that our model estimators are significant, since p-value of x1 coefficient\n","is lower than 0.05\n","Same for intercept, which is much significant with a very low p-value"],"metadata":{"id":"cxC2ka8ONBmG"}},{"cell_type":"markdown","source":["### 3) Multiple Linear Regression (y on x1 and x2)"],"metadata":{"id":"L1FbMpG9NMO4"}},{"cell_type":"code","source":["#fit the multiple regression model\n","model1_2 <- lm(y ~ x1 + x2, data = ourdata)\n","\n","#Get the coefficients\n","model1_2$coefficients"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":34},"id":"Dp8CNRE0NRG7","executionInfo":{"status":"ok","timestamp":1705841229708,"user_tz":-120,"elapsed":83,"user":{"displayName":"Data Science","userId":"07742026815209387612"}},"outputId":"c0bfcf3c-2616-4ce1-ee93-a8b3fb612d96"},"execution_count":35,"outputs":[{"output_type":"display_data","data":{"text/html":["<style>\n",".dl-inline {width: auto; margin:0; padding: 0}\n",".dl-inline>dt, .dl-inline>dd {float: none; width: auto; display: inline-block}\n",".dl-inline>dt::after {content: \":\\0020\"; padding-right: .5ex}\n",".dl-inline>dt:not(:first-of-type) {padding-left: .5ex}\n","</style><dl class=dl-inline><dt>(Intercept)</dt><dd>35.5546417000172</dd><dt>x1</dt><dd>0.0789503424556111</dd><dt>x2</dt><dd>0.470729196487979</dd></dl>\n"],"text/markdown":"(Intercept)\n:   35.5546417000172x1\n:   0.0789503424556111x2\n:   0.470729196487979\n\n","text/latex":"\\begin{description*}\n\\item[(Intercept)] 35.5546417000172\n\\item[x1] 0.0789503424556111\n\\item[x2] 0.470729196487979\n\\end{description*}\n","text/plain":["(Intercept)          x1          x2 \n","35.55464170  0.07895034  0.47072920 "]},"metadata":{}}]},{"cell_type":"markdown","source":["So the linear regression equation for this model is:\n","\n","y_hat = 35.55464 + 0.078950 x1 + 0.470729 x2\n"],"metadata":{"id":"RR8mKnIXOgiB"}},{"cell_type":"markdown","source":["**Anova Table:**"],"metadata":{"id":"uYafM4xxPNeF"}},{"cell_type":"code","source":["# Compute anova table\n","  tab <- anova(model1_2)\n","\n","  # Obtain number of predictors\n","  p <- nrow(tab) - 1\n","\n","  # Add predictors row\n","  predictorsRow <- colSums(tab[1:p, 1:2])\n","  predictorsRow <- c(predictorsRow, predictorsRow[2] / predictorsRow[1])\n","\n","  # F-quantities\n","  Fval <- predictorsRow[3] / tab[p + 1, 3]\n","  pval <- pf(Fval, df1 = p, df2 = tab$Df[p + 1], lower.tail = FALSE)\n","  predictorsRow <- c(predictorsRow, Fval, pval)\n","\n","  # Simplified table\n","  tab <- rbind(predictorsRow, tab[p + 1, ])\n","  row.names(tab)[1] <- \"Predictors\"\n","  return(tab)"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":161},"id":"EYwpECOwPaqJ","executionInfo":{"status":"ok","timestamp":1705841229709,"user_tz":-120,"elapsed":79,"user":{"displayName":"Data Science","userId":"07742026815209387612"}},"outputId":"6e223b89-5153-4691-e6e5-d6e39afe7a5b"},"execution_count":36,"outputs":[{"output_type":"display_data","data":{"text/html":["<table class=\"dataframe\">\n","<caption>A anova: 2 × 5</caption>\n","<thead>\n","\t<tr><th></th><th scope=col>Df</th><th scope=col>Sum Sq</th><th scope=col>Mean Sq</th><th scope=col>F value</th><th scope=col>Pr(&gt;F)</th></tr>\n","\t<tr><th></th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th></tr>\n","</thead>\n","<tbody>\n","\t<tr><th scope=row>Predictors</th><td> 2</td><td>1876.075</td><td>938.0374</td><td>2.994262</td><td>0.07406679</td></tr>\n","\t<tr><th scope=row>Residuals</th><td>19</td><td>5952.289</td><td>313.2784</td><td>      NA</td><td>        NA</td></tr>\n","</tbody>\n","</table>\n"],"text/markdown":"\nA anova: 2 × 5\n\n| <!--/--> | Df &lt;dbl&gt; | Sum Sq &lt;dbl&gt; | Mean Sq &lt;dbl&gt; | F value &lt;dbl&gt; | Pr(&gt;F) &lt;dbl&gt; |\n|---|---|---|---|---|---|\n| Predictors |  2 | 1876.075 | 938.0374 | 2.994262 | 0.07406679 |\n| Residuals | 19 | 5952.289 | 313.2784 |       NA |         NA |\n\n","text/latex":"A anova: 2 × 5\n\\begin{tabular}{r|lllll}\n  & Df & Sum Sq & Mean Sq & F value & Pr(>F)\\\\\n  & <dbl> & <dbl> & <dbl> & <dbl> & <dbl>\\\\\n\\hline\n\tPredictors &  2 & 1876.075 & 938.0374 & 2.994262 & 0.07406679\\\\\n\tResiduals & 19 & 5952.289 & 313.2784 &       NA &         NA\\\\\n\\end{tabular}\n","text/plain":["           Df Sum Sq   Mean Sq  F value  Pr(>F)    \n","Predictors  2 1876.075 938.0374 2.994262 0.07406679\n","Residuals  19 5952.289 313.2784       NA         NA"]},"metadata":{}}]},{"cell_type":"markdown","source":[" **Test whether the addition of the variable X to the equation\n","found in question 1**\n","\n","H0 : β2 = 0\n","\n","We can see from the anova table above that p-value of is greater than 0.05, so we\n","cannot reject H0\n","\n","Thus, it does not seem to add anything to the model created in part 1"],"metadata":{"id":"bsdmpyHtPxiy"}},{"cell_type":"markdown","source":["### 4) Multiple Linear Regression (y on x1 , x2 and x3)"],"metadata":{"id":"Zksz6XqGQDAl"}},{"cell_type":"code","source":["#Fit the multiple regression model\n","model1_2_3 <- lm( y ~ x1+x2+x3, data=ourdata)\n","\n","#Get the coefficients\n","model1_2_3$coefficients"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":34},"id":"EJEVGYKzQYMV","executionInfo":{"status":"ok","timestamp":1705841230235,"user_tz":-120,"elapsed":601,"user":{"displayName":"Data Science","userId":"07742026815209387612"}},"outputId":"81356d8b-b354-422c-d783-e93bcf8a70a5"},"execution_count":37,"outputs":[{"output_type":"display_data","data":{"text/html":["<style>\n",".dl-inline {width: auto; margin:0; padding: 0}\n",".dl-inline>dt, .dl-inline>dd {float: none; width: auto; display: inline-block}\n",".dl-inline>dt::after {content: \":\\0020\"; padding-right: .5ex}\n",".dl-inline>dt:not(:first-of-type) {padding-left: .5ex}\n","</style><dl class=dl-inline><dt>(Intercept)</dt><dd>23.9995661494815</dd><dt>x1</dt><dd>-0.00617344667456764</dd><dt>x2</dt><dd>-0.479869472682723</dd><dt>x3</dt><dd>8.48350016891916</dd></dl>\n"],"text/markdown":"(Intercept)\n:   23.9995661494815x1\n:   -0.00617344667456764x2\n:   -0.479869472682723x3\n:   8.48350016891916\n\n","text/latex":"\\begin{description*}\n\\item[(Intercept)] 23.9995661494815\n\\item[x1] -0.00617344667456764\n\\item[x2] -0.479869472682723\n\\item[x3] 8.48350016891916\n\\end{description*}\n","text/plain":[" (Intercept)           x1           x2           x3 \n","23.999566149 -0.006173447 -0.479869473  8.483500169 "]},"metadata":{}}]},{"cell_type":"markdown","source":["So the linear regression equation for this model is:\n","\n","y_hat = 23.999 + (-0.00617)x1 + (-0.479869)x2 + 8.48350 x3"],"metadata":{"id":"EeiT4DjXQ8MW"}},{"cell_type":"markdown","source":["### 5) 95% confidence interval for β3 in this equation"],"metadata":{"id":"olcRrO1JRUV9"}},{"cell_type":"code","source":["#Get the Confidence Interval\n","confint(model1_2_3, parm=\"x3\")"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":98},"id":"f8-tvI42RoTn","executionInfo":{"status":"ok","timestamp":1705841230236,"user_tz":-120,"elapsed":172,"user":{"displayName":"Data Science","userId":"07742026815209387612"}},"outputId":"6cbc1a76-ed36-4dcf-fecd-48412f27d6eb"},"execution_count":38,"outputs":[{"output_type":"display_data","data":{"text/html":["<table class=\"dataframe\">\n","<caption>A matrix: 1 × 2 of type dbl</caption>\n","<thead>\n","\t<tr><th></th><th scope=col>2.5 %</th><th scope=col>97.5 %</th></tr>\n","</thead>\n","<tbody>\n","\t<tr><th scope=row>x3</th><td>0.402924</td><td>16.56408</td></tr>\n","</tbody>\n","</table>\n"],"text/markdown":"\nA matrix: 1 × 2 of type dbl\n\n| <!--/--> | 2.5 % | 97.5 % |\n|---|---|---|\n| x3 | 0.402924 | 16.56408 |\n\n","text/latex":"A matrix: 1 × 2 of type dbl\n\\begin{tabular}{r|ll}\n  & 2.5 \\% & 97.5 \\%\\\\\n\\hline\n\tx3 & 0.402924 & 16.56408\\\\\n\\end{tabular}\n","text/plain":["   2.5 %    97.5 %  \n","x3 0.402924 16.56408"]},"metadata":{}}]},{"cell_type":"markdown","source":["0 does not belong to this interval, so we can reject , and confidently say that is\n","not equal to zero"],"metadata":{"id":"acmFGROERyge"}},{"cell_type":"markdown","source":["### 6) 95% confidence interval for at the point x1 = 221, x2 = 39 and x3 = 7."],"metadata":{"id":"PNpZwY4LR6rV"}},{"cell_type":"code","source":["test_data <- data.frame(x1=c(221),x2=c(39),x3=c(7))\n","predict(model1_2_3, interval='prediction', newdata = test_data)"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":98},"id":"RI6_CQmxSBSG","executionInfo":{"status":"ok","timestamp":1705841230237,"user_tz":-120,"elapsed":164,"user":{"displayName":"Data Science","userId":"07742026815209387612"}},"outputId":"fff425c5-1b4f-4ff2-a4cd-714065d2c477"},"execution_count":39,"outputs":[{"output_type":"display_data","data":{"text/html":["<table class=\"dataframe\">\n","<caption>A matrix: 1 × 3 of type dbl</caption>\n","<thead>\n","\t<tr><th></th><th scope=col>fit</th><th scope=col>lwr</th><th scope=col>upr</th></tr>\n","</thead>\n","<tbody>\n","\t<tr><th scope=row>1</th><td>63.30483</td><td>25.27463</td><td>101.335</td></tr>\n","</tbody>\n","</table>\n"],"text/markdown":"\nA matrix: 1 × 3 of type dbl\n\n| <!--/--> | fit | lwr | upr |\n|---|---|---|---|\n| 1 | 63.30483 | 25.27463 | 101.335 |\n\n","text/latex":"A matrix: 1 × 3 of type dbl\n\\begin{tabular}{r|lll}\n  & fit & lwr & upr\\\\\n\\hline\n\t1 & 63.30483 & 25.27463 & 101.335\\\\\n\\end{tabular}\n","text/plain":["  fit      lwr      upr    \n","1 63.30483 25.27463 101.335"]},"metadata":{}}]},{"cell_type":"markdown","source":["y_hat for x1 = 221, x2 = 39 and x3 = 7 falls within 28.39797 and 103.5925"],"metadata":{"id":"_fQuTC7hSXK6"}},{"cell_type":"markdown","source":["### 7) Regress x1 on x2 and x3"],"metadata":{"id":"69evJ3LQS_B7"}},{"cell_type":"code","source":["#Fit the Model\n","newlinear_model= lm(x1 ~ x2 +x3, data=ourdata)\n","\n","#Get the coefficients\n","summary(newlinear_model)"],"metadata":{"id":"zjJR39SyTXvR","colab":{"base_uri":"https://localhost:8080/","height":347},"executionInfo":{"status":"ok","timestamp":1705841230238,"user_tz":-120,"elapsed":156,"user":{"displayName":"Data Science","userId":"07742026815209387612"}},"outputId":"9c591435-1fe0-4608-e41f-6c8f59d3d2e5"},"execution_count":40,"outputs":[{"output_type":"display_data","data":{"text/plain":["\n","Call:\n","lm(formula = x1 ~ x2 + x3, data = ourdata)\n","\n","Residuals:\n","    Min      1Q  Median      3Q     Max \n","-64.734 -31.628  -0.529  29.773  90.961 \n","\n","Coefficients:\n","            Estimate Std. Error t value Pr(>|t|)   \n","(Intercept) -117.910     36.076  -3.268  0.00404 **\n","x2             2.597      2.052   1.266  0.22087   \n","x3            22.459      9.553   2.351  0.02967 * \n","---\n","Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1\n","\n","Residual standard error: 45.53 on 19 degrees of freedom\n","Multiple R-squared:  0.6715,\tAdjusted R-squared:  0.6369 \n","F-statistic: 19.42 on 2 and 19 DF,  p-value: 2.553e-05\n"]},"metadata":{}}]},{"cell_type":"markdown","source":["Multiple R-squared=0.6797 , means that the model explains about 67.97% of the\n","variance in the dependent variable (x1).\n","p-value suggests that at least one of the predictors (x2 or x3) is significantly related\n","to x1.\n","The overall model is statistically significant."],"metadata":{"id":"tZRZH0FATnvC"}},{"cell_type":"markdown","source":["### 8) 95% confidence interval for the coefficients of the linear regression of X1 on X3 ."],"metadata":{"id":"4V9r0mglTsp3"}},{"cell_type":"code","source":["conf_interval <- confint(newlinear_model,  level = 0.95)\n","conf_interval"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":161},"id":"jm4XEyGnT0zC","executionInfo":{"status":"ok","timestamp":1705841230238,"user_tz":-120,"elapsed":150,"user":{"displayName":"Data Science","userId":"07742026815209387612"}},"outputId":"a5c4dd44-faa8-4424-edec-b773a05d567a"},"execution_count":41,"outputs":[{"output_type":"display_data","data":{"text/html":["<table class=\"dataframe\">\n","<caption>A matrix: 3 × 2 of type dbl</caption>\n","<thead>\n","\t<tr><th></th><th scope=col>2.5 %</th><th scope=col>97.5 %</th></tr>\n","</thead>\n","<tbody>\n","\t<tr><th scope=row>(Intercept)</th><td>-193.418001</td><td>-42.402527</td></tr>\n","\t<tr><th scope=row>x2</th><td>  -1.696950</td><td>  6.890841</td></tr>\n","\t<tr><th scope=row>x3</th><td>   2.464409</td><td> 42.452738</td></tr>\n","</tbody>\n","</table>\n"],"text/markdown":"\nA matrix: 3 × 2 of type dbl\n\n| <!--/--> | 2.5 % | 97.5 % |\n|---|---|---|\n| (Intercept) | -193.418001 | -42.402527 |\n| x2 |   -1.696950 |   6.890841 |\n| x3 |    2.464409 |  42.452738 |\n\n","text/latex":"A matrix: 3 × 2 of type dbl\n\\begin{tabular}{r|ll}\n  & 2.5 \\% & 97.5 \\%\\\\\n\\hline\n\t(Intercept) & -193.418001 & -42.402527\\\\\n\tx2 &   -1.696950 &   6.890841\\\\\n\tx3 &    2.464409 &  42.452738\\\\\n\\end{tabular}\n","text/plain":["            2.5 %       97.5 %    \n","(Intercept) -193.418001 -42.402527\n","x2            -1.696950   6.890841\n","x3             2.464409  42.452738"]},"metadata":{}}]},{"cell_type":"markdown","source":["## Exercice 5"],"metadata":{"id":"HydIDZxkak7L"}},{"cell_type":"markdown","source":["### 1. Load Data"],"metadata":{"id":"ob-p2Nwkauu4"}},{"cell_type":"code","source":["data <- data.frame( y= c(37.8, 22.5, 17.1, 10.8, 7.2, 42.3, 30.2, 19.4, 14.8, 9.5, 32.4, 21.6),\n","                    x1= c(4, 4, 3, 2, 1, 6, 4, 4, 1, 1, 3, 4),\n","                    x2= c(4.0, 3.6, 3.1, 3.2, 3.0, 3.8, 3.8, 2.9, 3.8, 2.8, 3.4, 2.8)\n","                  )\n","data"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":474},"id":"HUnw-ob1ay43","executionInfo":{"status":"ok","timestamp":1705841230240,"user_tz":-120,"elapsed":145,"user":{"displayName":"Data Science","userId":"07742026815209387612"}},"outputId":"4cbd151b-0847-4f87-df54-e7551ae9de0d"},"execution_count":42,"outputs":[{"output_type":"display_data","data":{"text/html":["<table class=\"dataframe\">\n","<caption>A data.frame: 12 × 3</caption>\n","<thead>\n","\t<tr><th scope=col>y</th><th scope=col>x1</th><th scope=col>x2</th></tr>\n","\t<tr><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th></tr>\n","</thead>\n","<tbody>\n","\t<tr><td>37.8</td><td>4</td><td>4.0</td></tr>\n","\t<tr><td>22.5</td><td>4</td><td>3.6</td></tr>\n","\t<tr><td>17.1</td><td>3</td><td>3.1</td></tr>\n","\t<tr><td>10.8</td><td>2</td><td>3.2</td></tr>\n","\t<tr><td> 7.2</td><td>1</td><td>3.0</td></tr>\n","\t<tr><td>42.3</td><td>6</td><td>3.8</td></tr>\n","\t<tr><td>30.2</td><td>4</td><td>3.8</td></tr>\n","\t<tr><td>19.4</td><td>4</td><td>2.9</td></tr>\n","\t<tr><td>14.8</td><td>1</td><td>3.8</td></tr>\n","\t<tr><td> 9.5</td><td>1</td><td>2.8</td></tr>\n","\t<tr><td>32.4</td><td>3</td><td>3.4</td></tr>\n","\t<tr><td>21.6</td><td>4</td><td>2.8</td></tr>\n","</tbody>\n","</table>\n"],"text/markdown":"\nA data.frame: 12 × 3\n\n| y &lt;dbl&gt; | x1 &lt;dbl&gt; | x2 &lt;dbl&gt; |\n|---|---|---|\n| 37.8 | 4 | 4.0 |\n| 22.5 | 4 | 3.6 |\n| 17.1 | 3 | 3.1 |\n| 10.8 | 2 | 3.2 |\n|  7.2 | 1 | 3.0 |\n| 42.3 | 6 | 3.8 |\n| 30.2 | 4 | 3.8 |\n| 19.4 | 4 | 2.9 |\n| 14.8 | 1 | 3.8 |\n|  9.5 | 1 | 2.8 |\n| 32.4 | 3 | 3.4 |\n| 21.6 | 4 | 2.8 |\n\n","text/latex":"A data.frame: 12 × 3\n\\begin{tabular}{lll}\n y & x1 & x2\\\\\n <dbl> & <dbl> & <dbl>\\\\\n\\hline\n\t 37.8 & 4 & 4.0\\\\\n\t 22.5 & 4 & 3.6\\\\\n\t 17.1 & 3 & 3.1\\\\\n\t 10.8 & 2 & 3.2\\\\\n\t  7.2 & 1 & 3.0\\\\\n\t 42.3 & 6 & 3.8\\\\\n\t 30.2 & 4 & 3.8\\\\\n\t 19.4 & 4 & 2.9\\\\\n\t 14.8 & 1 & 3.8\\\\\n\t  9.5 & 1 & 2.8\\\\\n\t 32.4 & 3 & 3.4\\\\\n\t 21.6 & 4 & 2.8\\\\\n\\end{tabular}\n","text/plain":["   y    x1 x2 \n","1  37.8 4  4.0\n","2  22.5 4  3.6\n","3  17.1 3  3.1\n","4  10.8 2  3.2\n","5   7.2 1  3.0\n","6  42.3 6  3.8\n","7  30.2 4  3.8\n","8  19.4 4  2.9\n","9  14.8 1  3.8\n","10  9.5 1  2.8\n","11 32.4 3  3.4\n","12 21.6 4  2.8"]},"metadata":{}}]},{"cell_type":"markdown","source":["### 2. Plot the data"],"metadata":{"id":"5T4LSRNjhxqX"}},{"cell_type":"code","source":["par(mfrow=c(1,2))\n","plot(data$x1 , data$y , xlab = \"X1\" , ylab = \"Y\")\n","plot(data$x2 , data$y , xlab = \"X2\" , ylab = \"Y\")"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":437},"id":"iiUjqibeaqqn","executionInfo":{"status":"ok","timestamp":1705841230245,"user_tz":-120,"elapsed":143,"user":{"displayName":"Data Science","userId":"07742026815209387612"}},"outputId":"92f179f6-7099-4df6-c4ab-ee9cf9f4229a"},"execution_count":43,"outputs":[{"output_type":"display_data","data":{"text/plain":["plot without title"],"image/png":"iVBORw0KGgoAAAANSUhEUgAAA0gAAANICAMAAADKOT/pAAADAFBMVEUAAAABAQECAgIDAwME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\ngAGEBBhASIABhAQYQEiAAf8fT1+F4+VYULgAAAAASUVORK5CYII="},"metadata":{"image/png":{"width":420,"height":420}}}]},{"cell_type":"markdown","source":["### 3. Regress y on x1"],"metadata":{"id":"8oIJfLrWkdh0"}},{"cell_type":"code","source":["model1 <- lm(y ~ x1 , data = data)\n","summary(model1)"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":329},"id":"jajl_BSKktKX","executionInfo":{"status":"ok","timestamp":1705842099556,"user_tz":-120,"elapsed":404,"user":{"displayName":"Data Science","userId":"07742026815209387612"}},"outputId":"17e3b526-a1a5-4dd0-998e-f2264541bc18"},"execution_count":58,"outputs":[{"output_type":"display_data","data":{"text/plain":["\n","Call:\n","lm(formula = y ~ x1, data = data)\n","\n","Residuals:\n","   Min     1Q Median     3Q    Max \n","-8.266 -4.887 -1.208  3.232 10.770 \n","\n","Coefficients:\n","            Estimate Std. Error t value Pr(>|t|)    \n","(Intercept)    3.523      4.383   0.804 0.440237    \n","x1             6.036      1.279   4.721 0.000816 ***\n","---\n","Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1\n","\n","Residual standard error: 6.633 on 10 degrees of freedom\n","Multiple R-squared:  0.6903,\tAdjusted R-squared:  0.6593 \n","F-statistic: 22.29 on 1 and 10 DF,  p-value: 0.0008155\n"]},"metadata":{}}]},{"cell_type":"code","source":["anova(model1)"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":161},"id":"OB0befIklL4_","executionInfo":{"status":"ok","timestamp":1705841230248,"user_tz":-120,"elapsed":135,"user":{"displayName":"Data Science","userId":"07742026815209387612"}},"outputId":"ed65a988-dca5-43b9-9d19-495e9999c910"},"execution_count":45,"outputs":[{"output_type":"display_data","data":{"text/html":["<table class=\"dataframe\">\n","<caption>A anova: 2 × 5</caption>\n","<thead>\n","\t<tr><th></th><th scope=col>Df</th><th scope=col>Sum Sq</th><th scope=col>Mean Sq</th><th scope=col>F value</th><th scope=col>Pr(&gt;F)</th></tr>\n","\t<tr><th></th><th scope=col>&lt;int&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th></tr>\n","</thead>\n","<tbody>\n","\t<tr><th scope=row>x1</th><td> 1</td><td>980.6347</td><td>980.6347</td><td>22.28553</td><td>0.0008155066</td></tr>\n","\t<tr><th scope=row>Residuals</th><td>10</td><td>440.0320</td><td> 44.0032</td><td>      NA</td><td>          NA</td></tr>\n","</tbody>\n","</table>\n"],"text/markdown":"\nA anova: 2 × 5\n\n| <!--/--> | Df &lt;int&gt; | Sum Sq &lt;dbl&gt; | Mean Sq &lt;dbl&gt; | F value &lt;dbl&gt; | Pr(&gt;F) &lt;dbl&gt; |\n|---|---|---|---|---|---|\n| x1 |  1 | 980.6347 | 980.6347 | 22.28553 | 0.0008155066 |\n| Residuals | 10 | 440.0320 |  44.0032 |       NA |           NA |\n\n","text/latex":"A anova: 2 × 5\n\\begin{tabular}{r|lllll}\n  & Df & Sum Sq & Mean Sq & F value & Pr(>F)\\\\\n  & <int> & <dbl> & <dbl> & <dbl> & <dbl>\\\\\n\\hline\n\tx1 &  1 & 980.6347 & 980.6347 & 22.28553 & 0.0008155066\\\\\n\tResiduals & 10 & 440.0320 &  44.0032 &       NA &           NA\\\\\n\\end{tabular}\n","text/plain":["          Df Sum Sq   Mean Sq  F value  Pr(>F)      \n","x1         1 980.6347 980.6347 22.28553 0.0008155066\n","Residuals 10 440.0320  44.0032       NA           NA"]},"metadata":{}}]},{"cell_type":"markdown","source":[" What percentage of variation in breaking strength is explained by each of the regressions?\n","\n","\n","---\n","\n","\n","\n","From the summary we find that for this regression   $R^2$ = 0.6903 , wich mean that 69.03%  of variation in breaking strength is explained by the regression of y on x1"],"metadata":{"id":"IoynxHgXnAEU"}},{"cell_type":"markdown","source":["Residual mean squares(MSE)= $S^2$\n","\n","\n","---\n","\n","\n","\n"],"metadata":{"id":"SqWALle1-1Xq"}},{"cell_type":"code","source":["#mse = sse / n - (p + 1) = 12 - 2 =10\n","mse1 <- 440.03/10\n","mse1"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":34},"id":"yWiVynUVAFyB","executionInfo":{"status":"ok","timestamp":1705841230853,"user_tz":-120,"elapsed":732,"user":{"displayName":"Data Science","userId":"07742026815209387612"}},"outputId":"b63b58f9-5014-4433-e4c0-8775c0d0921b"},"execution_count":46,"outputs":[{"output_type":"display_data","data":{"text/html":["44.003"],"text/markdown":"44.003","text/latex":"44.003","text/plain":["[1] 44.003"]},"metadata":{}}]},{"cell_type":"markdown","source":["Residual standard deviations ( $ \\sqrt{MSE} $ ) = $S$\n","\n","\n","---\n","\n","*(we can also find it in model summary : \"Residual standard error\")*"],"metadata":{"id":"sIY8tP8wBEaV"}},{"cell_type":"code","source":["s1 <- sqrt(mse1)\n","s1"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":34},"id":"CJMN3lJRB2Cl","executionInfo":{"status":"ok","timestamp":1705841230854,"user_tz":-120,"elapsed":169,"user":{"displayName":"Data Science","userId":"07742026815209387612"}},"outputId":"de739f39-e1ba-4d98-b917-764cf946ab29"},"execution_count":47,"outputs":[{"output_type":"display_data","data":{"text/html":["6.63347571036482"],"text/markdown":"6.63347571036482","text/latex":"6.63347571036482","text/plain":["[1] 6.633476"]},"metadata":{}}]},{"cell_type":"markdown","source":["### 4. Regress y on x2"],"metadata":{"id":"VRFkYqOileVo"}},{"cell_type":"code","source":["model2 <- lm(y ~ x2 , data = data)\n","summary(model2)"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":329},"id":"7w4YUqbrliNU","executionInfo":{"status":"ok","timestamp":1705841230855,"user_tz":-120,"elapsed":166,"user":{"displayName":"Data Science","userId":"07742026815209387612"}},"outputId":"a00ecb99-36b5-4d56-c8d9-2e23b7190fa7"},"execution_count":48,"outputs":[{"output_type":"display_data","data":{"text/plain":["\n","Call:\n","lm(formula = y ~ x2, data = data)\n","\n","Residuals:\n","     Min       1Q   Median       3Q      Max \n","-15.1923  -5.1780  -0.2298   6.1123  12.3077 \n","\n","Coefficients:\n","            Estimate Std. Error t value Pr(>|t|)  \n","(Intercept)  -36.373     20.489  -1.775   0.1062  \n","x2            17.464      6.069   2.878   0.0164 *\n","---\n","Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1\n","\n","Residual standard error: 8.815 on 10 degrees of freedom\n","Multiple R-squared:  0.453,\tAdjusted R-squared:  0.3983 \n","F-statistic: 8.282 on 1 and 10 DF,  p-value: 0.01645\n"]},"metadata":{}}]},{"cell_type":"code","source":["anova(model2)"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":161},"id":"xB8IlDY7lv_m","executionInfo":{"status":"ok","timestamp":1705841230861,"user_tz":-120,"elapsed":163,"user":{"displayName":"Data Science","userId":"07742026815209387612"}},"outputId":"f91b3b0c-eef5-44ac-cc1c-124ef029640c"},"execution_count":49,"outputs":[{"output_type":"display_data","data":{"text/html":["<table class=\"dataframe\">\n","<caption>A anova: 2 × 5</caption>\n","<thead>\n","\t<tr><th></th><th scope=col>Df</th><th scope=col>Sum Sq</th><th scope=col>Mean Sq</th><th scope=col>F value</th><th scope=col>Pr(&gt;F)</th></tr>\n","\t<tr><th></th><th scope=col>&lt;int&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th></tr>\n","</thead>\n","<tbody>\n","\t<tr><th scope=row>x2</th><td> 1</td><td>643.5652</td><td>643.56517</td><td>8.28161</td><td>0.01644801</td></tr>\n","\t<tr><th scope=row>Residuals</th><td>10</td><td>777.1015</td><td> 77.71015</td><td>     NA</td><td>        NA</td></tr>\n","</tbody>\n","</table>\n"],"text/markdown":"\nA anova: 2 × 5\n\n| <!--/--> | Df &lt;int&gt; | Sum Sq &lt;dbl&gt; | Mean Sq &lt;dbl&gt; | F value &lt;dbl&gt; | Pr(&gt;F) &lt;dbl&gt; |\n|---|---|---|---|---|---|\n| x2 |  1 | 643.5652 | 643.56517 | 8.28161 | 0.01644801 |\n| Residuals | 10 | 777.1015 |  77.71015 |      NA |         NA |\n\n","text/latex":"A anova: 2 × 5\n\\begin{tabular}{r|lllll}\n  & Df & Sum Sq & Mean Sq & F value & Pr(>F)\\\\\n  & <int> & <dbl> & <dbl> & <dbl> & <dbl>\\\\\n\\hline\n\tx2 &  1 & 643.5652 & 643.56517 & 8.28161 & 0.01644801\\\\\n\tResiduals & 10 & 777.1015 &  77.71015 &      NA &         NA\\\\\n\\end{tabular}\n","text/plain":["          Df Sum Sq   Mean Sq   F value Pr(>F)    \n","x2         1 643.5652 643.56517 8.28161 0.01644801\n","Residuals 10 777.1015  77.71015      NA         NA"]},"metadata":{}}]},{"cell_type":"markdown","source":[" What percentage of variation in breaking strength is explained by each of the regressions?\n","\n","\n","---\n","\n","\n","\n","From the summary we find that for this regression   $R^2$ = 0.453 , wich mean that 45.3%  of variation in breaking strength is explained by the regression of y on x2"],"metadata":{"id":"dr5sDGS0nmEG"}},{"cell_type":"markdown","source":["Residual mean squares(MSE)= $S^2$\n","\n","\n","---\n","\n","\n","\n"],"metadata":{"id":"BQTTIfw2CRYI"}},{"cell_type":"code","source":["#mse = sse / n - (p + 1) = n - ( 1+ 1 ) = 12 - 2 = 10\n","mse2 <-777.1/10\n","mse2"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":34},"executionInfo":{"status":"ok","timestamp":1705841230865,"user_tz":-120,"elapsed":159,"user":{"displayName":"Data Science","userId":"07742026815209387612"}},"outputId":"6044fe1d-a246-47a9-9f34-716413408ce6","id":"qhb4fUEkCRYK"},"execution_count":50,"outputs":[{"output_type":"display_data","data":{"text/html":["77.71"],"text/markdown":"77.71","text/latex":"77.71","text/plain":["[1] 77.71"]},"metadata":{}}]},{"cell_type":"markdown","source":["Residual standard deviations ( $ \\sqrt{MSE} $ ) = $S$\n","\n","\n","---\n","\n","*(we can also find it in model summary : \"Residual standard error\")*"],"metadata":{"id":"PBIE4OjzCRYL"}},{"cell_type":"code","source":["s2 <- sqrt(mse2)\n","s2"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":34},"executionInfo":{"status":"ok","timestamp":1705841230866,"user_tz":-120,"elapsed":155,"user":{"displayName":"Data Science","userId":"07742026815209387612"}},"outputId":"1ee6e728-e78a-4129-cb8b-55547a6c3e09","id":"b1IDdgrqCRYL"},"execution_count":51,"outputs":[{"output_type":"display_data","data":{"text/html":["8.81532756056177"],"text/markdown":"8.81532756056177","text/latex":"8.81532756056177","text/plain":["[1] 8.815328"]},"metadata":{}}]},{"cell_type":"markdown","source":["### 5. Regress y on x1 and x2"],"metadata":{"id":"ccAHpHKhmAcq"}},{"cell_type":"code","source":["model <- lm(y ~ x1 + x2 , data = data)\n","summary(model)"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":347},"id":"HKJBnG-OmEQ-","executionInfo":{"status":"ok","timestamp":1705841230867,"user_tz":-120,"elapsed":147,"user":{"displayName":"Data Science","userId":"07742026815209387612"}},"outputId":"0799b9a2-7da9-48a4-d60f-26c6c43fb255"},"execution_count":52,"outputs":[{"output_type":"display_data","data":{"text/plain":["\n","Call:\n","lm(formula = y ~ x1 + x2, data = data)\n","\n","Residuals:\n","   Min     1Q Median     3Q    Max \n","-6.897 -2.135 -1.126  1.714 10.122 \n","\n","Coefficients:\n","            Estimate Std. Error t value Pr(>|t|)    \n","(Intercept)  -30.081     11.455  -2.626 0.027542 *  \n","x1             4.905      1.014   4.838 0.000923 ***\n","x2            11.072      3.621   3.058 0.013617 *  \n","---\n","Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1\n","\n","Residual standard error: 4.897 on 9 degrees of freedom\n","Multiple R-squared:  0.8481,\tAdjusted R-squared:  0.8143 \n","F-statistic: 25.12 on 2 and 9 DF,  p-value: 0.0002075\n"]},"metadata":{}}]},{"cell_type":"code","source":["# Compute anova table\n","  tab <- anova(model)\n","\n","  # Obtain number of predictors\n","  p <- nrow(tab) - 1\n","\n","  # Add predictors row\n","  predictorsRow <- colSums(tab[1:p, 1:2])\n","  predictorsRow <- c(predictorsRow, predictorsRow[2] / predictorsRow[1])\n","\n","  # F-quantities\n","  Fval <- predictorsRow[3] / tab[p + 1, 3]\n","  pval <- pf(Fval, df1 = p, df2 = tab$Df[p + 1], lower.tail = FALSE)\n","  predictorsRow <- c(predictorsRow, Fval, pval)\n","\n","  # Simplified table\n","  tab <- rbind(predictorsRow, tab[p + 1, ])\n","  row.names(tab)[1] <- \"Predictors\"\n","  return(tab)"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":161},"id":"NtC-OnMcmWM8","executionInfo":{"status":"ok","timestamp":1705841230873,"user_tz":-120,"elapsed":147,"user":{"displayName":"Data Science","userId":"07742026815209387612"}},"outputId":"cded1f85-81d0-4045-e579-8ac08ba08a6e"},"execution_count":53,"outputs":[{"output_type":"display_data","data":{"text/html":["<table class=\"dataframe\">\n","<caption>A anova: 2 × 5</caption>\n","<thead>\n","\t<tr><th></th><th scope=col>Df</th><th scope=col>Sum Sq</th><th scope=col>Mean Sq</th><th scope=col>F value</th><th scope=col>Pr(&gt;F)</th></tr>\n","\t<tr><th></th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th><th scope=col>&lt;dbl&gt;</th></tr>\n","</thead>\n","<tbody>\n","\t<tr><th scope=row>Predictors</th><td>2</td><td>1204.8576</td><td>602.42879</td><td>25.12341</td><td>0.0002075386</td></tr>\n","\t<tr><th scope=row>Residuals</th><td>9</td><td> 215.8091</td><td> 23.97879</td><td>      NA</td><td>          NA</td></tr>\n","</tbody>\n","</table>\n"],"text/markdown":"\nA anova: 2 × 5\n\n| <!--/--> | Df &lt;dbl&gt; | Sum Sq &lt;dbl&gt; | Mean Sq &lt;dbl&gt; | F value &lt;dbl&gt; | Pr(&gt;F) &lt;dbl&gt; |\n|---|---|---|---|---|---|\n| Predictors | 2 | 1204.8576 | 602.42879 | 25.12341 | 0.0002075386 |\n| Residuals | 9 |  215.8091 |  23.97879 |       NA |           NA |\n\n","text/latex":"A anova: 2 × 5\n\\begin{tabular}{r|lllll}\n  & Df & Sum Sq & Mean Sq & F value & Pr(>F)\\\\\n  & <dbl> & <dbl> & <dbl> & <dbl> & <dbl>\\\\\n\\hline\n\tPredictors & 2 & 1204.8576 & 602.42879 & 25.12341 & 0.0002075386\\\\\n\tResiduals & 9 &  215.8091 &  23.97879 &       NA &           NA\\\\\n\\end{tabular}\n","text/plain":["           Df Sum Sq    Mean Sq   F value  Pr(>F)      \n","Predictors 2  1204.8576 602.42879 25.12341 0.0002075386\n","Residuals  9   215.8091  23.97879       NA           NA"]},"metadata":{}}]},{"cell_type":"markdown","source":[" What percentage of variation in breaking strength is explained by each of the regressions?\n","\n","\n","---\n","\n","\n","\n","From the summary we find that for this regression   $R^2$ = 0.8481 , wich mean that 84.81%  of variation in breaking strength is explained by the regression of y on x1 and x2"],"metadata":{"id":"y6uhzjRLnngT"}},{"cell_type":"markdown","source":["Residual mean squares(MSE)= $S^2$\n","\n","\n","---\n","\n","\n","\n"],"metadata":{"id":"OVNzRMCaCtAP"}},{"cell_type":"code","source":["#mse = sse / n - (p + 1) = n - ( 2 + 1 ) = 12 - 3 = 9\n","mse3 <-215.81/9\n","mse3"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":34},"executionInfo":{"status":"ok","timestamp":1705841230874,"user_tz":-120,"elapsed":138,"user":{"displayName":"Data Science","userId":"07742026815209387612"}},"outputId":"112fcd52-a4d2-4785-b141-1faefb5bbe89","id":"Y1FFuA6CCtAQ"},"execution_count":54,"outputs":[{"output_type":"display_data","data":{"text/html":["23.9788888888889"],"text/markdown":"23.9788888888889","text/latex":"23.9788888888889","text/plain":["[1] 23.97889"]},"metadata":{}}]},{"cell_type":"markdown","source":["Residual standard deviations ( $ \\sqrt{MSE} $ ) = $S$\n","\n","\n","---\n","\n","*(we can also find it in model summary : \"Residual standard error\")*"],"metadata":{"id":"5wC6c0aGCtAR"}},{"cell_type":"code","source":["s3 <- sqrt(mse3)\n","s3"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":34},"executionInfo":{"status":"ok","timestamp":1705841230876,"user_tz":-120,"elapsed":135,"user":{"displayName":"Data Science","userId":"07742026815209387612"}},"outputId":"9918d321-6866-4315-8657-e0f1dbe01a27","id":"hZyE7ZmlCtAR"},"execution_count":55,"outputs":[{"output_type":"display_data","data":{"text/html":["4.8968243677805"],"text/markdown":"4.8968243677805","text/latex":"4.8968243677805","text/plain":["[1] 4.896824"]},"metadata":{}}]},{"cell_type":"markdown","source":["### 6. Test the hypothesis 𝐻0: 𝛽1 = 𝛽2 = 0 (against 𝐻1: at least one of 𝛽 ≠ 0) with a significance threshold 𝛼 = 5%. What is your conclusion?"],"metadata":{"id":"zagw1qCcE6kU"}},{"cell_type":"markdown","source":["From the anova table above we find that F-value = 25.12341\n","\n","Now we have to find f-critical"],"metadata":{"id":"m0F6i9PUFDSw"}},{"cell_type":"code","source":["#Find f critical with df1 = p = 2 and df2 = n - (p + 1) = 12 - (2 + 1) = 9 and 𝛼 = 0.05\n","f_critical <- qf(df1=2,df2=9,p=.05)\n","f_critical"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":34},"id":"aP2SvZVhFOny","executionInfo":{"status":"ok","timestamp":1705841709242,"user_tz":-120,"elapsed":516,"user":{"displayName":"Data Science","userId":"07742026815209387612"}},"outputId":"5b056922-0b81-4721-b64c-3c52072f892a"},"execution_count":56,"outputs":[{"output_type":"display_data","data":{"text/html":["0.0515867418432648"],"text/markdown":"0.0515867418432648","text/latex":"0.0515867418432648","text/plain":["[1] 0.05158674"]},"metadata":{}}]},{"cell_type":"markdown","source":["**Interpretation:** |F| = 25.12341 > f-critical (and also p-value < 0.05) ⇒ reject H0 ⇒ the model is significant"],"metadata":{"id":"10e0wDrtFuo-"}},{"cell_type":"markdown","source":["### 7. In the case of the linear regression model including only the thickness of the material as an explanatory variable, determine a 95% confidence interval for 𝛽1."],"metadata":{"id":"7v7bnejWGn5L"}},{"cell_type":"code","source":["#For model1 including only x1 as explanatory variable\n","confint(model1, level=0.95)"],"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":129},"id":"SDygLqTTGyp4","executionInfo":{"status":"ok","timestamp":1705842050878,"user_tz":-120,"elapsed":286,"user":{"displayName":"Data Science","userId":"07742026815209387612"}},"outputId":"dc1b08df-d6cc-4ec1-e748-6cd144e70796"},"execution_count":57,"outputs":[{"output_type":"display_data","data":{"text/html":["<table class=\"dataframe\">\n","<caption>A matrix: 2 × 2 of type dbl</caption>\n","<thead>\n","\t<tr><th></th><th scope=col>2.5 %</th><th scope=col>97.5 %</th></tr>\n","</thead>\n","<tbody>\n","\t<tr><th scope=row>(Intercept)</th><td>-6.242858</td><td>13.28806</td></tr>\n","\t<tr><th scope=row>x1</th><td> 3.187036</td><td> 8.88479</td></tr>\n","</tbody>\n","</table>\n"],"text/markdown":"\nA matrix: 2 × 2 of type dbl\n\n| <!--/--> | 2.5 % | 97.5 % |\n|---|---|---|\n| (Intercept) | -6.242858 | 13.28806 |\n| x1 |  3.187036 |  8.88479 |\n\n","text/latex":"A matrix: 2 × 2 of type dbl\n\\begin{tabular}{r|ll}\n  & 2.5 \\% & 97.5 \\%\\\\\n\\hline\n\t(Intercept) & -6.242858 & 13.28806\\\\\n\tx1 &  3.187036 &  8.88479\\\\\n\\end{tabular}\n","text/plain":["            2.5 %     97.5 %  \n","(Intercept) -6.242858 13.28806\n","x1           3.187036  8.88479"]},"metadata":{}}]},{"cell_type":"markdown","source":["### 8. With the confidence interval calculated in question 5., can we affirm, at the 𝛼 = 5% significance level, that the linear regression is significant between the breaking strength and the thickness of the material? Justify your conclusion.\n","\n"],"metadata":{"id":"VmpjUWykHUJu"}},{"cell_type":"markdown","source":["we have found in the previous part that the confidence interval for X1(thickness of the material) is [3.187036 , 8.88479] and since this interval don't include 0 then we can reject H0: β1 = 0 wich suggest that the linear regression is significant between the breaking strength and the thickness of the material"],"metadata":{"id":"NBgU7lzMHbrJ"}}]}