{"cells":[{"cell_type":"markdown","metadata":{"id":"00_I15xBesMU"},"source":["# **Exercice 1**\n","We want to model the average price of houses in Canada as a function of time and we obtain the following data (unfortunately incomplete...):\n","\n","An analysis of this data shows that a linear model could potentially be applied to explain house prices from year to year. We calculate the sample covariance and correlation, and we obtain the following quantities:\n","\n","cov(x,y) = 374225\n","\n","r = 0.77"]},{"cell_type":"markdown","metadata":{"id":"x_iLjH62e0XG"},"source":["## 1.\tWe decide to express the price of houses in thousands of dollars rather than in dollars. What happens to covariance and correlation?"]},{"cell_type":"markdown","metadata":{"id":"EdBNL10Sfavc"},"source":["\n","*  The covariance is affected by the scale of the data. If we express the price of houses in thousands of dollars, the covariance will decrease since the values of one variable (price) are scaled down. The formula for covariance involves the product of the differences from the mean, and when the values are scaled down, the product becomes smaller.\n","\n","\n","*   The correlation is a normalized measure and is not affected by the scale of the data. So, if we express the prices in thousands of dollars, the correlation will remain the same.\n","\n"]},{"cell_type":"markdown","metadata":{"id":"y41k83lnf6LZ"},"source":["##2.\tLet's keep the initial prices (in dollars). We now want to express time in number of years since 1980. What happens to the covariance and the correlation?"]},{"cell_type":"markdown","metadata":{"id":"qsonr3mDgcRP"},"source":["\n","\n","* Covariance: Shifting or scaling the time variable won't affect the covariance. Covariance measures the joint variability between two variables, and shifting or scaling one variable does not change the relationship between them.\n","\n","*   Correlation: Like covariance, correlation is also not affected by shifting or scaling. It only measures the strength and direction of the linear relationship between two variables, irrespective of any linear transformation applied to the variables.\n","\n"]},{"cell_type":"markdown","metadata":{"id":"a-DocBetg1Gp"},"source":["# Exercice2\n","R's Orange database contains information on 35 orange trees. We are interested in the following 2 variables:\n","\n","\n","\n","*   “age” represents the age of the trees, it is a continuous quantitative variable. It is measured in days, the youngest tree is 118 days old and the oldest 1582 days old.\n","\n","*   “circumference” represents the circumference of the shaft, it is a continuous quantitative variable, measured in mm. the smallest circumference is 30 mm and the largest is 214 mm.\n","\n","\n","***We will use the built-in dataset (orange) in R***\n"]},{"cell_type":"markdown","metadata":{"id":"Qf0XMUwghx0F"},"source":["## 1.\tDoes a linear adjustment seem justified? What coefficient should you calculate with R?"]},{"cell_type":"code","execution_count":null,"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":1000},"executionInfo":{"elapsed":339,"status":"ok","timestamp":1705748584045,"user":{"displayName":"Data Science","userId":"07742026815209387612"},"user_tz":-120},"id":"EOw_f2vPeg6b","outputId":"ce8143d3-75df-4b6d-c51b-6141a33af841","vscode":{"languageId":"r"}},"outputs":[{"data":{"text/html":["<table class=\"dataframe\">\n","<caption>A nfnGroupedData: 35 × 3</caption>\n","<thead>\n","\t<tr><th></th><th scope=col>Tree</th><th scope=col>age</th><th scope=col>circumference</th></tr>\n","\t<tr><th></th><th scope=col>&lt;ord&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>1</th><td>1</td><td> 118</td><td> 30</td></tr>\n","\t<tr><th scope=row>2</th><td>1</td><td> 484</td><td> 58</td></tr>\n","\t<tr><th scope=row>3</th><td>1</td><td> 664</td><td> 87</td></tr>\n","\t<tr><th scope=row>4</th><td>1</td><td>1004</td><td>115</td></tr>\n","\t<tr><th scope=row>5</th><td>1</td><td>1231</td><td>120</td></tr>\n","\t<tr><th scope=row>6</th><td>1</td><td>1372</td><td>142</td></tr>\n","\t<tr><th scope=row>7</th><td>1</td><td>1582</td><td>145</td></tr>\n","\t<tr><th scope=row>8</th><td>2</td><td> 118</td><td> 33</td></tr>\n","\t<tr><th scope=row>9</th><td>2</td><td> 484</td><td> 69</td></tr>\n","\t<tr><th scope=row>10</th><td>2</td><td> 664</td><td>111</td></tr>\n","\t<tr><th scope=row>11</th><td>2</td><td>1004</td><td>156</td></tr>\n","\t<tr><th scope=row>12</th><td>2</td><td>1231</td><td>172</td></tr>\n","\t<tr><th scope=row>13</th><td>2</td><td>1372</td><td>203</td></tr>\n","\t<tr><th scope=row>14</th><td>2</td><td>1582</td><td>203</td></tr>\n","\t<tr><th scope=row>15</th><td>3</td><td> 118</td><td> 30</td></tr>\n","\t<tr><th scope=row>16</th><td>3</td><td> 484</td><td> 51</td></tr>\n","\t<tr><th scope=row>17</th><td>3</td><td> 664</td><td> 75</td></tr>\n","\t<tr><th scope=row>18</th><td>3</td><td>1004</td><td>108</td></tr>\n","\t<tr><th scope=row>19</th><td>3</td><td>1231</td><td>115</td></tr>\n","\t<tr><th scope=row>20</th><td>3</td><td>1372</td><td>139</td></tr>\n","\t<tr><th scope=row>21</th><td>3</td><td>1582</td><td>140</td></tr>\n","\t<tr><th scope=row>22</th><td>4</td><td> 118</td><td> 32</td></tr>\n","\t<tr><th scope=row>23</th><td>4</td><td> 484</td><td> 62</td></tr>\n","\t<tr><th scope=row>24</th><td>4</td><td> 664</td><td>112</td></tr>\n","\t<tr><th scope=row>25</th><td>4</td><td>1004</td><td>167</td></tr>\n","\t<tr><th scope=row>26</th><td>4</td><td>1231</td><td>179</td></tr>\n","\t<tr><th scope=row>27</th><td>4</td><td>1372</td><td>209</td></tr>\n","\t<tr><th scope=row>28</th><td>4</td><td>1582</td><td>214</td></tr>\n","\t<tr><th scope=row>29</th><td>5</td><td> 118</td><td> 30</td></tr>\n","\t<tr><th scope=row>30</th><td>5</td><td> 484</td><td> 49</td></tr>\n","\t<tr><th scope=row>31</th><td>5</td><td> 664</td><td> 81</td></tr>\n","\t<tr><th scope=row>32</th><td>5</td><td>1004</td><td>125</td></tr>\n","\t<tr><th scope=row>33</th><td>5</td><td>1231</td><td>142</td></tr>\n","\t<tr><th scope=row>34</th><td>5</td><td>1372</td><td>174</td></tr>\n","\t<tr><th scope=row>35</th><td>5</td><td>1582</td><td>177</td></tr>\n","</tbody>\n","</table>\n"],"text/latex":["A nfnGroupedData: 35 × 3\n","\\begin{tabular}{r|lll}\n","  & Tree & age & circumference\\\\\n","  & <ord> & <dbl> & <dbl>\\\\\n","\\hline\n","\t1 & 1 &  118 &  30\\\\\n","\t2 & 1 &  484 &  58\\\\\n","\t3 & 1 &  664 &  87\\\\\n","\t4 & 1 & 1004 & 115\\\\\n","\t5 & 1 & 1231 & 120\\\\\n","\t6 & 1 & 1372 & 142\\\\\n","\t7 & 1 & 1582 & 145\\\\\n","\t8 & 2 &  118 &  33\\\\\n","\t9 & 2 &  484 &  69\\\\\n","\t10 & 2 &  664 & 111\\\\\n","\t11 & 2 & 1004 & 156\\\\\n","\t12 & 2 & 1231 & 172\\\\\n","\t13 & 2 & 1372 & 203\\\\\n","\t14 & 2 & 1582 & 203\\\\\n","\t15 & 3 &  118 &  30\\\\\n","\t16 & 3 &  484 &  51\\\\\n","\t17 & 3 &  664 &  75\\\\\n","\t18 & 3 & 1004 & 108\\\\\n","\t19 & 3 & 1231 & 115\\\\\n","\t20 & 3 & 1372 & 139\\\\\n","\t21 & 3 & 1582 & 140\\\\\n","\t22 & 4 &  118 &  32\\\\\n","\t23 & 4 &  484 &  62\\\\\n","\t24 & 4 &  664 & 112\\\\\n","\t25 & 4 & 1004 & 167\\\\\n","\t26 & 4 & 1231 & 179\\\\\n","\t27 & 4 & 1372 & 209\\\\\n","\t28 & 4 & 1582 & 214\\\\\n","\t29 & 5 &  118 &  30\\\\\n","\t30 & 5 &  484 &  49\\\\\n","\t31 & 5 &  664 &  81\\\\\n","\t32 & 5 & 1004 & 125\\\\\n","\t33 & 5 & 1231 & 142\\\\\n","\t34 & 5 & 1372 & 174\\\\\n","\t35 & 5 & 1582 & 177\\\\\n","\\end{tabular}\n"],"text/markdown":["\n","A nfnGroupedData: 35 × 3\n","\n","| <!--/--> | Tree &lt;ord&gt; | age &lt;dbl&gt; | circumference &lt;dbl&gt; |\n","|---|---|---|---|\n","| 1 | 1 |  118 |  30 |\n","| 2 | 1 |  484 |  58 |\n","| 3 | 1 |  664 |  87 |\n","| 4 | 1 | 1004 | 115 |\n","| 5 | 1 | 1231 | 120 |\n","| 6 | 1 | 1372 | 142 |\n","| 7 | 1 | 1582 | 145 |\n","| 8 | 2 |  118 |  33 |\n","| 9 | 2 |  484 |  69 |\n","| 10 | 2 |  664 | 111 |\n","| 11 | 2 | 1004 | 156 |\n","| 12 | 2 | 1231 | 172 |\n","| 13 | 2 | 1372 | 203 |\n","| 14 | 2 | 1582 | 203 |\n","| 15 | 3 |  118 |  30 |\n","| 16 | 3 |  484 |  51 |\n","| 17 | 3 |  664 |  75 |\n","| 18 | 3 | 1004 | 108 |\n","| 19 | 3 | 1231 | 115 |\n","| 20 | 3 | 1372 | 139 |\n","| 21 | 3 | 1582 | 140 |\n","| 22 | 4 |  118 |  32 |\n","| 23 | 4 |  484 |  62 |\n","| 24 | 4 |  664 | 112 |\n","| 25 | 4 | 1004 | 167 |\n","| 26 | 4 | 1231 | 179 |\n","| 27 | 4 | 1372 | 209 |\n","| 28 | 4 | 1582 | 214 |\n","| 29 | 5 |  118 |  30 |\n","| 30 | 5 |  484 |  49 |\n","| 31 | 5 |  664 |  81 |\n","| 32 | 5 | 1004 | 125 |\n","| 33 | 5 | 1231 | 142 |\n","| 34 | 5 | 1372 | 174 |\n","| 35 | 5 | 1582 | 177 |\n","\n"],"text/plain":["   Tree age  circumference\n","1  1     118  30          \n","2  1     484  58          \n","3  1     664  87          \n","4  1    1004 115          \n","5  1    1231 120          \n","6  1    1372 142          \n","7  1    1582 145          \n","8  2     118  33          \n","9  2     484  69          \n","10 2     664 111          \n","11 2    1004 156          \n","12 2    1231 172          \n","13 2    1372 203          \n","14 2    1582 203          \n","15 3     118  30          \n","16 3     484  51          \n","17 3     664  75          \n","18 3    1004 108          \n","19 3    1231 115          \n","20 3    1372 139          \n","21 3    1582 140          \n","22 4     118  32          \n","23 4     484  62          \n","24 4     664 112          \n","25 4    1004 167          \n","26 4    1231 179          \n","27 4    1372 209          \n","28 4    1582 214          \n","29 5     118  30          \n","30 5     484  49          \n","31 5     664  81          \n","32 5    1004 125          \n","33 5    1231 142          \n","34 5    1372 174          \n","35 5    1582 177          "]},"metadata":{},"output_type":"display_data"}],"source":["data <- Orange\n","\n","data"]},{"cell_type":"code","execution_count":null,"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":437},"executionInfo":{"elapsed":385,"status":"ok","timestamp":1705749260957,"user":{"displayName":"Data Science","userId":"07742026815209387612"},"user_tz":-120},"id":"OpxdqmAefe4W","outputId":"399328a5-e233-4a94-99ae-b5361e4e0d54","vscode":{"languageId":"r"}},"outputs":[{"data":{"image/png":"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","text/plain":["plot without title"]},"metadata":{"image/png":{"height":420,"width":420}},"output_type":"display_data"}],"source":["x <- data$age\n","y <- data$circumference\n","\n","\n","#Scatter Plot\n","plot(x , y, xlab = \"age\" , ylab = \"circumference\" )"]},{"cell_type":"markdown","metadata":{"id":"_4iRVJnajf9z"},"source":["The sactter plot above reaveal that points form a roughly straight line and there is a clear linear relationship,so a linear adjustment may be justified."]},{"cell_type":"markdown","metadata":{"id":"jmBiUpMPkR2b"},"source":["We should calculate the regression coefficients"]},{"cell_type":"code","execution_count":null,"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":34},"executionInfo":{"elapsed":293,"status":"ok","timestamp":1705751856095,"user":{"displayName":"Data Science","userId":"07742026815209387612"},"user_tz":-120},"id":"tuW0U0BDu2N3","outputId":"9ed701a4-c83b-4c3c-db14-709c5ba44c55","vscode":{"languageId":"r"}},"outputs":[{"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>17.3996502401635</dd><dt>x</dt><dd>0.106770325068761</dd></dl>\n"],"text/latex":["\\begin{description*}\n","\\item[(Intercept)] 17.3996502401635\n","\\item[x] 0.106770325068761\n","\\end{description*}\n"],"text/markdown":["(Intercept)\n",":   17.3996502401635x\n",":   0.106770325068761\n","\n"],"text/plain":["(Intercept)           x \n"," 17.3996502   0.1067703 "]},"metadata":{},"output_type":"display_data"}],"source":["#Fit a linear model\n","model <- lm(y ~ x , data = data)\n","\n","#Find beta1 and beta0\n","model$coefficients"]},{"cell_type":"markdown","metadata":{"id":"BCAKT9SWlhaR"},"source":["## 2.\tCalculate the residuals and verify the property that the residuals are normally distributed.\n","\n","H0: Residuals are normally distributed\n","\n","H1: Residuals are not normally distributed"]},{"cell_type":"code","execution_count":null,"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":104},"executionInfo":{"elapsed":407,"status":"ok","timestamp":1705749700702,"user":{"displayName":"Data Science","userId":"07742026815209387612"},"user_tz":-120},"id":"Lpg0ETLIlueH","outputId":"b1604d2e-3d02-413f-944c-f7c37d87b324","vscode":{"languageId":"r"}},"outputs":[{"data":{"text/plain":["\n","\tShapiro-Wilk normality test\n","\n","data:  residuals\n","W = 0.97289, p-value = 0.5273\n"]},"metadata":{},"output_type":"display_data"}],"source":["#Find the residuals\n","residuals <- model$residuals\n","\n","#Check the normality of residuals by performing the shapiro test\n","shapiro.test(residuals)"]},{"cell_type":"markdown","metadata":{"id":"NNgg4CpdmvJ6"},"source":["**Interpretation:** W = 0.97289 (close to 1) and p = 0.5273 > 0.05 then we fail to reject H0. This suggests that the residuals are normally distributed"]},{"cell_type":"markdown","metadata":{"id":"5YGxYbQEpvwU"},"source":["## 3.\tCalculate the coefficient of determination $R^2$ and interpret the result.\n","\n","$R^2$ = SSR/SST"]},{"cell_type":"code","execution_count":null,"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":34},"executionInfo":{"elapsed":320,"status":"ok","timestamp":1705751161673,"user":{"displayName":"Data Science","userId":"07742026815209387612"},"user_tz":-120},"id":"BcYcv5nLpr0d","outputId":"7139f213-cb22-4c10-f562-59682408597c","vscode":{"languageId":"r"}},"outputs":[{"data":{"text/html":["0.834516694588369"],"text/latex":["0.834516694588369"],"text/markdown":["0.834516694588369"],"text/plain":["[1] 0.8345167"]},"metadata":{},"output_type":"display_data"}],"source":["#Find SSR\n","ssr <- sum((model$fitted.values - mean(y))^2)\n","\n","#Find SST\n","sst <- sum((y - mean(y))^2)\n","\n","#Find the coefficinet of Determination\n","r_squared <- ssr / sst\n","r_squared"]},{"cell_type":"markdown","metadata":{"id":"g2KmwfcysgoF"},"source":["**Interpretation:** $R^2$ = 0.83 (close to 1)\n","\n","This suggests that approximately 83.45% of the variability in the 'circumference' (dependent variable) can be explained by the linear regression model with the 'age'( independent variable)"]},{"cell_type":"markdown","metadata":{"id":"ZnFrNxmguWCn"},"source":["## 4.\tCalculate the correlation coefficient r and interpret the result."]},{"cell_type":"code","execution_count":15,"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":34},"executionInfo":{"elapsed":319,"status":"ok","timestamp":1705753590499,"user":{"displayName":"Data Science","userId":"07742026815209387612"},"user_tz":-120},"id":"Ur5xqOfjklCP","outputId":"2e5d836d-6e01-418e-ce50-7304eb6ad7fd","vscode":{"languageId":"r"}},"outputs":[{"data":{"text/html":["0.913518852891591"],"text/latex":["0.913518852891591"],"text/markdown":["0.913518852891591"],"text/plain":["[1] 0.9135189"]},"metadata":{},"output_type":"display_data"}],"source":["#Find the correlation coefficient r\n","r <- cor(x , y)\n","r"]},{"cell_type":"markdown","metadata":{"id":"iLZ5D9dolMIF"},"source":["we found that the correlation coefficient: r = 0.9135 wich indicate a strong positive linear relationship between age and circumference"]},{"cell_type":"markdown","metadata":{"id":"TU-fE-KbvbKb"},"source":["## 5.\tSpecify the link between $R^2$ and r."]},{"cell_type":"markdown","metadata":{"id":"UW5T5V2BvjvF"},"source":["r is the squared root of $R^2$"]},{"cell_type":"markdown","metadata":{"id":"X7xngZWhx_Li"},"source":["## 6. Calculate the estimator of β1 using the correlation coefficient r\n","\n","Formula: β1 = r . (sy/sx)"]},{"cell_type":"code","execution_count":17,"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":34},"executionInfo":{"elapsed":327,"status":"ok","timestamp":1705753641050,"user":{"displayName":"Data Science","userId":"07742026815209387612"},"user_tz":-120},"id":"JBcZVbOt1Fh9","outputId":"bd06de1f-0d50-44d6-e5e7-fe0d8508ed2a","vscode":{"languageId":"r"}},"outputs":[{"data":{"text/html":["0.106770325068761"],"text/latex":["0.106770325068761"],"text/markdown":["0.106770325068761"],"text/plain":["[1] 0.1067703"]},"metadata":{},"output_type":"display_data"}],"source":["#Standard Devation of y\n","sy <- sd(y)\n","\n","#Standard Devation of x\n","sx <- sd(x)\n","\n","#Calculate the estimator of β1\n","b1 <- r * (sy/sx)\n","b1"]},{"cell_type":"markdown","metadata":{"id":"fWXQh5Qe2FGo"},"source":["## 7. Hypothesis Test on r\n","\n","H0: ρ(age,circumference)=0\n","\n","H1: ρ(age,circumference)≠0\n","\n","*with a significance threshold α=0.05*"]},{"cell_type":"code","execution_count":20,"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":225},"executionInfo":{"elapsed":298,"status":"ok","timestamp":1705754142867,"user":{"displayName":"Data Science","userId":"07742026815209387612"},"user_tz":-120},"id":"9bdLE1N22WhV","outputId":"3f6b221b-086e-4679-870a-1ab2cdb79e7f","vscode":{"languageId":"r"}},"outputs":[{"data":{"text/plain":["\n","\tPearson's product-moment correlation\n","\n","data:  x and y\n","t = 12.9, df = 33, p-value = 1.931e-14\n","alternative hypothesis: true correlation is not equal to 0\n","95 percent confidence interval:\n"," 0.8342364 0.9557955\n","sample estimates:\n","      cor \n","0.9135189 \n"]},"metadata":{},"output_type":"display_data"},{"name":"stdout","output_type":"stream","text":["t-crirical:  1.69236"]}],"source":["#t-test\n","test <- cor.test(x, y)\n","test\n","\n","#t-critical\n","t_critical <- qt(df= 33, p=0.05)\n","\n","cat( \"t-crirical: \" , abs(t_critical))"]},{"cell_type":"markdown","metadata":{"id":"1UnCC9xE25JZ"},"source":["**Interpretation:**\n","\n","t = 12.9 ⇒ |t| > t-critical ⇒ reject H0 ⇒ ρ(age,circumference) ≠ 0 : There exist a linear relationship between x and y\n","\n","\n"]},{"cell_type":"markdown","metadata":{"id":"87wwMIiz4YQK"},"source":["## 8. Test the hypothesis H0:β1=0 against H1:β1≠0 with α=0.05\n"]},{"cell_type":"code","execution_count":33,"metadata":{"colab":{"base_uri":"https://localhost:8080/"},"executionInfo":{"elapsed":270,"status":"ok","timestamp":1705754764107,"user":{"displayName":"Data Science","userId":"07742026815209387612"},"user_tz":-120},"id":"78qhHblR4nKi","outputId":"05e8f1f8-14d4-4d59-f21d-727d901fd8e1","vscode":{"languageId":"r"}},"outputs":[{"name":"stdout","output_type":"stream","text":["t-value: 12.90023\n","t-crirical:  1.69236"]}],"source":["#t-value\n","summary <- summary(model)\n","coef_table <- summary$coefficients\n","t_value <- coef_table[6]\n","\n","#ptint t value\n","cat(\"t-value:\" , t_value)\n","\n","#print t-critical\n","cat( \"\\nt-crirical: \" , abs(t_critical))"]},{"cell_type":"markdown","metadata":{"id":"yMQqC7RB6Skr"},"source":["**Interpretation:**\n","\n","t = 12.9 ⇒ |t| > t-critical ⇒ reject H0 ⇒ β1 ≠ 0 : slope significantly different from 0 , existance of linear relationship between x and y\n","\n","\n"]},{"cell_type":"markdown","metadata":{"id":"cPosAdEt6y2m"},"source":["## 9. Establish the ANOVA table associated with this regression. What can we conclude about parameter β1?"]},{"cell_type":"code","execution_count":34,"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":161},"executionInfo":{"elapsed":284,"status":"ok","timestamp":1705755008625,"user":{"displayName":"Data Science","userId":"07742026815209387612"},"user_tz":-120},"id":"jUJHu1BA67NM","outputId":"8a9a21c5-dfb7-4ccc-b5cf-40333a383d75","vscode":{"languageId":"r"}},"outputs":[{"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>x</th><td> 1</td><td>93771.54</td><td>93771.5413</td><td>166.4159</td><td>1.930596e-14</td></tr>\n","\t<tr><th scope=row>Residuals</th><td>33</td><td>18594.74</td><td>  563.4771</td><td>      NA</td><td>          NA</td></tr>\n","</tbody>\n","</table>\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","\tx &  1 & 93771.54 & 93771.5413 & 166.4159 & 1.930596e-14\\\\\n","\tResiduals & 33 & 18594.74 &   563.4771 &       NA &           NA\\\\\n","\\end{tabular}\n"],"text/markdown":["\n","A 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","| x |  1 | 93771.54 | 93771.5413 | 166.4159 | 1.930596e-14 |\n","| Residuals | 33 | 18594.74 |   563.4771 |       NA |           NA |\n","\n"],"text/plain":["          Df Sum Sq   Mean Sq    F value  Pr(>F)      \n","x          1 93771.54 93771.5413 166.4159 1.930596e-14\n","Residuals 33 18594.74   563.4771       NA           NA"]},"metadata":{},"output_type":"display_data"}],"source":["anova(model)"]},{"cell_type":"code","execution_count":35,"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":34},"executionInfo":{"elapsed":295,"status":"ok","timestamp":1705755099615,"user":{"displayName":"Data Science","userId":"07742026815209387612"},"user_tz":-120},"id":"PqhbBTKp7AAN","outputId":"07417c34-b569-4b4f-c7de-69a0c4024302","vscode":{"languageId":"r"}},"outputs":[{"data":{"text/html":["4.13925249555537"],"text/latex":["4.13925249555537"],"text/markdown":["4.13925249555537"],"text/plain":["[1] 4.139252"]},"metadata":{},"output_type":"display_data"}],"source":["#F-critical\n","f_critical <- qf(df1=1, df2=35-2, p=.95)\n","f_critical"]},{"cell_type":"markdown","metadata":{"id":"NXlQFaxN7bW4"},"source":["**Interpretation:**\n","\n","F = 166.4159 ⇒ F > F-critical ⇒ reject H0 ⇒ β1 ≠ 0 : slope significantly different from 0 , existance of linear relationship between x and y\n","\n","\n"]},{"cell_type":"markdown","metadata":{"id":"Rpy9tBlz7wJ-"},"source":["## 10. Construct a 95% confidence interval for parameter 1. What can we conclude about parameter"]},{"cell_type":"code","execution_count":38,"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":98},"executionInfo":{"elapsed":309,"status":"ok","timestamp":1705755569517,"user":{"displayName":"Data Science","userId":"07742026815209387612"},"user_tz":-120},"id":"fUi2aKiB757x","outputId":"89b0e7d9-2369-4024-cb31-378f1b68b2fd","vscode":{"languageId":"r"}},"outputs":[{"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>x</th><td>0.08993141</td><td>0.1236092</td></tr>\n","</tbody>\n","</table>\n"],"text/latex":["A matrix: 1 × 2 of type dbl\n","\\begin{tabular}{r|ll}\n","  & 2.5 \\% & 97.5 \\%\\\\\n","\\hline\n","\tx & 0.08993141 & 0.1236092\\\\\n","\\end{tabular}\n"],"text/markdown":["\n","A matrix: 1 × 2 of type dbl\n","\n","| <!--/--> | 2.5 % | 97.5 % |\n","|---|---|---|\n","| x | 0.08993141 | 0.1236092 |\n","\n"],"text/plain":["  2.5 %      97.5 %   \n","x 0.08993141 0.1236092"]},"metadata":{},"output_type":"display_data"}],"source":["confint(object=model, parm=\"x\", level=.95)"]},{"cell_type":"markdown","metadata":{"id":"pR7lJmO7-AtR"},"source":["# Exercice 3\n","\n","We recorded for different countries the GDP per capita in 2004 X (in dollars) and the gross enrollment rate of those under 24 in the same year Y (in percentage). The results are as follows:\n","\n","![Picture1.png](data:image/png;base64,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)\n","\n"]},{"cell_type":"markdown","metadata":{"id":"2q0jiFvX-zXg"},"source":["## 1.\tIdentify the variable to be explained and the explanatory variable."]},{"cell_type":"markdown","metadata":{"id":"u2_kGMZo_QmC"},"source":["\n","\n","*   Variable to be Explained (Dependent Variable): The gross enrollment rate  in percentage.\n","\n","*   Explanatory Variable (Independent Variable): The GDP (X) in dollars.\n","\n"]},{"cell_type":"markdown","metadata":{"id":"NzWo3yHiAxol"},"source":["## 2.\tSpecify the conditions for applying simple linear regression and give the equation of the theoretical model."]},{"cell_type":"markdown","metadata":{"id":"GCCoR5fVADda"},"source":["Simple linear regression is appropriate when certain conditions are met:\n","\n","\n","\n","\n","\n","*   Linearity: There should be a linear relationship between the explanatory variable (X) and the variable to be explained (Y). This can be assessed by plotting a scatterplot of the data.\n","\n","\n","*   Independence: The observations should be independent of each other. In the\n","context of this problem, this means that the data points for different countries should not be dependent on each other.\n","\n","\n","*   Homoscedasticity: The variability of the residuals (the differences between the observed and predicted values) should be roughly constant across all levels of the explanatory variable. This can be checked by plotting the residuals against the predicted values.\n","\n","\n","*  Normality of Residuals: The residuals should be approximately normally distributed. This can be assessed through a histogram or a normal probability plot of the residuals.\n","\n","\n","\n","\n","\n","Given these conditions, the theoretical model for simple linear regression is expressed as:\n","\n","y = β0 + β1⋅xi\n","\n"]},{"cell_type":"markdown","metadata":{"id":"7aADTtVFA25K"},"source":["## 3.\tCalculate the covariance between X and Y then the correlation coefficient r. Interpret the result.\n","\n","Formula:  \n","\n","cov(x,y) = Sxy / n-1\n","\n","r = cov(x,y) / (sx . sy)\n"]},{"cell_type":"code","execution_count":39,"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":34},"executionInfo":{"elapsed":304,"status":"ok","timestamp":1705756984909,"user":{"displayName":"Data Science","userId":"07742026815209387612"},"user_tz":-120},"id":"AHMF_9vxCRp2","outputId":"c5cde71e-02c1-4ef2-ab67-edf800779996","vscode":{"languageId":"r"}},"outputs":[{"data":{"text/html":["36176.1964285714"],"text/latex":["36176.1964285714"],"text/markdown":["36176.1964285714"],"text/plain":["[1] 36176.2"]},"metadata":{},"output_type":"display_data"}],"source":["n <- 8\n","x_bar <- 39457/n\n","y_bar <- 509/n\n","sumxy <- 2763685\n","sxy <- sumxy - n*x_bar*y_bar\n","\n","#Find the covariance\n","covxy <- sxy/(n-1)\n","covxy"]},{"cell_type":"code","execution_count":40,"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":34},"executionInfo":{"elapsed":299,"status":"ok","timestamp":1705757130897,"user":{"displayName":"Data Science","userId":"07742026815209387612"},"user_tz":-120},"id":"h89HR01BCohE","outputId":"9797afaa-0f29-4f8d-9f64-9cca8f867768","vscode":{"languageId":"r"}},"outputs":[{"data":{"text/html":["0.984798516287842"],"text/latex":["0.984798516287842"],"text/markdown":["0.984798516287842"],"text/plain":["[1] 0.9847985"]},"metadata":{},"output_type":"display_data"}],"source":["ssx <- 245474957 - (n*x_bar*x_bar)\n","sx <- sqrt(ssx/(n-1))\n","\n","ssy <- 33685 - (n*y_bar*y_bar)\n","sy <- sqrt(ssy/(n-1))\n","\n","#Find r\n","r <- covxy/(sx*sy)\n","r"]},{"cell_type":"markdown","metadata":{"id":"FjUWtYi0ElVC"},"source":["we found that the correlation coefficient: r = 0.9847 wich indicate a strong positive linear relationship between age and circumference"]},{"cell_type":"markdown","metadata":{"id":"lxbV6p0cDaPA"},"source":["## 4.\tGive the estimated values of the unknown coefficients β0 and β1."]},{"cell_type":"code","execution_count":41,"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":34},"executionInfo":{"elapsed":316,"status":"ok","timestamp":1705757307437,"user":{"displayName":"Data Science","userId":"07742026815209387612"},"user_tz":-120},"id":"XL2jIEXODtey","outputId":"4bae00e3-2cda-4459-9510-d9c228246bc6","vscode":{"languageId":"r"}},"outputs":[{"data":{"text/html":["0.00497823529174559"],"text/latex":["0.00497823529174559"],"text/markdown":["0.00497823529174559"],"text/plain":["[1] 0.004978235"]},"metadata":{},"output_type":"display_data"}],"source":["b1 <- r*(sy/sx)\n","b1"]},{"cell_type":"code","execution_count":42,"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":34},"executionInfo":{"elapsed":278,"status":"ok","timestamp":1705757326603,"user":{"displayName":"Data Science","userId":"07742026815209387612"},"user_tz":-120},"id":"xOKJoKFfDxJe","outputId":"96f72328-42fc-4318-b58c-46700d39b58f","vscode":{"languageId":"r"}},"outputs":[{"data":{"text/html":["39.0717212616993"],"text/latex":["39.0717212616993"],"text/markdown":["39.0717212616993"],"text/plain":["[1] 39.07172"]},"metadata":{},"output_type":"display_data"}],"source":["b0 <- y_bar - b1*x_bar\n","b0"]},{"cell_type":"markdown","metadata":{"id":"nC2fptkmD4Se"},"source":["## 5.\tDetermine the coefficient of determination $R^2$ and interpret the result."]},{"cell_type":"code","execution_count":43,"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":34},"executionInfo":{"elapsed":294,"status":"ok","timestamp":1705757379069,"user":{"displayName":"Data Science","userId":"07742026815209387612"},"user_tz":-120},"id":"gShMMwWWD_3v","outputId":"ad2d87ed-0379-4a74-89f6-a898348c9f3e","vscode":{"languageId":"r"}},"outputs":[{"data":{"text/html":["0.969828117682736"],"text/latex":["0.969828117682736"],"text/markdown":["0.969828117682736"],"text/plain":["[1] 0.9698281"]},"metadata":{},"output_type":"display_data"}],"source":["r_squared <- r**2\n","r_squared"]},{"cell_type":"markdown","metadata":{"id":"rZPY_yzgEFDi"},"source":["**Interpretation:** $R^2$ = 0.96(close to 1)\n","\n","This suggests that approximately 96.98% of the variability in the 'rate' (dependent variable) can be explained by the linear regression model with the 'GDP'( independent variable)"]},{"cell_type":"markdown","metadata":{"id":"Kte1jAPhEtX7"},"source":["## \tTest the hypothesis H0: ρ(X,Y) = 0 (against H1: ρ(X,Y) ≠0) with a significance threshold α=5%"]},{"cell_type":"code","execution_count":44,"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":34},"executionInfo":{"elapsed":300,"status":"ok","timestamp":1705757608962,"user":{"displayName":"Data Science","userId":"07742026815209387612"},"user_tz":-120},"id":"4eRAyoOtE3_D","outputId":"f723b3c6-3aad-48c1-e69a-c53f82122b0c","vscode":{"languageId":"r"}},"outputs":[{"data":{"text/html":["13.8874276132552"],"text/latex":["13.8874276132552"],"text/markdown":["13.8874276132552"],"text/plain":["[1] 13.88743"]},"metadata":{},"output_type":"display_data"}],"source":["# t value\n","r*sqrt(n-2)/sqrt(1-r_squared)"]},{"cell_type":"code","execution_count":45,"metadata":{"colab":{"base_uri":"https://localhost:8080/","height":34},"executionInfo":{"elapsed":292,"status":"ok","timestamp":1705757651751,"user":{"displayName":"Data Science","userId":"07742026815209387612"},"user_tz":-120},"id":"fjUIpVLRE5dC","outputId":"1ec6cd3b-9880-49cc-949f-136b1c7a1965","vscode":{"languageId":"r"}},"outputs":[{"data":{"text/html":["1.9431802805153"],"text/latex":["1.9431802805153"],"text/markdown":["1.9431802805153"],"text/plain":["[1] 1.94318"]},"metadata":{},"output_type":"display_data"}],"source":["# t critical\n","qt(df=n-2, p=.95)"]},{"cell_type":"markdown","metadata":{"id":"Jbd3NlZ4FBns"},"source":["**Interpretation:**\n","\n","t = 13.88 ⇒ |t| > t-critical ⇒ reject H0 ⇒ ρ(x,y) ≠ 0 : There exist a linear relationship between x and y\n","\n","\n"]}],"metadata":{"colab":{"authorship_tag":"ABX9TyOfiBqNdRBN8Iz4Evj0Ojti","provenance":[],"toc_visible":true},"kernelspec":{"display_name":"R","name":"ir"},"language_info":{"name":"R"}},"nbformat":4,"nbformat_minor":0}