[Music] assalamualaikum and hi everyone welcome to topic 8 correlation and regression before we start let's go through about what you will learn in the last topic for am025 which is topic 8. first of all there will be sub topic 8. 1 linear correlation and the linear correlation you need to find the Pearson product moment correlation coefficient which is denoted by R and then you need to interpret the value of r after that there will be 8.
2 simple linear regression and the 8. 2 simple linear regression you need to find the equation of regression line which is y hat equals to A Plus BX and then you need to determine A and B in the regression line using least Square method after that you need to interpret the value of a and b in the regression line more than that you need to determine the coefficient of determination denoted by r squared lastly you need to interpret the value of R square that you get so these are the lesson that you will learn in this topic please prepare your pencil your calculator and your module stay focused and in a formats topic 8 correlation and regression like to one of three subtopic 8. 1 linear correlation at the end of the lesson you should be able to 8.
1 a use a scatter diagram to describe the relationship between two variables B solve and interpret the patient product moment correlation coefficient which is denoted by R first of all let's recall what are the variables in this topic as you remember from your graph paper you will have the x-axis and the y-axis for the x-axis it is referring to X variable for the y-axis it will be y variable 4X variables we would say that it will be our independent variable in other words it is the explanatory variable for y-axis it is the dependent variable in other words it is the response variable the explanatory variable is the cause whereas for dependent variable which is the response variable is the effect from the cost let's look at example for independent variable and dependent variable for independent variable let's say we have temperature and then for dependent variable we have shares of ice cream from here if you can relate between these two variables between temperature and the sales of ice cream what will happen if temperature rises to the sales of ice cream because the independent variable is the one that caused the problem whereas for dependent variable is the one that will get affected so if you can see here as the temperature increases the sales of ice cream will increase as well it would make sense if you can read it like this when the temperature increase the desire of people in order to eat something cool is increase so that's why increase in temperature the sales of ice cream will increase as well as a conclusion we may write like this temperature may affect the sales of ice cream let's proceed with part a scatter diagram to describe the relationship between two variables which is the X variable and Y variable first a scatter diagram shows the relationship between two variables measured on the same individual secondly each individual in the data set is represented by a point in the scatter diagram thirdly the independent variable X is plotted on the horizontal axis and the dependent variable y Y is plotted on the vertical axis the examples of relationships are the first one your price of car maybe depend on your income so the price of car will be your y variable and then your income will be the X variable secondly your expenses may be determined by your salary which one is your y variable and X variable the expenses will be your variable whereas your salary will be your X variable thirdly the number of births in a country can be used to determine the number of population from here you can see that the number of births may affect the number of population which means that the number of births will be our X variable and then the number of population will be our y variable next is about the different type of relationships first you will have the positive relationship second negative relationship third no relationship if you can see here these are the examples of scatter diagram where you can see the point are scattered everywhere nevertheless positive relationship negative relationship or no relationship the points are scattered like this that's why these are called scatter diagram for the positive relationship if you can see the trend would be as the X increase the Y increase as well for negative relationship the trend will be as X increase the Y will decrease for no relationship we cannot say x increased y increase or X increase y decrease because this character point is everywhere so the trend cannot be determined from the scatter diagram we would know about whether there is any relationship between the variables B whether the relationship is linear or non-linear C whether the relationship is positive or negative next let's continue with Part B solve and interpret the patient product moment correlation coefficient first of all let's see about correlation first correlation is a statistical measurement of the relationship between two variables so in this case we will discussing about the X variable and the Y variable for example we have as dead temperature went up the series of ice cream went up as well the next one is coefficient of correlation which is denoted by r what is coefficient of correlation coefficient of correlation is a numerical value which measures the strength or degree of the relationship between two random variables which is given by R equals to s x y over square root as x x times s y y so these are the formula in order to find x x y s x x x y y for s x y it is equals to summation of x times y minus summation x times summation y over n whereas for s x x is equals to summation of x squared minus summation X in bracket Square Over N for x y y is equals to summation y Square minus summation Y in bracket Square over n n is the number of pad observation if you can see the pattern of X at y s x x and x y y from the formula it is just slightly different between them for summation X Y is 4 s x y 4 XX the summation will be summation x squared for x y y it is a summation of Y square if you can see the pattern or the difference between s x y s a x and xyy you can remember this formula easily the correlation coefficient has values varies from positive one to negative 1 inclusively of both values in other words we will see that the modulus are less than one for linear correlation are the interpretation of the values will be firstly we would say that the correlation is perfect linear if the value of R is positive 1 or negative 1. whereas we would say that the relationship between X and Y is strong relationship if the value is 0. 82 less than 1.
for moderate the value would be 0. 42 less than 0. 8 for weak relationship the value is more than 0 and less than 0.
4 lastly if the value is zero which means that X and Y has no relationship for a better understanding for you in order to define the value of R let me show you a picture of number line so here if you can see if the value of R is 0 it shows that there is no relationship between X and Y then from 0 to the right hand side it will be defined as positive side whereas from zero to the left hand side it will Define for [Music] negative leading okay then if you can see here from 0 to 0. 4 the value will be defined as V positive linear whereas for 0. 4 until 0.
8 the value is defined as moderate positive linear and for 0. 8 until 1 the value is defined for strong positive linear it will be the same as the negative side where for Value 0 until negative 0. 4 it will be defined as with negative linear from negative 0.
4 until negative 0. 8 it will Define as moderate negative linear and from the value negative 0. 8 to negative 1 it will Define as strong negative linear then the value that you get you need to interpret it so the interpretation of R would be there is a here you need to choose whether you are give you the definition of perfect or strong or moderate or weak or no so the colored one is the one that you need to choose the next one where you need to choose whether the value of R gives you positive linear or negative linear and then you need to write linear relationship between here you need to Define your X variable and here you need to Define your y variable so these are the scheme in order for you to interpret your r example one the following data indicates the level of sales for 10 models of pen sold by a particular company the cells together with the selling price of spend are given below calculate the coefficient of correlation and interpret your answer give your answer correct to four decimal places just before we start please watch and learn on how to use your calculator for this topic you may scan this respected QR code to watch the video about it if you are already done let's continue with the examples let's list out all the submission that you get from your calculator your n would be equals to 10 because of 10 models of pen the summation of X is one for zero point five the summation of Y is 193 and the summation of x square would be 2 7 2 3.
75 summation of Y square is four four eight nine and in the summation of X Y would be 2 O 8 7. 5 what you have to do is to list out all the summation that you get from your calculator after you list out all the summation from your calculator then you need to know what are the formula involved in your question the question asking for the coefficient of correlation denoted by R okay and then you have to interpret your value of r the formula of R is R equals to s x y over square root x x times x y y the formula for xxy XXX x y y ah here so you have to find as x y s x x x x y y let's start with x x y as x y equals to just put all the values in the formula so you will get [Music] 2087. 5 minus summation X is one for 0.
5 summation of Y one nine three divided by 10 then just press your calculator you will get your sxy is equals to negative 624. 15 and then you need to find as x x x x equals to just put the values that you get into the formula you will get [Music] 2723. 75 minus with one four zero point five square over 10.
just press your calculator and you will get your sxx is equals to 749. 72 5 the next one is your s y y s y y is equals to summation y Square 4 4 8 9 minus which one nine three Square divided by 10. press your calculator and you will get the answer as y y is 764.
1 then just put all the values that you get as x y s x x s y y into the formula of coefficient of correlation R is equals to negative 624. 15 divided by square root of XXX is seven four nine point seven two five times three seven six four point one okay and then just press your calculator correctly you will get your value of R is equal to negative zero point eight two four six so this will be your value of r the next part is you need to interpret your answer but before that please ensure that your answer is correct to four decimal places so the interpretation of R would be there is a strong negative linear relationship between the price and the sales why the interpretation is negative because here we have negative value that's why it said negative linear relationship and then the value is negative 0. 8246 from the scale just now we can see that this will be strong negative linear in other words we would say that selling cheaper pen will increase the level of sales which means that the price of pen decreases the sales of ban will increase and these examples is about inverse relationship let's continue with example 2 the following table shows the annual profit obtained by eight small Industries which referring to our value of n is equals to 8 for a particular State versus the amount invested different from example one from the example one we already know what is X and what is y for example 2 the question detested about X variable or Y variable so here firstly you need to Define your X and your y from this statement it showed that the annual profit obtained which means that the amount invested will affect the amount of profit so from here we know that the annual profit will be our sorry will be our y variable and then the amount of invested will be our X variable so Student please take note you need to Define your X and your y firstly if not the value of the summation that you get from your calculator will be wrong and then the question asks you to find or to calculate the coefficient of correlation denoted by r and interpret your answer give your answer correct to four decimal places first of all list out all the submission from your calculator let's substitute n is equals to A and then you will get the value of the summation would be equals to for summation X is 263 summation Y is 386 summation x square 9195 summation y square is nineteen thousand two hundred forty two summation x y 13 000.
245. and then please remember the formula for R is s x y divided by square root x x times x y y and then the formula for the respected s is okay so let's find 4 x x y s x y is equals to put the values inside the formula you will have such that one thirteen thousand two hundred forty-five minus this summation X 263 times 386 divided by eight pressure calculated correctly you will get your answer 555. 25 and then proceed with sxx sxx is equals to 9195 minus [Music] 263 power of 2 divided by 8.
press your calculator you will get the value of XXX is equals to 548. 875 the next one is s x y sorry s y y is equals to 19242 minus [Music] 386 power of 2 divided by eight [Music] pressure calculator correctly you will get the answer for x y y is 617. 5 and then the formula for R is equals to the value of x at y [Music] 555.
25 divided by square root sxx five four eight point eight seven five times three six one seven point five is equals to 0.
