A Level Maths Tuition Lesson 19 Statistics: Correlation and Regression Exam Practice!

hey hello and welcome back to another one of these a-level maths uh videos so again we still want statistics and one like one of the last parts of the data section of Statistics which is going to be correlation and regression so let's go through okay so if we have two pieces of data right let's call them X Y for now something similar to that right what we normally do is we can draw a line of best fit in theory it should be a straight line and what this will help us do hello Nathan how are you and what this essentially allows us to do is have a look at the data and tell us if it's a positive correlation negative correlation if there's no correlation at all so what correlation means is that in this sense as X goes up does y increase decrease or stay the same So today we're going to be doing the correlation and regression part of a level statistics a level math statistics so I'm doing this as slowly as possible because statistics can be quite confusing but we have all of the exam questions from the past few years ready to go I'm just going to go through some of the notes for this at the beginning so essentially this is what correlation tells us it tells us what happens as we increase X okay now X is normally called our independent variable in science or even in maths beforehand however with Statistics we instead call it the explanatory variable and why we sometimes call that the dependent variable or at least we used to but now in a level statistics we're going to be calling it The response variable okay so sometimes in a question it might just be one mark and it might literally just say what is the response variable what's the explanatory variable so it's good to just know that so that's pretty much it however the whole thing about regression is that when we draw a line of best fit it's an approximation in the same way if you remember back at GCSE you know remember when you had to find the gradient of a curve at a point we drew a tangent and it was a um approximation because we haven't calculated it right like I could change this line of best fit slightly and it would still be correct but there's obviously a mathematically correct way of doing it so the line of regression is simply the same as the line of best fit but we are going to calculate it numerically we're actually going to have a way of calculating it and again it's going to take the form technically the form is just y equals MX plus b however in a level statistics sometimes they change it round to B plus ax as an example if you look that's just the exact same thing right okay the next thing we need to do is obviously be able to work out if something's positive correlation negative correlation or no correlation and the best way for us to do that is there is a number called the pmcc I'm not going to go through too many of the notes because I don't think it's necessary we're going to go into exam questions in literally two seconds and that is the product moment correlation coefficient so it's it's an extremely long name and I don't think it's really necessary okay but essentially it's just a number that goes from minus one all the way up to plus one where plus one is perfect positive correlation perfect positive correlation and minus one is perfect negative correlation and obviously right in the middle there's going to be no correlation so the closer the number is to zero the less correlation there is the more negative it is the more negative correlation there is the more positive is the more positive correlation and that's pretty much all the notes you need to do they will give you the product moment correlation coefficient however I would like to warn you that if you do older past papers you did have to calculate this in the old spec but in this spec you don't need to do it and but essentially all that is for that so if you're going to some exam questions so the first one thing here so football coach measures the Heights and weights of 12 players they give this data here height and weight and you get that and it says draw a scatter graph for this information so drawing an accurate scatter graph on this is pretty tricky so I just found a way of plotting it here so this is the scatter graph you should get shouldn't be too tricky for you to do I imagine um but again let me know if you would like me to go through it but they normally give you the axes in the exam but I didn't put that on here just yet okay now for Part B it says give an interpretation of the correlation between the height and weight of the footballers so if I have a look at this as height increases so does the weight right it seems like a very positive correlation okay so when they say given interpretation all you're going to say is you know what does it mean the fact that these increase well all you have to do is literally write out what it says so I'm gonna actually write this by text so as the Heights of the players increases the height of the players increases their weight also increases you could also say something like um taller players tend to weigh more and both those would give you the mark maybe that's a fairly obvious to you at the moment however it is the equation of the regression line is W equals 1.37 H minus 173 okay so the equation for the weight is just equal to 1.37 times the height minus 173 okay and remember this regression line is going to give us our line of best fit so something that well that was terrible something that looks something similar to that um for Part B so as the height of the play increases this bit here yeah okay so all they're asking you to do is to have a look at the graph and give your interpretation of the correlation so what I would say first is is it positive or negative correlation this is all you should think in your head right positive correlation is when the as the X variable increases so this variable here the Y should also increase so basically if the graph is going up and to the right it's positive correlation okay so that's the first thing I'll do so I looked at this graph and I saw as the height increases so does the weight right so then you have to interpret that the positive correlation well all it's saying is as the heights of the players increases their weight also increases does that make a bit more sense yeah okay perfect Nathan uh where did the Discord go uh I haven't finished it yet um I could just post the link though maybe I'll put the I'll put the link in the chat at the end of the lesson and that should be fine perfect so for C give an interpretation of the gradient of the regression line well think of it as just the gradient in a normal sense right so we have this straight line going through here all we need to do is say what the gradient represents so the gradient all it represents is so first of all the the weight is the so w is the weight right and H is the height so with the gradient of 1.37 always saying is for every centimeter gained in height the player gains 1.37 kilograms in weight when you're going over modeling uh what kind of modeling like in in Pure or in statistics it's not like binomial distribution and things like that that should be next week or the week after so that's all you have to do for the gradient okay all you have to do is look at the number in front right so I'm looking at the 1.37 modeling for differentiation um it's a bit later on the ratio will be X centimeters yeah pretty much so every time the height increases by one we're gaining 1.37 kilograms in weight yeah so we have to do is look at the number in front of the X variable and use that to your advantage right so weight is 1.37 times height height okay so modeling for differentiation will come up a bit later on not just yet although I have um videos for cie a level maths differentiation that are coming out soon so Part D it says use the equation of the regression line to estimate the weight of a player who's 170 centimeters tall all you have to do is substitute in h is 170 that's all you have to do so then weight is equal to 1.37 times 170 minus 173. okay then you can use a calculator which is 59.9 kilograms okay hopefully that's okay so far lastly for part a it says comment on the reliability of your estimate in part D so this is a question that they're always going to ask you in a level statistics is comment on the reliability so all you have to do is this isn't the answer this is just going to be some notes if the value is inside the data range it's reliable if the value is outside the data range unreliable that's all you have to say okay so we subbed in 170 centimeters is 170 centimeters in this data range it is right 170 is right here so because it's inside this data right we have data from around 170. don't see noisy okay I'll try and uh sort that out I was pressing the keys a bit hard so when I say data range look at the range of values we have we go from 160 centimeters to 200 centimeters which means 170 centimeters is perfectly fine so um it is reliable because 170 centimeters is inside the range of okay does that make sense so far Nathan are you okay with this so far not really which bit is um a bit shaky so I'm trying to teach through using the exam questions so was the substitution okay for Part D was it putty that you're a bit um confused on okay so I'll just go over this last bit again it asks us to comment on the reliability of the estimate in part D which was when we used the regression line to estimate the weight of a play is 170 centimeters tall okay so here's your conditions if the value that you substituted in inside the data range it's a reliable estimate if it's outside it's unreliable so in this case we subbed in 170 centimeters now if I look at my range of data I go from 160 centimeters to 200 so it is reliable if they asked me for 210 centimeters or even 150 centimeters those are not reliable because they're not on the graph okay that's all you have to remember if it's inside the range of values then you're fine it's outside then you're not fine okay and again you're only looking at the value that you're actually substituting in okay so for the next part we have so again these are like the questions that specifically involve linear regret regression in um in this um but obviously I've had to take out parts that are like not related to it so 2021 paper three I just realized that I don't actually have the uh graph give me one second so there's a scatter graph and he took a random sample of 16 students from a school and for each student recorded the number of letters in their last name the number letters in their first name his results are shown in the static diagram on the next page describe the correlation between X and Y so if we look here can we see any kind of correlation well it's ever so slightly negative right like if I were to draw a little line going through this maybe move it down a little bit that would be roughly where your line of best fit would be as you can see the lines going down therefore it is negative correlation Mark suggests that parents with long last names tend to give their children shorter first names use the scatter diagram to comment on Mark's suggestion giving a reason so if you look here x is the last name so I'm going to write last name next to it just to make it a bit clearer first name so as the length of the last name increases the length of the first name does decrease so it's sensible as there is negative correlation between the variables okay so all that means is that as the length of last name increases the length of the first name does seem to decrease again we can't be sure and what tends to happen in the next couple of parts is that you're going to actually work out um hypothesis testing so you'd actually test whether or not it is indeed true but that's coming up a bit later so we're not going to do it today the next question so Tessa owns a small clothes shop in a Seaside town she records the weekly sales figures pound w and the average weekly temperature 2 degrees for eight weeks during the summer the product moment correlation coefficient for this data is minus 0.915 if you remember it goes from -1 two plus one and this thing here is very close to negative one so it's strong negative correlation that's what that's telling me stating your hypotheses clearly and using a five percent level of significance so this is hypothesis testing so we're not going to do this however I'm just going to write that it is indeed it is actually negative okay this is just a possible reason for this correlation so why would um so as the temperature increases why would the sales figures decrease well it could be that as the temperature it could be as the temperature increases people tend to spend more time uh at the beach rather than shopping right it could be something like that so they'd rather be outside than inside shopping so that could be one reason for it but it's basically any sensible reason so as temperature drops people are more likely to go to um shopping instead of the beach as the temperature increases people would rather be outside um if the temperatures large then maybe they don't want to go shopping or let's close less clothes needed as the temperature increases any of those is completely fine any valid suggestion now this is just a linear regression model could be used to model these data States given a reason whether or not the Coalition correlation coefficient is consistent with this well potentially this is going to be like a straight line graph so if R is away from zero so the magnitude of R is closer to one then you can indeed use regression models in this case r is close to negative one so it is consistent and then what would be the explanatory variable so which one is the independent variable well that's going to be the temperature why because sales are likely to depend on the temperature right it makes more sense that the sales are affected by the temperature then the temperature being affected by the sales right when I buy more clothes it doesn't make the world hotter whereas if the world's hotter it might affect how many clothes I buy and lastly we have again given interpretation of the gradient of this regression equation well for every one degree or every degree rise in temperature leads to a drop in weekly sales of 171 pounds again if you look at the thing here we have the variable of temperature it's 171 times it's negative 171 so it's decreasing by 171 for every increase of T of 1. lastly the sample paper so meteorologist believes that there is a relationship between daily mean wind speed and the daily temperature a random sample of nine consecutive days is taken from past record to return in the UK in July and the relevant data is given in the table below so the meteorologists calculate the product moment correlation coefficient for the last nine days and obtain 0.609 so explain why a linear regression model based on these data is unreliable on a day when the mean temperature is 24 degrees well if we look at the temperatures here none of them are anywhere near 24 degrees which means it's outside of the data range so all you'd have to do is write 24 degrees outside the data range hence extrapolation will be needed anytime you extrapolate it's unreliable okay puppy says Statewide is measured by the product moment correlation coefficient it's just the relationship between uh two variables right so how strongly they are tied together Posse again is a hypothesis tester so we're going to skip that because we're doing that next week so next week's going to be just doing probability distribution in general for binomial then it's going to be hypothesis testing then we move on to mechanics and that will be okay but yeah that's pretty much everything for correlation and linear regression from the past few years um it shouldn't be too bad it's mainly interpreting data and looking at different data sets but again next week we're definitely going to be doing the um so actually let me double check exactly what we're doing so yeah next week's going to probability then discrete random variables sampling hypothesis testing and then we're going to move on to as mechanics it should be quite fun and then we got into a2pr yeah until then have a good week