A Level Maths Grade A Normal Distribution Problem Solving Explained

hey guys welcome back to another a Lev maths video this time we're finally starting to go through kind of hard problem solving exam questions um so again my students have sent me various questions that they have found difficult and when I've been writing explanations for them I found that yeah it does require some weird Nuance to it it's a little bit trickier than the normal question so this here is a normal distribution question 14 marks in total is pretty fat but remember there's only around four or so stats questions and four or so mechanics questions in your actual exam so they are very fat questions now it's part A and Part D I don't think are too bad as long as you know how to do a hypothesis test and for part A as long as you know how to do the normal distribution but part B and C where you have to use quite a bit of problem solving techniques so I'm going to go through that and well we're going to we're going to see so for part A we have the lifetime L hours of a battery is has a normal distribution so we have L approximated by a normal distribution and has a mean of 18 a standard deviation of four so comma 4 squ and Alice's calculator requires four batteries and we stop working when any one of the batteries reaches the end of its lifetime this believe it or not this is um useful information for the later part in the question for part A just says find the probability that it's any selected battery will last longer than 16 hours so you're looking for the probability that L is greater than 16 which if you're doing that on a normal distributions quite nice you just go to your menu you find wherever your distributions are I went too far down then you do the normal cumulative distribution and you enter in your values now the lower value is of course where we're starting from which is 16 the upper value you just need to enter in a really really big number mu uh mu is 18 standard deviation whoops got that the wrong way around Sigma is four mu is 18 I could have swore that said me and we get 0. 6915 I normally give it to four or five decimal places it doesn't really matter though so I'll just give it to four decimal places 0.691 five to 4 DP so again that part wasn't meant to be the Super tricky part the tricky part is for Part B and C and you see Part B is a probability question for five marks which is it should be illegal um but let's have a look at it so if we have a look at this it says at the start of her exams Alice put four new batteries in her calculator and she has used her calculator for 16 hours already but has another 4 hours of exam to sit find the probability that her calculator will not stop working for Alice's remaining exams now remember it will stop working if any one of the four batteries stops running runs out so again you only need one battery to fail now if we go through how how this will look I'm going to write it out in terms of the expression first so I can explain it so we want the probability that the lifetime of any One battery is going to be greater than 20 now where does that come from well it says here she's used to calculate for 16 hours and has another 4 hours of exams so in total it needs to last at least 20 hours right so 20 21 hours 22 hours 23 hours and so on keep that in mind now here's the bit that is quite tricky CU I'm going to use conditional probability now how do I know that it's conditional well the first way is I'm doing a bit of meta analysis here is they've told me she's used it for 16 hours that has to be relevant in some way and if you look at part A it says find the probability that the battery will last longer than 16 hours so you know it's kind of there's a bit of a a mystery going on here but the second reason is it's conditional because the longer that you use a battery the greater the probability it will run out in any given hour so the way I look at it is imagine your phone had no battery counter and you're using it now in the first hour you wouldn't expect your phone to randomly die if you know it's been fully charged but as the hours go on it's more and more likely that your phone will die at any moment if you use it for 10 hours it's more likely to die than if it was at 5 hours or 2 hours or 1 hour so in other words the probability changes which means it's conditional probability and it's given that it has already been alive been alive lasted for 16 hours so we know that it's lasted longer than 16 hours already but we want the probability that it will last more than 20 hours so I hope that I've made that somewhat clear uh to work that out you need the probability that L is greater than 20 and you divide it by the probability that L is greater than 6 so we already know the probability that L's greater than 16 because we've already worked that out so we can just write that in as 0. 6915 the probity that's greater than 20 we can work that out nice and quickly using our calculator if we go back the lower now is 20 the higher doesn't change the standard deviation doesn't change the mean doesn't change so 03857 blah blah blah I might just use the answer button for that because again it make my answer eight five dot dot dot so that means your final answer if we go back to calculate we have answer over 0. 6915 gives me the answer of 04462 to four decimal places now now the question I usually get given where sorry I get asked not get given uh when we do these kind of Applied questions is is that it because if we have a look at that you could technically do this from here you could jump straight to here you've worked out one probability and you're dividing it by another probability there's only two steps and so people do ask is that really it the same is true with mechanics sometimes in moments questions you can do one or two lines of working out and get five marks the reason why they're giving you five marks is because it takes a lot of effort and logic and deep understanding to know that this is what you're supposed to do so again just to go through it it's already been used for 16 hours so it's given that it's lasted more than 16 hours what's the probability it will last more than 20 because again her exam is at least or her exam's plural is 20 hours so we want it to last more than 20 hours and it's already last 16 hours so it's given that it's it's going to last more than 16 hours I hope hope that makes sense but for part C again this is also quite tricky even though it's only three marks instead of the five marks on the previous one I think this is also a somewhat tricky question so with part C Alice only has two new batteries so these batteries are completely new okay which means they have they haven't lasted any time they are 100% full it says show that the probability that her calculator will not stop working for the remainder of her exam is1 199 to 3sf so here's the thing we want it to not stop working so we're looking for the probability that it it doesn't run out of battery so here's what I would do we have four batteries in total we're going to work out the probability that each of them doesn't fail okay so let's just assume that we have new battery so NB 1 mb2 and then we have the two old batteries again it doesn't actually matter the order in which the batteries are because it doesn't make a difference right but the idea is we have two new batteries and we have two old batteries so the First new battery she it only has to last for 4 hours right because she's already done the 16 hours of exams she's replaced two batteries those batteries only need to last up until hour 20 so that's four more hours what's the probability of that well all we're going to do is for new battery 1 it's going to be the probability that L is greater than four and that's also going to be true for the second battery because remember they're both brand new batteries that's fairly nice that's that's all right I might spread these out because the next two are going to be a bit longer if I'm being spoilers now for the other two it's actually just a probability we've already worked out because they need to last 20 hours or more given that they've already lasted 16 hours or more so in other words it's just going to be 04462 twice okay so all we need to do is work out this so if I quickly just whip out the calculator again man that lagged for a second there I got scared uh menu down to distribution normal CD we have the lower bound of four the upper bound of again just a really really big number Sigma was four mu is 18 cool so I get a really really big number that's nice 0.999 77 and that's for both of these right and then all we're doing is well we've got these four probabilities we need them all to be true so what does that mean it means we times them all together so we do 0.99 977 squared because there's two batteries time 0.446 2^ squar again we should have four probabilities because there are four batteries that can fail and we want the probability that each of them fails we times them all together because again we want or it's only one of them okay because only one of them has to fail the only way that the calculator doesn't stop working is if all of them last for the rest of the exam so we've worked out the probability that the first battery lasts until the end of the exam it's here the probability that the second battery last until the end of the exam is here probity that the third battery lasts until the end of the exam probability that the fourth battery lasts until the end of the exam but we want all four of those to be true so we want this one and this one and this one and this one when we say and in maths what do we mean we mean times which is why we're doing this so I hope that makes a bit more sense and it gives me 0.199 so that's basically a 20% chance that her calculator will actually survive for the whole exam which is a bit depressing to be honest but let's move on to Part D which is the hypothesis test honest ly this isn't a tricky hypothesis testing if you've done one or two with the normal distribution you can do all of them so for Part D it says after our exams Alice believed that the lifetime of the batteries is more than 18 hours okay so that means that's our alternate hypothesis she took a random sample of 20 and found that their mean was 19.2 stating your hypothesis clearly so again this is very similar to the hypothesis testing from binomial the first step is always to write your hypotheses H not is the null hypothesis this is what we're going to assume to be true initially which is that our mean is equal to 18 because that's our pre-established mean and our alternate hypothesis she's saying that the average lifetime should be well it says she believed that the lifetime should be more than 18 which means the average lifetime should be more than 18 according to what she said she takes a sample of 20 and finds their mean lifetime was 19.2 so let's write out a normal distribution now it tells us already what we have the mean of 18 and in terms of your Sigma or your Sigma squar all you need to do is you take the pre-existing standard deviation 4 and you divide it by the square root of the sample size so if that sample size was 30 then it'd be bigger if it was 50 100 a million whatever and that's of course all squared although we're not really going to use the squared to be fair and all we're going to do is actually work out what the probability using this different method um maybe I should write this as L1 or something using this alternate um distribution that we've written we're going to find out what the probability of it lasting longer than 19.2 hours is because again we're testing if it's more than 18 hours we want to find the probability that it lasted more than 19.2 so just to write that out we're finding the probability that L1 is bigger than uh 19.2 just use the uh good old fashioned normal distribution CD the lower value is 19.2 again our upper value is just a really big number with Sigma typing this in some people find it a bit tricky you can just literally write it as it you know as a division and when you press equals it works at the value for you and 18 we get 0.089 86 it doesn't really matter matter only the first kind of nonzero digit really matters because we're comparing it to 5% significance level because this is greater than 0.05 that means it is it means it is outside the critical region which means there is insufficient evidence at 5% blah blah blah so there is insufficient evidence at the 5% significance level to support Alice's claim and of course the S key decided not to press s there we are and that's it for five marks I do think that hypothesis testing is one of those really really strong topics which is quite nice it's five marks even for binomial and I mean I mean look at that it's not too bad right and as long as you remember this part which is some part that some students find a bit tricky but you can just use it as an equation and that's quite nice but yeah that brings us to the end of that kind of difficult question I will show you the marks quickly just to show you why this was submitted in the first place um if we have a look at the Mark scheme it's not designed to teach you so when you have something like this it's very hard to read and it's very hard to understand unless you have a really strong understanding of the maths anyway so I mean maybe this would be okay but have a look at this I mean the way they write it isn't necessarily the clearest but hopefully I've gone through exactly how you'd approach a problem solving question like that it is very difficult in a level math far harder than GCSE maths because it requires some level of logic or like real world experience to kind of understand what's going on but again with practice you should be very very fine with that but I'll leave you to your revision I'll see you in the next one