Small sample hypothesis test

the mean emission of all engines of a new design needs to be below 20 parts per million if the design is to meet new emission requirements 10 engines are manufactured for testing purposes and the emission level of each is determined the emission data is and they give us 10 data points for the 10 test engines and I went ahead and calculated the mean of these data points the sample mean is 17 point one seven and the standard deviation of these 10 data points right here is two point nine eight the sample standard deviation does the data supply sufficient evidence to conclude that this type of engine meets the new standard assume we are willing to take a type or we are willing to risk a type 1 error with a probability of 0.01 and we'll touch on this in a second before we do that let's just define what our null hypothesis and our alternative hypothesis are willing to be our null hypothesis our null hypothesis can be that we don't meet the standards that we just barely don't meet the standards that the mean of our new engines is exactly 20 parts 20 parts per million and you essentially want the best possible value where we still don't meet or the lowest possible value where we still don't meet the standard and then our alternative hypothesis our alternative hypothesis no we do meet the standard that then the true mean for our new engines is below 20 parts per million and to see if the data that we have is sufficient what we're going to do is assume we're going to assume that this is true we are going to assume we are going to assume that is true and given that this is true if we assume this is true and the probability of this occurring and the probability of getting a sample mean of that is less than 1% then we will reject the null hypothesis so we are going to reject we are going to reject our null hypothesis if if the probability of getting a sample mean of 17 point 1 7 given the null hypothesis is true is less than 1 percent and notice if we do it this way there will be less than a 1% chance that we are making a type 1 error a type 1 errors that were ejecting it even though it's true here there's only a 1% chance that or less than a 1% chance that we will reject it if it is true now the next thing we have to think about is what type of distribution we should think about and the I guess the first thing that rings in my brain is we only have 10 samples here we only have 10 samples we have a small sample size right over here so we're going to be dealing with a t-distribution and a t-statistic so with that said so let's think of it this way there's going to have some we can come up with a t statistic that is based on these statistics right over here so the T statistic is going to be 17 point one seven our sample mean minus the assumed population mean minus 20 parts per million minus 20 parts per million over our sample standard deviation over our sample standard deviation two point nine eight two point nine eight this is really the definition of the T statistic and hopefully we see now that this really comes from a z-score and we the T distribution is kind of an engineered version of the normal distribution using T statistics two point nine eight divided by the square root of our sample size we have ten samples so it's divided by the square root of ten and so this value right here let me get the calculator out just to get a value in place there just get a value in place so this is going to be 17 point one seven minus twenty close parenthesis divided by divided by two point nine eight divided by the square root the square root that's not what I wanted let me complete that let me go back divided by the square root of ten and then close parenthesis and then close parenthesis it is almost exactly negative three our teeth that statistic is almost exactly negative three negative three point zero zero and what we need to figure out because T statistics have a T distribution so what we need to figure out is the probability of getting this T statistic or a value of t equal to this or less than this is that less than 1% so the way we can think about it is we have a T distribution we have a T distribution let's say we have a normalized t-distribution right the distribution of all the T statistics would be a normalized t-distribution this is the mean of the T distribution there's going to be some threshold there's going to be some threshold t-value some threat threshold T value right here so this is our threshold t-value Thresh threshold threshold and my writing isn't that easy to view well let's just take it this is some threshold T value right over here and we want a threshold T value such that any p value less than that or the probability of getting a t-value less than that is 1% the probability of getting a t-value less than that is 1% so that entire area in yellow is 1% we need to figure out a threshold T value there and this is for a T distribution this is for a T distribution that has that has n equal to 10 or 10 minus 1 equals 9 degrees of freedom degrees of freedom so what is that threshold value over there and notice that this is a one-sided distribution we care about this is 1% and then all of this stuff over here all of this stuff over here is going to be 99% and just the way most tea tables are set up they don't set up a negative T value that is oriented like this they'll just give you a positive T value that's oriented the other way so the way tea tables and I have one that we're going to use in a second right over here the way tea tables are set up is you have your distribution like this and they will just give a positive T value they will give a positive T value over here some threshold value where the probability of getting a t-value above that is going to be 1% and the probability of getting a t-value below that below that is going to be is going to be 99% and you can see that well we know T distributions are ar-ar-ar symmetric around there mean so whatever value this is if this number is 2 then this value is just going to be negative 2 so we just have to keep that in mind but the T tables actually help us figure out this value so let's figure out a T value where the probability of getting a t-value below that is 99% once again this is a one-sided this is going to be a one-sided district or one-sided the one-sided situation so let's look at that over here so with one-sided this is just straight from Wikipedia one side we want the cumulative distribution below that T value to be 99% we have it right over here 99% we have 9 degrees of freedom we have 10 data points 10 minus 1 is 9 9 degrees of freedom so a team our threshold t-value here is 2.8 to 1 so our threshold t-value in the case that we care about is just the flip this over it's completely symmetric is negative two point eight to one so what this tells us is the probability of getting a t-value less than negative two point eight to one is going to be one percent now we got a value that's a good bit less than that we got a T value we got of T value of negative three we got a T value right here our T statistic of negative three right over here so that definitely goes into our our I guess you could call it our area of of rejection that the problem this this is even less probable than the 1% we can even figure it out that the the area the area over here the the probability of getting a t statistic less than negative three is even less than it's a subset of this yellow area right over here so so because the probability of getting the T statistic that we actually got is let is less than one percent we can safely reject the null hypothesis we can safely reject the null hypothesis and feel and feel pretty good about our alternate hypothesis right over here that we do meet the emission standards and we know that we are we have a lower than 1% chance of actually making a type 1 error in this circumstance