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Main content
Current time:0:00Total duration:11:53
AP.STATS:
DAT‑3 (EU)
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DAT‑3.F (LO)
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DAT‑3.F.1 (EK)
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DAT‑3.F.2 (EK)
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VAR‑7 (EU)
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VAR‑7.B (LO)
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VAR‑7.B.1 (EK)
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VAR‑7.C (LO)
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VAR‑7.C.1 (EK)
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VAR‑7.D (LO)
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VAR‑7.D.1 (EK)
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VAR‑7.E (LO)
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VAR‑7.E.1 (EK)

Video transcript

regulations require that product labels on containers of food that are available for sale to the public accurately state the amount of food in those containers specifically if milk containers are labeled to have 128 fluid ounces and the mean number of fluid ounces of milk in the containers is at least 128 the milk processor is considered to be in compliance with the regulations the filling machines could be set to the labeled amount variability in the filling process causes the actual contents of milk containers to be normally distributed a random sample of 12 containers of milk was drawn from the milk processing line in a plant and the amount of milk and each container was recorded the sample mean and standard deviation of this sample of 12 containers of milk were 127 point 2 ounces and 2.1 ounces respectively is there sufficient evidence to conclude that the packaging plant is not in compliance with the regulations provides statistical justification for your answer so pause this video and see if you can have a go at it alright now let's do this together so first let's say what we're talking about so let me define mu and this is going to be the mean mean amount amount of milk in population population of containers containers at the plant that we care about and so then we can set up our hypotheses our null hypothesis over here is that we are in compliance we could say that the mean for our population of containers is actually 128 that's our minimum we need to be in compliance and that our alternative hypothesis that we are not in compliance so that's that our mean the true population mean is less than 128 fluid ounces and so this is a situation where we are not in compliance not in compliance compliance in the alternative hypothesis now if you're going to do a significance test you need to set a significance level so let's do that over here significance level and if you haven't noticed I'm doing I'm trying to do in this video what would be expected of you on a test and this is an actual question from an AP exam so our significance level here I'll just pick it to be 0.05 because well that's a fairly typical one and since they didn't give it one to us it's important to set one ahead of time and now we want to check our conditions for inference so let me do that over here conditions conditions for inference and this is to feel good that the sample that we're using to make our inference to do our significance test that it's a reasonable one to make make inferences from and so the first one is are the random condition and do we meet that well they tell us here it's a random sample of 12 containers of milk if I was doing this on the AP exam I would write it out here so I would say in the passage or in question in the question they say they say a random a random sample of 12 and then they go on to say more things and so I would say that meets condition meets condition now the next one we want to care about is our normal condition and this is to feel good that our sampling distribution is roughly normal now there's a couple of ways that we could do that one is if our sample size is greater than 30 or greater than or equal to 30 then we'd say okay our sampling distribution is going to be roughly normal but in this situation our sample size n so sample size sample size is less than 30 but but there's another way to meet the normal condition and that's if the underlying parent data is normally distributed and they actually say it right over your variability in the filling process causes the actual contents of milk to be nor Emily distributed so we could say in passage in passage says and we'll see I could quote part of this so actual contents actual contents and dot dot dot normally distributed normally distributed so that meets condition meets condition and then the last condition we want to think about is the independence condition in the pendants and this is to feel good that the observations of the individual observations in our sample can be considered to be roughly independent now one ways if they were sampling with replacement which they're not doing here looks like they took all twelve containers at once but another way is if this is less than ten percent of the overall population then you could say okay they're gonna you can view them as roughly independent and so you say didn't didn't sample with replacement with a replacement but but assume assume that twelve is less than ten percent of the population and in that case you would meet condition meet this condition as well so it looks like we are we've met these three conditions that we need to make for inference or we can assume we've done it they haven't given us any information to the contrary and so now what we can do is calculate a t statistic and then from that calculate our p-value compare our p-value to our significance level and see what kind of conclusions we can make and so our T statistic right over here and once again if at any point you're inspired and if you haven't done so already try to do it on your own our T statistic is going to be our sample mean minus the assumed mean from the null hypothesis and let me since I'm introducing this notation this little sub zero I'll say that's the assumed assumed mean from my null hypothesis so I'll do that and then I'll divide ideally if I was doing a Z statistic I would divide by the standard deviation of the sampling distribution of the sample mean which is often known as the standard error of the mean but the real reason why I'm doing a T statistic is well I don't know exactly what that is but I could estimate the standard deviation of the sampling distribution of the sample mean using the sample standard deviation divided by the square root of N and once again it's always good if you're doing this on a test to explain what n is or what some of these things are if you're using standard notation it's people might assume what they are but if you have time on these tests you can always explain more of what these actual variables are but in this case this is going to be 120 7.2 that is our sample mean minus our assumed mean from our null hypothesis minus 128 all of that over our sample standard deviation is 2.1 divided by the square root of 12 and so this is going to be approximately equal to get a calculator out here and so we have it's in the numerator we have 127 0.2 minus 128 and then we're gonna divide that by I'll do another parentheses to point 1 divided by the square root of 12 and then let me close my parentheses that I typed that in correctly yeah that looks right click enter and so this is negative I'll say it's approximately negative 1 point 3 2 so negative 1 point 3 2 and now we can figure out our p value our p value which is the same thing as the probability of getting a t statistic this low or lower so we could say t is less than or equal to negative 1 point 3 2 is equal to so I'll get my calculator back out and so here what I would use is I would use the cumulative distribution function for T statistic so that's that right over there and so I do care about the left tail so I care about the area under the curve from negative infinity up to an including negative one point three two so let's do a negative negative one point three two and then my degrees of freedom well it's going to be my sample size minus one my sample size was twelve so that minus one is eleven and then I do paste and so I have this T CDF from negative e99 to negative one point three two comma eleven and actually you would want to write this down on your exam if you were doing it just so they know where you got that from and so this is this is equal to zero point one zero seven so let me write it this is approximately zero point one zero seven and it's important to say how you calculated this so used used T C D F and we went from negative one times ten to the 99th power and we went up to negative one point three two and then we had eleven degrees of freedom to get this result right over here and it also might be good practice to draw your T distribution right over here so that's our T distribution that's the mean of our T distribution so we say that this is the area that we care about so that is that right over there just to make sure people know what we're talking about and so here now we're ready to make a conclusion we can compare this to our significance level and so we can say since since 0.107 is greater than our significance level is greater than 0.05 which is alpha we fail we fail to reject reject the null hypothesis and so let's just make sure we read their question right is there sufficient evidence to conclude that the packaging plant is not in compliance with the regulations and so another way of saying is there is not there is not sufficient sufficient I'm gonna have to scroll down a little bit I'm trying to squeeze it on the page but I'm gonna have to go down there is not sufficient evidence sufficient evidence to conclude - to conclude that the plant is not in compliance with regulations and then we are done
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