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# Probability of sample proportions example

AP.STATS:
UNC‑3 (EU)
,
UNC‑3.M (LO)
,
UNC‑3.M.1 (EK)

## Video transcript

we're told suppose that 15% of the 1750 students at a school have experienced extreme levels of stress during the past month a high school newspaper doesn't know this figure but they are curious what it is so they decide to ask us a simple random sample of 160 students if they have experienced extreme levels of stress during the past month subsequently they find that 10% of the sample replied yes to the question assuming the true proportion is 15% which they tell us up here they say 15% of the population of the seventeen hundred and fifty students actually have experienced extreme levels of stress during the past month so that is the true proportion so let me just write that the true proportion for our population is 0.15 what is the approximate probability that more than 10% of the sample would report that they experienced extreme levels of stress during the past month so pause this video and see if you can answer it on your own and there are four choices here I'll scroll down a little bit and see if you can answer this on your own so the way that we're going to tackle this is we're going to think about the sampling distribution of our sample proportions and first we're going to say well is this sampling distribution approximately normal is it approximately normal and if it is then we can use its mean and standard deviation and create a normal distribution that has that same mean and standard deviation in order to approximate the probability that they're asking for so for the first this first part how do we decide this well the rule of thumb we have here and it is a rule of thumb is that if we take our sample size times our population proportion and that is greater than or equal to 10 and our sample size times 1 minus our population proportion is greater than or equal to 10 then if both of these are true then our sampling distribution of our sample proportions is going be approximately normal so in this case the newspaper is asking 160 students that's the sample size so 160 the true population proportion is 0.15 and that needs to be greater than or equal to 10 and so let's see this is going to be 16 plus 8 which is 24 and 24 is indeed greater than or equal to 10 so that checks out and then if I take our sample size times 1 minus P well 1 minus 1500s is going to be 85 hundredths and this is definitely going to be greater than or equal to 10 let's see this is going to be 24 less than 160 so this is going to be 136 which is way larger than 10 so that checks out and so the sampling distribution of our sample proportions is approximately going to be normal and so what is the mean and standard deviation of our sampling distribution so the mean of our sampling distribution is just going to be our population proportion we've seen that in other videos which is equal to 0.15 and our standard deviation of our sampling distribution of our sample proportions is going to be equal to the square root of P times 1 minus P over N which is equal to the square root of 0.15 times 0.85 all of that over our sample size 160 so now let's get our calculator out so I'm going to take the square root of 0.15 times 0.85 divided by 160 let me close those parentheses and so what is this going to give me so it's going to give me approximately 0.028 and I'll go to the thousands place here so this is approximately zero point zero two eight this is going to be approximately a normal distribution so you could draw your classic bell curve for a normal distribution so something like this and our normal distribution is going to have a mean it's going to have a mean right over here of so this is the mean of our sampling distribution so this is going to be equal to the same thing as our population proportion 0.15 and we also know that our standard deviation here is going to be approximately equal to zero point zero two eight and what we want to know is what is the approximate probability that more than 10% of the sample would report that they experienced extreme levels of stress during the past month so we could say that 10% would be right over here I'll say zero point one zero and so the probability that in a sample of 160 you get a proportion for that sample a sample proportion that is larger than 10% would be this area right over here so this right over here would be the probability that your sample proportion is greater than they say is more than 10% is more than 0.1 I could write one zero just like that and then to calculate it I can get out our calculator again so here I'm gonna go to my distribution menu right over there and then I'm going to do a normal cumulative distribution function so let me click enter there and so what is my lower bound well my lower bound is ten percent zero point one what is my upper bound well we'll just make this one because that is the highest proportion you could have for a sampling distribution of sample proportions now what is our mean well we already know that's 0.15 what is the standard deviation of our sampling distribution well it's approximately zero point zero to eight and then I can click enter and if you're taking an AP exam you actually should write this you should say you should tell the graders what you're actually typing in in your normal CDF function but if we click enter right over here and then enter there we have it it's approximately 96% so this is approximately zero point nine six and not out of our choices it would be this one right over here if you're taking this on the AP exam you would say that called called normal normal CDF where you have your lower bound lower bound and you would put in your zero point one zero you would say that use an upper bound upper bound of one you would say that you gave a mean of 0.15 and then you gave a standard deviation of 0.02 a just so people know that you knew what you were doing but hopefully this is helpful
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