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Conditions for confidence interval for a proportion worked examples

AP.STATS:
UNC‑4 (EU)
,
UNC‑4.B (LO)
,
UNC‑4.B.1 (EK)
,
UNC‑4.B.2 (EK)

Video transcript

Ali is in charge of the dinner menu for his senior prom and he wants to use a one-sample Z interval to estimate what proportion of seniors would order a vegetarian option he randomly selects 30 of the 150 total seniors and finds that seven of those sampled would order the vegetarian option which conditions for constructing this confidence interval did ali's sample meet so pause this video and you can select more than one of these all right now let's work through this together so one thing that you might be wondering is well what is a one sample Z interval well you could really interpret that as he's gonna take one sample and then construct a confidence interval based on that the reason why it might be called a Z interval is the whole idea behind a confidence interval is you're going to pick a number of standard deviations above and below the true parameter that you are actually trying to estimate and then use that to make your inferences and a one way of thinking about the number of standard deviations people will often call that a z-score or a Z is often used as a variable for the number of standard deviations above or below something so really he's just trying to construct a confidence interval but remember in order to construct a confidence interval we have to make some assumptions he's taking there's a hundred fifty students right over here he's finding it impractical to survey all hundred fifty to figure out the true population proportion so instead he samples 30 of the seniors so n is equal to 30 and from that he calculates a sample proportion and looks like seven out of the 30 are they want the vegetarian option and he's going to determine some confidence level and then you construct a confidence interval but remember the conditions that we've talked about in previous videos the first thing is we have to be confident that is this a random sample so that would be the random condition and that's what choice a is telling us the data is a random sample from the population of interest do we know that well it tells us in the passage here he randomly selects 30 of the total seniors so I guess we'll take their word for it we don't know his methodology of what he considers random but we'll take their word for it that yes this has been met the data is a random random sample if it said he sampled the football team well that would not have been a random sample the next condition here it looks all mathematical but this is really the normal condition and the idea behind the normal condition is that in order to construct these confidence intervals we're assuming that the sampling distribution of the sample proportions is roughly normal and it is not skewed to the right or skewed to the left like this and so right here it says look the sample size times our sample proportion has to be greater than or equal to 10 or our sample size times 1 minus our sample sample proportion has to be greater than or equal to 10 well another way to think about this is our successes our successes in our sample need to be greater than or equal to 10 and our failures need to be greater than or equal to 10 well how many successes were they there were seven seven and you could even say look our n is thirty times our sample proportion is seven over 30 which is going to be seven so our set our successes is less than 10 so actually we violate the normal condition and once again this is a rule of thumb but this is telling us that our actual sampling distribution might be skewed remember this is just based on one sample what we're able to figure out this is one samples the interval we might be wrong but we wouldn't feel good that we're meeting the normal condition here so I would rule this one out individual observations can be considered independent well if he randomly selected people with replacement then they could be independent or if the people he is selective if his sample size is less than 10% of the total population then it could be considered independent even though it wouldn't be perfectly independent but we see here that he sampled 30 people out of 150 so his sample size was 30 out of 150 which the same thing as one-fifth of the population which is the same thing as 20% and since this is greater than 10% we are violating the independence condition we could have met the independence condition if he was sampling with replacement which it doesn't seem like he is or if this thing right over here was less than 10% but we're not meeting that so we cannot feel good about that constraint and so since we're not meeting two of the three constraints for I would say valid confidence intervals or confidence intervals we would feel confident in this is not so good of an analysis on alleys part
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