The national center for health statistics nchs reports that...

Name: the national center for health statistics nchs reports that 70 of us adults aged 65 and over have ever received a pneumococcal vaccination suppose that you obtain an independent sample of 20 46064
Uploaded: 2023-07-30T13:06:12-08:00
Duration: 11 min 31 s
Channel: Ramon Kryzhan
Description: the national center for health statistics nchs reports that 70 of us adults aged 65 and over have ever received a pneumococcal vaccination suppose that you obtain an independent sample of 20 46064

The National Center for Health Statistics (NCHS) reports that 70% of U.S. adults aged 65 and over have ever received a pneumococcal vaccination. Suppose that you obtain an independent sample of 20 adults aged 65 and over who visited the emergency room and the pneumococcal vaccination rate applies to the sample. Determine the probability that exactly 14 members of the sample received a pneumococcal vaccination. Let X represent the number of successes in a binomial setting with n trials and probability p of success in each trial. The probability of obtaining exactly k successes is given by the formula $P(X = k) = \binom{n}{k} p^k (1 - p)^{n-k}$ where $\binom{n}{k}$ is the binomial coefficient, also represented as $_nC_k$, and is defined as $\binom{n}{k} = \frac{n!}{k!(n - k)!}$ Follow the steps to compute $P(X = 14)$. First determine the binomial coefficient. Do not round your answer. 0.1934 $_nC_k$ Incorrect Answer

Transcript

00:01 Okay, so they're giving you a binomial distribution here and they're asking you to solve for various situations they do give you the formula which i'm going to put up on the board which is if we're trying to find the probability that some variable has exactly k values the way we do that is we do a combination where we take the total number of values with the number that we want which is the k and we then do the probability raised to the kth power and we do one minus the probability raised to the n minus k power i'll explain what all this is in a minute and how this all works but this is the formula they give you and then this part the c of n k they give you that that's called a combination and they give you that formula.

00:53 It's n factorial over k factorial times uh n minus k factorial fix that n minus k factorial, um, and i'll explain how that all works in a minute also, so let's go to the first problem they give you three scenarios.

01:17 The first one is they want to know we have a total of 20 people the probability of them being vaccinated is 0 .7 and we want to know the probability that exactly two of them are vaccinated and so if we go to this first formula and start filling it in we're going to get a combination of out of 20 people.

01:45 What's how many different ways can we select two of them? that's what a combination does is it takes a scenario and it says how many different ways can this happen? um, so if we have 20 people and we just have them in a room and we randomly pick two of them how many combinations of two people can we pick this p to the k power? that is if you're vaccinated, it's a 70 chance that you're vaccinated and we want two people to be vaccinated so out of those two people 70 of the chance of them being vaccinated.

02:16 This is how we find the probability of that happening okay, and then this next one is for the other 18 people uh who are not vaccinated that would be 30 chance that they're not vaccinated and there's 18 of those people so if you see n minus k that's 20 minus 2 is 18 these two numbers should always add up to whatever the sample size is in this case 20 people.

02:40 Okay, and so this is uh probability of two people being vaccinated and this is the probability of the other 18 people not being vaccinated and we combine that all together to figure out the probability of this occurring this is the number of ways this can happen how we randomly have two people who are vaccinated and the other 18 are not okay so now the next part is once we have this set up is we got to figure this out we got to figure out what this combination is.

03:10 So they give you the formula for that so we go say combination of 20 people selecting two of them that is 20 factorial on the top and the bottom you're going to have two factorial and then they say n minus k that's 20 minus 2 which is 18 factorial and these two numbers right here should add up to how many people we have in the in the sample size? and so this is our setup and then we go ahead and figure this out one way of doing this.

03:41 I mean you can go into calculator if you know where the factorial button is you can use that but one way of doing this a trick that we use is 20 factorial by definition is 20 times 19 times 18 and so forth and we can write it as 20 times 19 times 18 factorial and that's how we do factorials and then two factorials just one two times one and then 18 factorial i'm going to leave alone so that's times 18 factorial because this 18 factorial and that 18 factorial are going to cancel so what we literally have here is we have 20 times 19 on the top 2 times 1 on the bottom plug that in a calculator figure out what that is and i got an answer of 190 so there's 190 different ways of this situation happening within this room if we have 20 people to randomly have two people that could be vaccinated there's 190 ways of that happening.

04:40 So we say 190 from this times the 0 .70 squared times the 0 .30 to the 18th power plug that in a calculator and get our answer and this number is so small um, i get 3 .6 0 6 8 8 but then my calculator says e negative 8 and what that means is that's 3 .6 0 6 8 8 times 10 to the negative 8th power, which is 0 .000000 should have seven zeros and then one more decimal place would be the three six zero six eight eight an extremely small possibility of this happening where only two of them are vaccinated the reason for that is because if we expect 70 of them to be vaccinated on average we expect it to be a lot higher number of people that are actually vaccinated in this room.

05:45 Not just two people we expect it to be a lot bigger to only have two people vaccinated.

05:51 It's a very small probability of this happening basically down to zero.

05:55 It's so small um, okay, so that's for the first one for the second one when they ask us probability that there are 18 people vaccinated.

06:06 So we go back to the original setup when i say, okay the combination of 20 people and out of those 20 people 18 are vaccinated we have a 70 chance that someone's vaccinated.

06:20 We want 18 of those people to be vaccinated there's a 30 chance that they're not vaccinated.

06:26 So we expect only two people to be not vaccinated and this is what we do to figure out this situation.

06:32 So we use that same formula it's p to the k power and then one minus p to the n minus k power but it helps to understand in context what this mean 18 people are vaccinated 70 chance that someone's vaccinated two people not vaccinated.

06:48 There's a 30 chance that they're not vaccinated...

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