More than a decade ago, high levels of lead in the blood put...

More than a decade ago, high levels of lead in the blood put $88 \%$ of children at risk. A concerted effort was made to remove lead from the environment. Now, according to the Third National Health and Nutrition Examination Survey (NHANES III) conducted by the Centers for Disease Control, only $9 \%$ of children in the United States are at risk of high blood-lead levels. (a) In a random sample of 200 children taken more than a decade ago, what is the probability that 50 or more had high blood-lead levels? (b) In a random sample of 200 children taken now, what is the probability that 50 or more have high blood-lead levels?

Transcript

00:01 So we have that more than a decade ago, the kids were at risk of having like high levels of blood, like the risk was 88%.

00:12 So the probability of having high blood here, high level of blood for kids were 88 % more than a decade ago.

00:21 But now, recently, let's say that this number now is 9%.

00:26 9 % of the children in the united states are at risk of 100%.

00:30 High blood lead levels.

00:33 So as you can see, this probability here changed a lot or this proportion he changed a lot.

00:41 So for the first part of the question, we want to compute for a sample of 200 kids more than a decade ago.

00:50 What is the probability that, and i'm going to use the letter x to express the number of kids with high blood levels was greater than, oh, it's greater or more.

01:02 So we have the equal sign here than 50.

01:05 So to compute this, because the simple size is large, we can use the approximation to the normal distribution.

01:13 So to use this, what we need to do.

01:14 First, we need to consider a correction here factor.

01:19 So the correction factor here, because we have the greater and equal, is given by x, greater than 50 minus 0 .5.

01:29 So this means here that we are computing x being greater than 49 .5.

01:35 And to approximate this to the normal distribution, we need to compute the mean number of people, in this case kids, who had like high blood levels, which is 200.

01:48 And the probability before was 88%.

01:51 So this number here is 1 .76.

01:55 And we also need to compute, this is the mean, so i'm going to use the new letter.

02:01 And we also need to compute the standard deviation of the number of kids with high blood, which is given by the square root of np, 1 minus p.

02:09 So in our case, here we just need to put this 176, multiplied by 1, in this case, 1 minus 0 .88.

02:22 Because this number here is already np, so now we just need 1 .76, multiply by 1 .1 .1 .1.

02:26 In this case, 1 minus 0 .8.

02:26 Because this number here is 0, minus p.

02:26 So if you compute this, you're going to find 4 .5956.

02:31 So now that we have this information, we can rewrite this probability to be in a standard normal distribution.

02:38 To do this, we need to subtract the mean.

02:41 So i'm going to put z to express that now we are talking about this standard number distribution...

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