1. (12.5 points) Based on data collected by the National...

1. (12.5 points) Based on data collected by the National Center for Health Statistics and made available to the public, an estimate of the percentage of adults who at some point in their life have been told they have hypertension is 23.53%. If we select a simple random sample of 20 US adults and assume that the probability that each has been told that he or she has hypertension is 0.20, find the probability that the number of people in the sample who have been told that they have hypertension will be: (a) exactly three (b) at least three (c) at most five (d) between six and ten, inclusive (e) Using the actual percentage of 23.53%, what is the mean and standard deviation?

Transcript

00:01 For this exercise, we have been told to assume that the probability that a person has hypertension is 0 .24, and we consider a random selection of 20 people, and we're asked for the probabilities that certain numbers of these 20 have hypertension.

00:18 So let's first define a random variable x as the number of people in the sample who have hypertension.

00:26 Here, each of the people in the sample can be viewed as a bernoulli trial, which is to say there is two outcomes of interest that either have high -intention, hypertension or not.

00:34 And since it's a random sample, their outcomes are independent.

00:39 The number of successes and a given number of independent brunuli trials is a binomial random variable.

00:45 So we can say here that x is a binomial random variable.

00:50 And the probability mass function for a binomial random variable is given by this formula.

01:05 Now for part a we want the probability that exactly three of the 20 have hypertension.

01:11 This is the probability that x is equal to equal to 3.

01:15 And using the probability mass function, for x equals 3, we have 20 choose 3 times 0 .24 to the exponent 3 times 0 .76 to the exponent 17.

01:33 And this comes out to a probability of approximately 0 .1484.

01:44 And then for b, we want the probability that x is at least 3, 3 or more.

01:50 This is equal to 1 minus the probability that x is at most 2, and the probability that x is at most 2 is equal to the probability that x equals 0 plus the probability that x equals 1 plus the probability that x equals 2.

02:08 So for x equals 0 the probability mass function simplifies to 1 minus p to the exponent n, and then for x equals 1 we have the following, and then for x equals 2 we have the following.

02:45 And this comes out to 0 .8915.

02:55 And for c, we want the probability that fewer than 3 out of the 20 have hypertension.

03:01 This is the probability that x is smaller than 3, and 3 that is, which is equal to the probability that x is at most 2, which is everything inside these square brackets here...

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