Question

Statistics Homework:

Diabetes and hypertension are two of the most common diseases in Western, industrialized nations. In the United States, approximately 9% of the population has diabetes, while about 30% of adults have high blood pressure. The two diseases frequently occur together: an estimated 6% of the population has both diabetes and hypertension.

1. Are having diabetes and having hypertension disjoint?

2. Draw a Venn diagram summarizing the variables and their associated probabilities.

3. Let A represent the event of having diabetes, and B in the event of having hypertension. Calculate P (A or B).

4. What percent of Americans have neither hypertension nor diabetes?

5. Is the event of someone being hypertensive independent of the event that someone has diabetes?

Answer 1

Solution:-

According to question, in the United States, approximately 9% of the population has diabetes, about 30% of adults have high blood pressure and about 6% of the population has both diabetes and hypertension.

(1)

Two sets are said to be disjoint if there is no elements in common.

People having diabetes and having hypertension are not disjoint because 6% of people are common have both diabetes and hypertension.

(2)

Let A represents the event having diabetes and B represents the event having hypertension.

Since 9% of people have diabetes.

So, P(A) = 0.09 ....(1)

Since 30% of people have hypertension.

So, P(B) = 0.30 .....(2)

Since 6% of people have both diabetes and hypertension.

So, P(AB) =0.06 .....(3)

Now,

Probability of people having only diabetes is equal to

P(A) - P(AB) = 0.09 - 0.06 = 0.03 .....(4)

and probability of people having hypertension is equal to

P(B) - P(AB) = 0.30 - 0.06 = 0.24 .....(5)

Now, the probability of people having diabetes or hypertension is given by

P(AB) = P(A) + P(B) - P(AB)

So, P(AB) = 0.09 + 0.30 - 0.06 = 0.39-0.06 = 0.33 ...(6)

And probability of having neither diabetes nor hypertension is

= 1- P(AB) = 1 - 0.33= 0.67 ....(7)

Using the above information, the venn diagram is drawn as shown below-

(3)

If A represent the event of having diabetes, and B in the event of having hypertension than P (A or B) = P(AB)

And as calculated above from equation (6), we get

P(A or B) = P(AB) = 0.33

HENCE, P(A or B) = 0.33

(4)

Probability of people having neither hypertension nor diabetes is = 1- P(A or B) = 1- 0.33 =0.67

So, percentage of Americans have neither hypertension nor diabetes = 0.67×100 = 67%

Hence, 67% of Americans having neither hypertension nor diabetes.

(5)

Two events A and b are said to be independent when the occurrence of one is not depend on occuranve of other.

In this case

P(AB) = P(A)×P(B)

But here, P(A)×P(B) = 0.09×0.30= 0.027

And P(AB) = 0.06.

So, the event of someone being hypertensive not independent of the event that someone has diabetes.

Hence, the two events are not independent.