Answer to Question #288499 in Statistics and Probability for amy

Question

For a certain group of individuals, the av- erage heart rate is 72 beats per minute. Assume the vari- able is normally distributed and the standard deviation is 3 beats per minute. If a subject is selected at random, find the probability that the person has the following heart rate.

a. Between 68 and 74 beats per minute b. Higher than 70 beats per minute

c. Less than 75 beats per minute 

Answer 1

Let $X=$ the heart rate: $X\sim N(\mu, \sigma^2).$

Given $\mu=72, \sigma=3.$

a.

$$P(68<X<74)=P(X<74)-P(X\leq 68)$$

$$=P(Z<\dfrac{74-72}{3})-P(Z\leq \dfrac{68-72}{3})$$

$$\approx P(Z<0.666667)-P(Z\leq -1.333333)$$

$$\approx0.74751-0.09121=0.6563$$

b.

$$P(X>70)=1-P(X\leq 70)$$

$$=1-P(Z\leq\dfrac{70-72}{3})\approx1-P(Z\leq-0.666667)$$

$$\approx0.7475$$

c.

$$P(X<75)=P(Z<\dfrac{75-72}{3})$$

$$=P(Z<1)\approx0.8413$$