Suppose that diastolic blood pressures (DBPs) for men aged 35-44 are normally distributed with a mean of 80 (mm Hg) and a standard deviation of 10. What is the probability that a random 35-44 year old has a DBP less than 70? Express your answer as a percentage to the nearest percentage point.

Question

kirbykirby · Answer

Say $X$ is the DPB of 35-44 year olds . According to the question, you are interested in $P(X<70)$. Since the data normally distributed ($X$ is normal), you can find this probability by standardizing $X$ will will allow you to use a standard normal table to find the probability.
$$ P(X<70)=P\left( \frac{X-80}{10}<\frac{70-80}{10}\right)=P(Z<-1)$$
Now, you can use either a standard normal table, or realize that a Z-score is -1 is 1 standard deviation from the mean (to the left). 

Now you know that between -1 and 1 standard deviation, you have approximately 68% = 0.68 of your data. So, outside of that range, you have $(1-0.68)/2=0.32 $ outside of that range, separated into 2 areas. Since you want the area less than -1, divide again by 2 to get 0.16 = 16%|dw:1400859037489:dw|