Question

Researchers at a pharmaceutical company have found that the effective time duration of a safe dosage of pain relief drug is normally distributed with the mean 1.5 hours and standard deviation 0.8 hours For a patient selected at random
a) what is the probability that the drug will be effective between 1 and 3 hours after taken
?
b) what is the probability the drug will be effective for 4 hours or less?

Answer 1

Answer:

a) $P(1$

b) $P(X$

Step-by-step explanation:

1) Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

2) Part a

Let X the random variable that represent the duration of a safe dosage of pain relief drug, and for this case we know the distribution for X is given by:

$X \sim N(1.5,0.8)$  

Where $\mu=1.5$ and $\sigma=0.8$

We are interested on this probability

$P(1$

And the best way to solve this problem is using the normal standard distribution and the z score given by:

$z=\frac{x-\mu}{\sigma}$

If we apply this formula to our probability we got this:

$P(1$

And we can find this probability on this way:

$P(-0.625$

And in order to find these probabilities we can find tables for the normal standard distribution, excel or a calculator.  

$P(-0.625$

3) Part b

We are interested on this probability

$P(X$

And the best way to solve this problem is using the normal standard distribution and the z score given by:

$z=\frac{x-\mu}{\sigma}$

If we apply this formula to our probability we got this:

$P(X$

And we can find this probability on this way:

$P(z$