Researchers at a pharmaceutical company have found that the effective time duration of a safe dosage of a pain relief drug is normally distributed with mean 2 hours and standard deviation 0.3 hour. For a patient selected at random:

a) What is the probability that the drug will be effective for 2 hours or less?

b) What is the probability that the drug will be effective for 1 hour or less?

c) What is the probability that the drug will be effective for 3 hours or more?

#### Solution

Let $X$ denote the effective time duration of a safe dosage of a pain relief drug.

Given that the population mean `$\mu = 2$`

and the population standard deviation `$\sigma = 0.3$`

.

a. The probability that the drug will be effective for 2 hours or less is

` $$ \begin{aligned} P(X\leq 2) & = P\bigg(\frac{X-\mu}{\sigma} \leq \frac{2-2}{0.3} \bigg)\\ & = P\big(Z\leq 0 \big)\\ &= 0.5 \end{aligned} $$ `

b. The probability that the drug will be effective for 1 hours or less is

` $$ \begin{aligned} P(X\leq 1) & = P\bigg(\frac{X-\mu}{\sigma} \leq \frac{1-2}{0.3} \bigg)\\ & = P\big(Z\leq -3.33 \big)\\ &= 0.0004 \end{aligned} $$ `

c. The probability that the drug will be effective for 3 hours or more

` $$ \begin{aligned} P(X\geq 3) & = 1-P\bigg(\frac{X-\mu}{\sigma} \leq \frac{3-2}{0.3} \bigg)\\ & = 1-P\big(Z\leq 3.33 \big)\\ &= 1- 0.9996\\ &=0.0004 \end{aligned} $$ `

#### Further Reading

- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators