On the average, there are 500 mg of aspirin in an "Extra Strength" aspirin tablet. Suppose the random variable assigning amount of aspirin in a tablet is normally distributed with standard deviation 30 mg.

What proportion of extra strength aspirin tablets can be expected to contain between 475 and 525 mg of aspirin?

If you purchase a 100-tablet bottle of extra strength aspirin, about how many of them would contain more than 530 mg of aspirin?

How much aspirin would a tablet have to contain to be in the top 5% of the aspirin content range?

### Solution

Given that Suppose the random variable assigning amount of aspirin in a tablet is normally distributed with standard deviation 30 mg..

That is mean $\mu=500$ and standard deviation $=\sigma = 30$.

The proportion of extra strength aspirin tablets can be expected to contain between 475 and 525 mg of aspirin is

` $$ \begin{aligned} P(475 < X < 525) &= P(X < 525) - P(X < 475)\\ &= P\bigg(\frac{X-\mu}{\sigma} < \frac{525-500}{30}\bigg)-P\bigg(\frac{X-\mu}{\sigma} < \frac{475-500}{30}\bigg)\\ &=P\bigg(Z < 0.833\bigg)-P\bigg(Z < -0.833\bigg)\\ &=0.7977-0.2023\\ &= 0.5953 \end{aligned} $$ `

The probability that an extra strength aspirin tablets contains more than $530$ is

` $$ \begin{aligned} P(X>530) &= 1-P(X\leq530)\\ &= 1- P\bigg(\frac{X-\mu}{\sigma} \leq \frac{530-500}{30}\bigg)\\ &=1- P\bigg(Z \leq 1\bigg)\\ &=1-0.8413\\ &= 0.1587 \end{aligned} $$ `

Thus if you purchase a 100-tablet bottle of extra strength aspirin, about $100* 0.1587=16$ of them would contain more than 530 mg of aspirin.

Let the top 5% percentile of aspirin content range be "a" or more. Then we have

` $$ \begin{aligned} & P(X > a)=0.05\\ &\Rightarrow P(X < a) =0.95\\ &\Rightarrow P\big(\frac{X-\mu}{\sigma} < \frac{a-500}{30}\big)=0.95\\ &\Rightarrow P(Z < \frac{a-500}{30}\big)=0.95\\ &\Rightarrow \frac{a-500}{30}= 1.645\\ &\Rightarrow a = 500 + 1.645* 30\\ &\Rightarrow a = 549.35 \end{aligned} $$ `

Thus the tablet would have to contain $549.35$ of aspirin to be in the top 5% of the apsirin content range.

#### Further Reading

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