Porphyrin is a pigment in blood protoplasm and other body fluids that is significant in body energy and storage. Let x be a random variable that represents the number of milligrams of porphyrin per deciliter of blood. In healthy circles, x is approximately normally distributed with mean $\mu = 42$ and standard deviation $\sigma = 13$. Find the following probabilities. (round your answers to four decimal places.)
(a) x is less than 60
(b) x is greater than 16
(c) x is between 16 and 60
(d) x is more than 60 (this may indicate an infection, anemia, or another type of illness.)
Solution
Given that $\mu = 42$, $\sigma = 13$.
(a) The probability that $X$ is less than $60$ is
$$ \begin{aligned} P(X < 60) &= P\bigg(\frac{X-\mu}{\sigma} < \frac{60-42}{13}\bigg)\\ &= P\big(Z < 1.38 \big)\\ &= 0.9169 \end{aligned} $$
(b) The probability that $X$ is greater than $16$ is
$$ \begin{aligned} P(X > 16) & = 1- P(X < 16)\\ &= 1- P\bigg(\frac{X-\mu}{\sigma} < \frac{16-42}{13} \bigg)\\ & = 1- P\big(Z < -2 \big)\\ &= 1- 0.0228\\ &= 0.9772 \end{aligned} $$
(c) The probability that $X$ is between $16$ and $60$ is
$$ \begin{aligned} P(16 < X < 60) & = P(X < 60) -P(X < 16)\\ & = P\bigg(\frac{X- \mu}{\sigma} < \frac{60-42}{13} \bigg)- P\bigg(\frac{X-\mu}{\sigma} < \frac{16-42}{13} \bigg)\\ & = P\big(Z < 1.38 \big)-P\big(Z < -2 \big)\\ &= 0.9169 -0.0228\\ &= 0.8942 \end{aligned} $$
(d) The probability that $X$ is more than $60$ is
$$ \begin{aligned} P(X > 60) & = 1- P(X < 60)\\ & = 1- P\bigg(\frac{X-\mu}{\sigma} < \frac{60-42}{13} \bigg)\\ & = 1- P\big(Z < 1.38 \big)\\ &= 1- 0.9169\\ &= 0.0831 \end{aligned} $$
Further Reading
- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators
