A group of students at a school takes a history test. The distribution is normal with a mean of 25, and a standard deviation of 4.

(a) Everyone who scores in the top 30% of the distribution gets a certificate. What is the lowest score someone can get and still earn a certificate?

(b) The top 5% of the scores get to compete in a statewide history contest. What is the lowest score someone can get and still go onto compete with the rest of the state?

#### Solution

Let $X$ denote the history score. Given that $\mu = 25$ and $\sigma = 4$.

`$X\sim N(25, 4^2)$`

.

(a) Let the lowest score for top 30% of the students be $a$.

` $$ \begin{aligned} & P(X > a) =0.3\\ & \Rightarrow P(X < a) =0.7\\ &\Rightarrow P\big(\frac{X-\mu}{\sigma} < \frac{a-25}{4}\big)=0.7\\ &\Rightarrow P(Z < \frac{a-25}{4}\big)=0.7\\ &\Rightarrow \frac{a-25}{4}= 0.52\\ & \qquad \text{(From normal statistical table)}\\ &\Rightarrow a = 25 + 0.52* 4\\ &\Rightarrow a = 27.08 \end{aligned} $$ `

(b) Let the lowest score for top 5% of the students be $b$.

` $$ \begin{aligned} & P(X > b) =0.05\\ & \Rightarrow P(X < b) =0.95\\ &\Rightarrow P\big(\frac{X-\mu}{\sigma} < \frac{b-25}{4}\big)=0.95\\ &\Rightarrow P(Z < \frac{b-25}{4}\big)=0.95\\ &\Rightarrow \frac{b-25}{4}= 1.64\\ & \qquad \text{(From normal statistical table)}\\ &\Rightarrow b = 25 + 1.64* 4\\ &\Rightarrow b = 31.56 \end{aligned} $$ `

#### Further Reading

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