The annual precipitation amounts in a certain mountain range are normally distributed with a mean of 91 inches, and a standard deviation of 14 inches. What is the probability that the mean annual precipitation during 49 randomly picked years will be at least 92.5 inches?

#### Solution

Let $X$ denote the annual precipitation amounts in a certain mountain range.

Given that population mean `$\mu = 91$`

, population standard deviation `$\sigma = 14$`

and the sample size `$n = 49$`

.

`$X\sim N(91, 14^2)$`

. Thus `$\overline{X}\sim N(91, 14^2/49)$`

.

The probability that the mean annual precipitation during 49 randomly picked years will be at least 92.5 inches is

` $$ \begin{aligned} P(\overline{X}\geq 92.5) & =1- P(\overline{X} < 92.5)\\ & = 1- P\bigg(\frac{\overline{X}-\mu}{\sigma/\sqrt{n}} < \frac{92.5-91}{14/\sqrt{49}} \bigg)\\ & = 1-P\big(Z < 0.75 \big)\\ &= 1- 0.7734\\ & = 0.2266\\ \end{aligned} $$ `

#### Further Reading

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