Answer to Question #346761 in Statistics and Probability for Bethsheba Kiap

Question

City planners wish to estimate the mean lifetime of the most commonly planted trees in urban settings. A sample of 16 recently felled trees yielded mean age 32.7 years with standard deviation 3.1 years. Assuming the lifetimes of all such trees are normally distributed, construct a 99.8% confidence interval for the mean lifetime of all such trees.

Answer 1

The critical value for $\alpha = 0.002, df=n-1=15$ degrees of freedom is $t_c​=z_{1−α/2;n−1}= 3.73283$

The corresponding confidence interval is computed as shown below:

$$CI=(\bar{x}-t_c\times\dfrac{s}{\sqrt{n}}, \bar{x}+t_c\times\dfrac{s}{\sqrt{n}})$$ $$=(32.7- 3.73283\times\dfrac{3.1}{\sqrt{16}},$$ $$32.7+ 3.73283\times\dfrac{3.1}{\sqrt{16}})$$

$$=(29.807, 35.593)$$

Therefore, based on the data provided, the 99.8% confidence interval for the population mean is $29.807 < \mu < 35.593,$ which indicates that we are 99.8% confident that the true population mean $\mu$ is contained by the interval $(29.807, 35.593).$