Question

The average growth of a certain variety of pine tree is 10.1 inches in three years. A biologist claims that a new variety will have a greater three-year growth. A random sample of 25 of the new variety has an average three-year growth of 10.8 inches and a standard deviation of 2.1 inches.

(a). State the null and alternative hypotheses.

Answer 1

Null hypothesis
`(\(H_0\))`: The average three-year growth of the new pine tree variety is less than or equal to 10.1 inches. Alternative hypothesis
`(\(H_a\))`: The average growth is greater than 10.1 inches based on a biologist's claim.

In this scenario, you are testing whether the new variety of pine tree has a greater three-year growth than the known average of 10.1 inches for the current variety. The null and alternative hypotheses can be formulated as follows:

Let
`\( \mu \)` be the average three-year growth for the new variety.

The null hypothesis (H0) represents the status quo or the current belief, and the alternative hypothesis (Ha) represents the biologist's claim:

`\[ H_0: \mu \leq 10.1 \]`

This states that the average three-year growth of the new variety is less than or equal to the average growth of the current variety.

`\[ H_a: \mu > 10.1 \]`

This states that the average three-year growth of the new variety is greater than the average growth of the current variety.

So, the null and alternative hypotheses are:

`\[ H_0: \mu \leq 10.1 \]\[ H_a: \mu > 10.1 \]`