Question

if we are investigating p-hat is less than and greater than p-naught, we use the alternative hypothesis:

Answer 1

The appropriate form of the alternative hypothesis when investigating if the sample proportion
`(\( \hat{p} \))` is significantly different from or either less than or greater than a hypothesized population proportion
`(\( p_0 \))` would be:
`\[ H_a: \hat{p} \\eq p_0 \]` for a two-tailed test, or
`\[ H_a: \hat{p} < p_0 \]`for a left-tailed test, or
`\[ H_a: \hat{p} > p_0 \]` for a right-tailed test.

Step-by-step explanation:

The appropriate form of the alternative hypothesis, when investigating whether the sample proportion
`(\( \hat{p} \))` is significantly less than or greater than a hypothesized population proportion
`(\( p_0 \)),` would be:

`\[ H_a: \hat{p} \\eq p_0 \text{ (Two-tailed test)} \]`

or

`\[ H_a: \hat{p} < p_0 \text{ (Left-tailed test)} \]`

or

`\[ H_a: \hat{p} > p_0 \text{ (Right-tailed test)} \]`

These alternatives allow for testing in three different directions: less than, greater than, or not equal to the hypothesized population proportion.

Full Question:

If we are investigating whether the sample proportion
`(\( \hat{p} \))` is significantly less than or greater than a hypothesized population proportion
`(\( p_0 \)`), what is the appropriate form of the alternative hypothesis?