To find the critical value for a hypothesis test where the null hypothesis \( H_0: p \geq 0.78 \) and the alternative hypothesis \( H_1: p < 0.78 \), you need to determine the critical value corresponding to an alpha level of \( 0.005 \) in a left-tailed test.
Since you're assuming that the normal distribution applies, you can use the standard normal (Z) distribution to find the critical value.
Steps to find the critical value:
- Identify the significance level: \( \alpha = 0.005 \).
- Find the critical Z-value: Because this is a left-tailed test, you want to find the Z-value such that the area to the left of it equals \( 0.005 \).
Using a standard Z-table (or a statistical calculator), you would look for the Z-value that corresponds to the area of \( 0.005 \):
- From the Z-table, the Z-value that corresponds to an area of \( 0.005 \) in the left tail is approximately \( -2.576 \).
So, the critical value is:
\[ \text{The critical value} = -2.576 \]
If you are using \( p \) for hypothesis testing (proportion), you can also convert the Z-value back to the sample proportion using the formula:
\[ \hat{p} = p_0 + Z \times \sqrt{\frac{p_0(1-p_0)}{n}} \]
Where:
- \( p_0 \) is the hypothesized population proportion (0.78).
- \( n \) is the sample size.
However, in the context of just finding the critical Z-value for your test, \( -2.576 \) is sufficient.
Let me know if you need further assistance!