Sure, let's break down the calculation in Step 6, where we compute the z-score:

Step 6: Calculation of the z-score

The formula for the z-score is:

$z = \frac{\bar{x} - \mu}{\sqrt{\frac{r}{n}}}$

where:

• $\bar{x} = 0.80$ (sample proportion)
• $\mu = 0.90$ (hypothesized proportion)
• $r = 0.09$ (calculated variance)
• $n = 200$ (sample size)

Detailed Calculation

1. Calculate the Standard Error:

   The standard error of the proportion is:

   $\text{Standard Error} = \sqrt{\frac{r}{n}} = \sqrt{\frac{0.09}{200}}$

   Calculating the value inside the square root:

   $\frac{0.09}{200} = 0.00045$

   Then, take the square root:

   $\text{Standard Error} = \sqrt{0.00045} \approx 0.0212$

2. Substitute Values into the z-score Formula:

   Now, plug in the values:

   $z = \frac{0.80 - 0.90}{0.0212}$

   Compute the numerator:

   $0.80 - 0.90 = -0.10$

   Divide by the standard error:

   $z = \frac{-0.10}{0.0212} \approx -4.71$

This results in a z-score of approximately $-4.71$. This detailed calculation shows how the components come together to determine the z-score.

Would you like further clarification or have any more questions? Here are some related topics for deeper understanding:

1. What is the relationship between the standard error and the sample size?
2. How do you interpret a negative z-score in hypothesis testing?
3. What are common critical values for z-scores at various significance levels?
4. What happens if the sample proportion exactly matches the hypothesized proportion?
5. How does increasing the sample size affect the z-score?

Tip: Always verify each step of the calculation to ensure accuracy, especially when working with small standard errors, as they significantly impact the z-score.