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.

Understand how to calculate the z-score step-by-step in hypothesis testing. This example involves a sample proportion, standard error calculation, and interpretation of z-score results. Ideal for advanced high school and college students studying statistics.

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