The **z-score** measures how many standard deviations a data point (or sample mean) is from the population mean. In the context of hypothesis testing, the z-score is used when the population standard deviation is known, or when the sample size is large enough (typically \( n > 30 \)) to approximate the t-distribution with a normal distribution.

Given the problem statement, let's compute the z-score using the formula:

\[
z = \frac{\bar{x} - \mu_0}{\frac{\sigma}{\sqrt{n}}}
\]

However, in this problem, we are using the sample standard deviation \( s \) instead of the population standard deviation \( \sigma \). Since the sample size is large, we can use the z-score as an approximation. So:

\[
z = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}}
\]

Where:
- \(\bar{x} = 495495 \, \text{mg}\) (sample mean)
- \(\mu_0 = 500500 \, \text{mg}\) (population mean)
- \(s = 2020 \, \text{mg}\) (sample standard deviation)
- \(n = 5555\) (sample size)

Plug in the values:

\[
z = \frac{495495 - 500500}{\frac{2020}{\sqrt{5555}}}
\]

Now let's compute it:

\[
z = \frac{-5005}{\frac{2020}{\sqrt{5555}}}
\]
\[
z \approx \frac{-5005}{26.95} \approx -185.66
\]

So the **z-score** is approximately \(-185.66\).

This is an extremely large (negative) z-score, indicating that the sample mean is far below the hypothesized population mean.

Would you like more details or have any further questions?

### Further Questions:
1. What does such an extreme z-score imply about the data?
2. How does the z-score relate to the p-value in hypothesis testing?
3. What is the critical z-score for a 0.05 significance level in a two-tailed test?
4. How would the z-score change if the sample mean were closer to 500500 mg?
5. What are the implications of using the z-score instead of the t-score in this case?

**Tip:** The z-score can be used as an approximation for the t-score when the sample size is large, making the results more straightforward to interpret using the standard normal distribution.

Explore how to calculate and interpret the z-score in statistics with a detailed example. This problem involves determining the z-score for a sample mean and its significance in relation to the population mean using the standard normal distribution. Learn more about hypothesis testing and the implications of large z-scores.

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