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This problem involves using properties of the normal distribution to find probabilities and numbers of workers in various earnings ranges. Here's how we can approach it:

### Given:
- Mean (\(\mu\)) = ¥1000
- Standard deviation (\(\sigma\)) = ¥200
- Total number of workers = 10,000

Since the earnings follow a **normal distribution**, we can use the **z-score formula** to calculate probabilities:
\[
z = \frac{x - \mu}{\sigma}
\]
Where:
- \(z\) is the standard normal variable.
- \(x\) is the value of interest.

Let's calculate the z-scores and corresponding probabilities for each part.

### (a) Workers earning less than ¥1000:
\[
z = \frac{1000 - 1000}{200} = 0
\]
From standard normal distribution tables, \(P(Z < 0) = 0.5\). Hence, 50% of the workers earn less than ¥1000:
\[
\text{Number of workers} = 0.5 \times 10,000 = 5,000
\]

### (b) Workers earning more than ¥600 but less than ¥800:
For \(x = 600\):
\[
z = \frac{600 - 1000}{200} = -2
\]
From standard normal tables, \(P(Z < -2) = 0.0228\).

For \(x = 800\):
\[
z = \frac{800 - 1000}{200} = -1
\]
From standard normal tables, \(P(Z < -1) = 0.1587\).

Now, to find the probability that earnings fall between ¥600 and ¥800:
\[
P(600 < X < 800) = P(Z < -1) - P(Z < -2) = 0.1587 - 0.0228 = 0.1359
\]
Thus, the number of workers in this range:
\[
\text{Number of workers} = 0.1359 \times 10,000 = 1,359
\]

### (c) Workers earning more than ¥1000 but less than ¥1200:
For \(x = 1200\):
\[
z = \frac{1200 - 1000}{200} = 1
\]
From the tables, \(P(Z < 1) = 0.8413\).

Now, to find the probability between ¥1000 and ¥1200:
\[
P(1000 < X < 1200) = P(Z < 1) - P(Z < 0) = 0.8413 - 0.5 = 0.3413
\]
Thus, the number of workers:
\[
\text{Number of workers} = 0.3413 \times 10,000 = 3,413
\]

### (d) Workers earning above ¥1200:
\[
P(X > 1200) = 1 - P(Z < 1) = 1 - 0.8413 = 0.1587
\]
Thus, the number of workers:
\[
\text{Number of workers} = 0.1587 \times 10,000 = 1,587
\]

### Summary:
- (a) Less than ¥1000: 5,000 workers
- (b) More than ¥600 but less than ¥800: 1,359 workers
- (c) More than ¥1000 but less than ¥1200: 3,413 workers
- (d) Above ¥1200: 1,587 workers

Would you like more details or clarifications on any part? Here are some follow-up questions to expand on the topic:

1. How would the number of workers change if the standard deviation was reduced to ¥100?
2. What percentage of workers earn more than ¥900 but less than ¥1100?
3. Can we estimate the probability of a worker earning more than ¥1500?
4. What does the shape of the normal distribution tell us about extreme earnings?
5. How would these calculations change if the earnings were not normally distributed?

**Tip:** When working with normal distributions, z-scores help transform values into the standard normal form, making it easier to find probabilities.

The average monthly earnings of a group of 10,000 unskilled engineering workers employed by firms in northeast China in 1997 was ¥1000 and the standard deviation was ¥200. Assuming a normal distribution of earnings, determine the number of workers who earned (a) less than ¥1000; (b) more than ¥600 but less than ¥800; (c) more than ¥1000 but less than ¥1200; (d) above ¥1200.

Determine the number of workers earning within specific ranges using normal distribution and z-scores. This problem involves statistical analysis of a group of 10,000 workers with an average income of ¥1000 and a standard deviation of ¥200, suitable for Grades 11-12.

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