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To solve these questions, we need to utilize the properties of a normal distribution. The heights of males in the Netherlands are normally distributed with:

- Mean \( \mu = 183 \, \text{cm} \)
- Standard deviation \( \sigma = 10.5 \, \text{cm} \)

We'll standardize the heights using the Z-score formula:

\[
Z = \frac{X - \mu}{\sigma}
\]

Where:
- \( Z \) is the Z-score
- \( X \) is the height in question
- \( \mu \) is the mean (183 cm)
- \( \sigma \) is the standard deviation (10.5 cm)

Then, we can use standard normal distribution tables or a calculator to find the probabilities.

### a. Probability that a Dutch male is shorter than 175 cm:
\[
Z = \frac{175 - 183}{10.5} = \frac{-8}{10.5} \approx -0.7619
\]
Using the Z-table, we can find the corresponding probability.

### b. Probability that a Dutch male is taller than 195 cm:
\[
Z = \frac{195 - 183}{10.5} = \frac{12}{10.5} \approx 1.1429
\]
We need to find the probability for Z > 1.1429.

### c. Probability that a Dutch male is between 173 cm and 193 cm:
We need to calculate two Z-scores for this range.
- For \( X = 173 \):
  \[
  Z = \frac{173 - 183}{10.5} = \frac{-10}{10.5} \approx -0.9524
  \]
- For \( X = 193 \):
  \[
  Z = \frac{193 - 183}{10.5} = \frac{10}{10.5} \approx 0.9524
  \]
Now, find the probability between these two Z-scores.

### d. Out of a random sample of 1,000 Dutch men, how many would we expect to be taller than 190 cm?
Calculate the Z-score for 190 cm:
\[
Z = \frac{190 - 183}{10.5} = \frac{7}{10.5} \approx 0.6667
\]
Use the Z-table to find the probability for \( Z > 0.6667 \), and multiply by 1,000 to get the expected number.

I'll calculate these probabilities now.Here are the results for each part of the question:

### a. Probability that a Dutch male is shorter than 175 cm:
\[
P(X < 175) \approx 0.2231
\]

### b. Probability that a Dutch male is taller than 195 cm:
\[
P(X > 195) \approx 0.1265
\]

### c. Probability that a Dutch male is between 173 cm and 193 cm:
\[
P(173 < X < 193) \approx 0.6591
\]

### d. Out of a random sample of 1,000 Dutch men, how many would we expect to be taller than 190 cm?
We expect approximately **252 men** to be taller than 190 cm.

Would you like more details or have any further questions?

Here are 5 related questions:
1. How would the probabilities change if the standard deviation increased?
2. What is the probability that a Dutch male is taller than 200 cm?
3. How many Dutch males out of 500 would be shorter than 170 cm?
4. Can you calculate the probability for a height of exactly 183 cm?
5. How does changing the sample size in part d affect the expected value?

**Tip**: When solving normal distribution problems, always check if you need to calculate cumulative or complementary probabilities based on the context.

What is the probability that a Dutch male is shorter than 175 cm, taller than 195 cm, or between 173 and 193 cm, assuming the average height of males in the Netherlands is 183 cm with a standard deviation of 10.5 cm? Additionally, out of a sample of 1,000 Dutch males, how many are expected to be taller than 190 cm?

Calculate the probabilities for Dutch male height based on the normal distribution with an average height of 183 cm and a standard deviation of 10.5 cm. Determine the probability that a Dutch male is shorter than 175 cm, taller than 195 cm, or between 173 and 193 cm, and find how many out of 1,000 Dutch males are expected to be taller than 190 cm.

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