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The image shows a problem involving a normal distribution \( N(\mu, \sigma^2) \) with mean \(\mu = 136\) and variance \(\sigma^2 = 9\) (hence standard deviation \(\sigma = 3\)). It asks to match the given probability statements to specific values based on standard properties of the normal distribution:

1. **68.3% of values lie within \(\mu \pm \sigma\):**
   - This means between \( 136 - 3 = 133 \) and \( 136 + 3 = 139 \).
   - **Corresponding probability: 68.3%** for the interval [133, 139].

2. **95.4% of values lie within \(\mu \pm 2\sigma\):**
   - This means between \( 136 - 6 = 130 \) and \( 136 + 6 = 142 \).
   - **Corresponding probability: 95.4%** for the interval [130, 142].

3. **99.7% of values lie within \(\mu \pm 3\sigma\):**
   - This means between \( 136 - 9 = 127 \) and \( 136 + 9 = 145 \).
   - **Corresponding probability: 99.7%** for the interval [127, 145].

Additional statements:

4. **Probability of a value being greater than 136:**
   - For a normal distribution, the probability is exactly **50%** since 136 is the mean.

5. **Probability of a value being less than 136:**
   - Same as above, the probability is **50%**.

6. **Probability of a value being exactly 136:**
   - For a continuous distribution, the probability of taking any exact value is **0%**.

### Summary:
- [133, 139]: **68.3%**
- [127, 145]: **99.7%**
- [130, 142]: **95.4%**
- Greater than 136: **50%**
- Less than 136: **50%**
- Exactly 136: **0%**

Would you like more details or have any other questions?

### Related Questions:
1. What are the properties of the standard normal distribution?
2. How do we calculate probabilities for non-standard intervals in a normal distribution?
3. What is the empirical rule and how does it apply to normal distributions?
4. How do you calculate z-scores and their corresponding probabilities?
5. How can the normal distribution be used in hypothesis testing?

### Tip:
For normally distributed data, about 68%, 95%, and 99.7% of values lie within 1, 2, and 3 standard deviations from the mean, respectively. This is known as the Empirical Rule or the 68-95-99.7 rule.

Normaalijakauman N(μ, σ²) nyrkkisääntöjen soveltaminen. Tarkastellaan normaalijakaumaa N(136; 3²). Todennäköisyydet eri väleille ja arvoille, kuten 68.3%, 95.4%, 99.7%, 50%, ja 0%.

Learn how to apply the empirical rule to a normal distribution with mean μ = 136 and standard deviation σ = 3. Understand key probability intervals, including 68.3%, 95.4%, 99.7%, and the probabilities of values being greater than, less than, or equal to 136.

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