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:

• 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.