To solve for the standard deviation ($\sigma$) of the normal distribution, we can use the given probability $P(X > 52) = 0.288$, the mean ($\mu = 45$), and the standard properties of the normal distribution.

Steps:

1. Convert the probability to a z-score: The probability corresponds to the area under the standard normal curve to the right of $x = 52$. Using standard normal tables or a calculator: $P(Z > z) = 0.288 \implies z \approx 0.56$

   Here, $Z$ is the standard normal random variable.

2. Standardize the given value: The z-score is calculated as: $z = \frac{x - \mu}{\sigma}$ Substituting $z = 0.56$, $x = 52$, and $\mu = 45$: $0.56 = \frac{52 - 45}{\sigma}$

3. Solve for $\sigma$: $\sigma = \frac{52 - 45}{0.56} = \frac{7}{0.56} \approx 12.5$

Final Answer:

The standard deviation ($\sigma$) is approximately 12.5.

Would you like a more detailed explanation or help with related concepts? Here are 5 follow-up questions:

1. How do you interpret z-scores in a normal distribution?
2. Can you explain the relationship between standard deviation and variance?
3. How would the probability change if the mean were higher?
4. What are the key properties of a normal distribution?
5. How do you find probabilities for normal distributions without a table?

Tip: Remember, variance is the square of the standard deviation ($\sigma^2$), so it provides a measure of dispersion.