https://file.aiquickdraw.com/mathbot/file/1725524975465f2z83hhi.jpg

The text in the image is in Greek and seems to deal with a mathematical problem, specifically concerning the area of probability and statistical distribution. The formulas and references to distributions indicate it's likely dealing with a normal distribution problem or some related statistical inference.

Here is the transcription of the relevant mathematical content from the image, which I can analyze for you:

### Key Equations:
1. **Formula for the normal distribution area:**

   \[
   P\left( Z \leq z \right) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{z} e^{- \frac{t^2}{2}} dt
   \]

2. **Central Limit Theorem approximation for large samples:**

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

3. **Sample Proportion Distribution:**

   \[
   \hat{p} = \frac{x}{n}
   \]

4. **Confidence interval for proportion:**

   \[
   \hat{p} \pm Z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}
   \]

The formulas involve standard statistical procedures such as normal distribution calculations, Central Limit Theorem, and confidence intervals.

If you have any specific questions or need further details or clarification on any particular part of the text or equations, feel free to ask!

### Would you like details on:

1. The derivation of the normal distribution area?
2. How the Central Limit Theorem is applied in this context?
3. Interpretation of the sample proportion distribution formula?
4. Explanation of confidence intervals?
5. Steps for solving a similar statistical problem?

### Tip:
When dealing with confidence intervals, remember to verify the sample size, as larger samples provide more reliable estimates.

Consider the solution to the problem involving the distribution of values when assessing the area of probability from a random sample.

This problem involves solving a probability distribution problem using the normal distribution and the Central Limit Theorem. The solution includes key formulas such as the probability density function of the normal distribution and confidence intervals for sample proportions.

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