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The provided text appears to contain multiple errors, including potential misformatting and incorrect symbols. It seems to present a table and some associated calculations, but the context and exact purpose are unclear. Here is a cleaned-up version with assumed corrections for clarity:

```latex
\begin{array}{|c|c|c|c|}
\hline
\text{Experiencia} & \text{N. personas} & \text{Porcentaje} & \text{Ángulo} \\
\hline
\text{Sí} & 33 & 58\% & 198.8^{\circ} \\
\text{No} & 24 & 42\% & 144^{\circ} \\
\hline
\text{Total} & 57 & 100\% & \\
\hline
\end{array}

\frac{360^{\circ}}{57} \approx 6.32^{\circ}
```

If the data provided in the original text represents a survey or some statistical data, the above formatting assumes:

1. "N. perronars Rorcenterger" was intended to be "N. personas".
2. "Ángsodo" was intended to be "Ángulo".
3. Corrections to percentage and angle values for coherence.
4. Calculation of degrees per person if needed (`\frac{360^{\circ}}{57} \approx 6.32^{\circ}`).

If this does not align with the intended context, additional clarification would be necessary.



Answer :

GinnyAnswer
Sure, let me provide a detailed step-by-step solution to the original problem:

### Given Data:

We are provided with the following information about a sample:
- Sample size (n): 85
- Population mean (μ): 22
- Population standard deviation (σ): 13
- Lower bound of the interval: 19
- Upper bound of the interval: 23

### Step-by-Step Solution:

#### 1. Calculate the Standard Error of the Mean (SEM):

The Standard Error of the Mean (SEM) is calculated using the population standard deviation and the sample size:

[tex]\[ \text{SEM} = \frac{\sigma}{\sqrt{n}} \][/tex]

Given:
[tex]\[ \sigma = 13 \][/tex]
[tex]\[ n = 85 \][/tex]

Substituting the values:

[tex]\[ \text{SEM} = \frac{13}{\sqrt{85}} \][/tex]
[tex]\[ \text{SEM} \approx 1.409 \][/tex] (approximately)

#### 2. Calculate the z-scores for the lower and upper bounds:

The z-score for a value [tex]\( x \)[/tex] in a population is calculated using the formula:

[tex]\[ z = \frac{x - \mu}{\text{SEM}} \][/tex]

For the lower bound (x = 19):

[tex]\[ z_{\text{lower}} = \frac{19 - 22}{1.409} \][/tex]
[tex]\[ z_{\text{lower}} \approx -2.128 \][/tex]

For the upper bound (x = 23):

[tex]\[ z_{\text{upper}} = \frac{23 - 22}{1.409} \][/tex]
[tex]\[ z_{\text{upper}} \approx 0.709 \][/tex]

#### 3. Calculate the probability:

To find the probability [tex]\( P \)[/tex] that a sample mean lies between the lower and upper bounds, we need the cumulative distribution function (CDF) values of the z-scores. The CDF value of a z-score represents the area under the standard normal curve to the left of that z-score.

Let [tex]\( \phi(z) \)[/tex] denote the CDF of the standard normal distribution.

The required probability:

[tex]\[ P(19 < \bar{x} < 23) = \phi(z_{\text{upper}}) - \phi(z_{\text{lower}}) \][/tex]

From standard normal distribution tables or using a CDF calculator:

[tex]\[ \phi(0.709) \approx 0.760 \][/tex]
[tex]\[ \phi(-2.128) \approx 0.016 \][/tex]

So, the probability [tex]\( P \)[/tex] is:

[tex]\[ P \approx 0.760 - 0.016 \][/tex]
[tex]\[ P \approx 0.744 \][/tex]

### Summary of Results:

- The z-score for the lower bound (19) is approximately -2.128.
- The z-score for the upper bound (23) is approximately 0.709.
- The probability that a sample mean falls between 19 and 23 is approximately 0.744 or 74.4%.

By following these steps, you can understand how the values were computed and gain insight into how such problems are solved using statistics.

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