https://r2.erweima.ai/mathbot/image/1726358925888lk2y28ha.jpg

Let's address the questions from the image step by step based on the sales data for the 5 vendors:

### Sales Data:
- Vendor 1: 1 sale
- Vendor 2: 15 sales
- Vendor 3: 37 sales
- Vendor 4: 125 sales
- Vendor 5: 14 sales
- Total Sales = 192

### A) **Cantidad de muestras tamaño 2**
This refers to how many different combinations of samples of size 2 can be made from the 5 vendors.

This is a **combination** problem, calculated as:
\[
C(n, k) = \frac{n!}{k!(n-k)!}
\]
where \(n = 5\) (vendors) and \(k = 2\) (sample size).

\[
C(5, 2) = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10
\]
So, there are **10 different samples** of size 2.

### B) **Calcule la media muestral**
To find the sample mean, sum all the sales and divide by the number of vendors.

\[
\text{Mean} = \frac{\text{Total Sales}}{n} = \frac{192}{5} = 38.4
\]
So, the sample mean is **38.4** sales.

### C) **Determine la distribución muestral**
This refers to the frequency of sales per vendor. We already have the sales per vendor, so the **distribution** is simply:
- Vendor 1: 1 sale
- Vendor 2: 15 sales
- Vendor 3: 37 sales
- Vendor 4: 125 sales
- Vendor 5: 14 sales

This distribution is highly skewed because Vendor 4 has significantly more sales compared to the others.

### D) **Probabilidad de tener más de 100 ventas**
We have only one vendor (Vendor 4) with more than 100 sales (125). So, the probability is calculated as:

\[
P(\text{more than 100 sales}) = \frac{1}{5} = 0.2
\]

### E) **Probabilidad de tener 40 ventas o menos**
There are 4 vendors with sales of 40 or less (Vendors 1, 2, 3, and 5). The probability is:

\[
P(\text{40 sales or less}) = \frac{4}{5} = 0.8
\]

### F) **¿Por qué es confiable la muestra de estos 5 vendedores?**
This sample may be considered reliable because it includes all 5 vendors from the company, representing the full population of salespeople. The data is exhaustive (complete) for this particular group, providing a full picture of the sales distribution.

---

### Would you like more details on any specific question?  
Here are some related questions you might consider:
1. What would the variance and standard deviation of the sales be?
2. How would the results change if one more vendor was added with 50 sales?
3. What does the skewness in the sales distribution tell us about the data?
4. How would you compute a confidence interval for the mean sales?
5. Could we model this distribution with a known probability distribution (e.g., normal, binomial)?

#### Tip:
For highly skewed data, it's often useful to calculate the **median** or use non-parametric tests to better understand the central tendency.

El corporativo de Palacio de Hierro tiene 5 vendedores estrella. En su reunión mensual, el gerente le solicita a cada vendedor sus ventas mensuales, que son las siguientes: Vendedor 1: 1, Vendedor 2: 15, Vendedor 3: 37, Vendedor 4: 125, Vendedor 5: 14. Con base en esta información, responda las siguientes preguntas:
A) Cantidad de muestras tamaño 2
B) Calcule la media muestral
C) Determine la distribución muestral
D) Probabilidad de tener más de 100 ventas
E) Probabilidad de tener 40 ventas o menos
F) ¿Por qué es confiable la muestra de estos 5 vendedores?

This problem involves analyzing sales data from 5 vendors using concepts like combinations, sample mean, probability, and distribution. Learn how to calculate the number of samples, find the mean, and determine sales probabilities in a detailed step-by-step solution. Suitable for advanced high school students or college learners.

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