In order to collect a representative sample from your company, how many employees would you have to survey? Explain how you arrived at that figure.

To determine the number of employees that need to be surveyed in order to collect a representative sample from the company, we need to consider the size of the company and the desired level of confidence and margin of error for the survey results.

One common approach is to use a formula called the sample size formula, which takes into account the population size, desired confidence level, and margin of error. The formula is:

n = (Z^2 * p * q) / E^2

Where:
n = sample size
Z = Z-score corresponding to the desired confidence level (e.g. 1.96 for a 95% confidence level)
p = estimated proportion of the population that has a particular characteristic
q = 1 - p
E = margin of error

For example, if we have a company with 500 employees and we want to survey with a 95% confidence level and a margin of error of 5%, we can use the formula to calculate the sample size:

n = (1.96^2 * 0.5 * 0.5) / 0.05^2
n = (3.8416 * 0.25) / 0.0025
n = 0.9604 / 0.0025
n = 384.16

Therefore, we would need to survey approximately 385 employees from the company to collect a representative sample with a 95% confidence level and a 5% margin of error.

Determining Sample Size for Employee Surveys

The sample size required to collect a representative sample from a company depends on several factors, including:

- Confidence level: This represents the desired level of accuracy or confidence in the results. A higher confidence level requires a larger sample size.
- Margin of error: This is the amount of error that can be tolerated in the results. A smaller margin of error requires a larger sample size.
- Population size: The total number of employees in the company. A larger population size generally requires a smaller sample size as a proportion of the total population.

Formula:

One commonly used formula to calculate the sample size is:

```
n = (Z^2 - p - q) / (e^2 - N)
```

where:

- n = Sample size
- Z = Confidence level factor (e.g., 1.96 for 95% confidence)
- p = Estimated proportion of the population that has a particular characteristic (e.g., 0.5 for a 50% estimate)
- q = 1 - p (e.g., 0.5 for a 50% estimate)
- e = Margin of error (e.g., 0.05 for a 5% margin of error)
- N = Population size

Example:

Let's say we want to collect a representative sample from a company with 1,000 employees, with a 95% confidence level and a 5% margin of error.

```
n = (1.96^2 - 0.5 - 0.5) / (0.05^2 - 1000) = 96.04
```

Round Up:

Since the sample size should be a whole number, we round up to the nearest integer, which gives us:

Sample Size = 97

This indicates that we would need to survey at least 97 employees to collect a representative sample from the company under the given assumptions.