To use samples within a population to make valid inferences about the population at large, several key conditions must be met:

1. Random Sampling

The sample should be a random sample from the population to ensure that every individual has an equal chance of being selected. This helps make the sample representative of the population and supports the independence of observations

2. Sample Size

• The sample size should be sufficiently large, typically n≥30n\geq 30n≥30, to invoke the Central Limit Theorem, which states that the sampling distribution of the sample mean will be approximately normal regardless of the population distribution

• For smaller samples (less than 15), the population should be approximately normally distributed, and the sample data should not contain strong skewness or outliers

• If the population is normal, even small sample sizes can be used for inference

3. Independence of Observations

Samples should be independent and identically distributed (i.i.d.). This means the selection of one individual should not influence the selection of another, which is usually ensured by random sampling

4. Population Distribution

• The population should have a finite variance

• If the population is not normal, a larger sample size is necessary to ensure the sampling distribution of the mean is approximately normal

5. Sample Size Relative to Population

When sampling without replacement, the sample size should not exceed 10% of the population to maintain independence

6. Absence of Bias and Outliers

The sample data should not have strong skewness or outliers, especially for small samples, as these can distort the inference about the population mean

7. Precision and Confidence

The sample size should be chosen to achieve desired precision (margin of error) and confidence level (e.g., 95%), balancing practical constraints like cost and time

Summary Table

Condition| Description
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Random Sampling| Sample must be randomly selected to represent the population
Sample Size| Typically n≥30n\geq 30n≥30 for normal approximation; smaller if population is normal
Independence| Observations must be independent and identically distributed
Population Distribution| Population should have finite variance; normality helps with smaller samples
Sample Size vs Population| Sample size ≤ 10% of population when sampling without replacement
Data Quality| No strong skewness or outliers, especially in small samples
Precision & Confidence| Sample size chosen to meet desired confidence level and margin of error

When these conditions are met, statistical inference methods such as confidence intervals and hypothesis tests can be validly applied to draw conclusions about the population from the sample