Statistical Inference

1. Introduction

Statistical inference is the process of using data analysis to deduce properties of an underlying probability distribution. It allows us to make conclusions about population parameters based on sample statistics.

2. Key Concepts

• Population: The entire set of individuals or instances about which we hope to learn.
• Sample: A subset of the population used to represent the group.
• Parameter: A numerical characteristic of a population (e.g., mean, variance).
• Statistic: A numerical characteristic of a sample (e.g., sample mean).

3. Types of Inference

Statistical inference can be broadly classified into two types:

1. Estimation: Estimating population parameters based on sample statistics.
2. Hypothesis Testing: Testing assumptions or claims about a population parameter.

4. Hypothesis Testing

The hypothesis testing process involves the following steps:

1. Formulate the null hypothesis (H0) and alternative hypothesis (H1).
2. Select a significance level (α).
3. Calculate the test statistic.
4. Determine the p-value or critical value.
5. Make a decision: Reject H0 or fail to reject H0.

5. Confidence Intervals

A confidence interval provides a range of values that likely contain the population parameter. The formula for a confidence interval for the population mean is:

CI = x̄ ± Z*(σ/√n)

Where:

• x̄ = sample mean
• Z = Z-value from the standard normal distribution for the desired confidence level
• σ = population standard deviation
• n = sample size

6. Best Practices

• Ensure a representative sample is collected.
• Choose the appropriate statistical tests based on data characteristics.
• Always report confidence intervals along with point estimates.
• Avoid over-reliance on p-values; consider effect sizes.

7. FAQ