9  Statistical Inference

Statistical inference is the process of drawing conclusions about a population based on information obtained from a sample. It allows researchers and analysts to make generalizations, predictions, and decisions under uncertainty, bridging the gap between observed data and the broader population [1].

This mind map illustrates the core structure of statistical inference (see Figure Figure 9.1), highlighting its three main components: Statistical Hypotheses, Hypothesis Testing Methods, and Statistical Decision Making. Key elements such as Null Hypothesis (H₀), Alternative Hypothesis (H₁), T-Test, Z-Test, Chi-Square Test, and P-Value for decision making are included, providing a concise overview of how hypotheses are formulated, tested, and used to guide statistical decisions.

Figure 9.1: Statistical Inference

Statistical inference explains how conclusions about a population can be drawn from sample data by systematically accounting for uncertainty and variability. This topic serves as a bridge between descriptive statistics and formal decision-making methods, such as parameter estimation and hypothesis testing. The following video is designed to provide a clear and intuitive introduction to statistical inference, helping students build conceptual foundations that will support their learning of more advanced statistical techniques.

Statistical Inference

9.1 Statistical Hypotheses

Statistical hypotheses are formal statements about a population parameter that can be tested using sample data. They provide a framework for making objective decisions based on evidence, helping researchers determine whether observed effects are due to random variation or represent a true phenomenon. In hypothesis testing, we compare the Null Hypothesis (H₀) and the Alternative Hypothesis (H₁) to decide which statement is more consistent with the observed data.

The following video provides a clear and concise conceptual explanation of the relationship between statistical hypotheses, parameter estimation, and confidence intervals, helping you build strong intuition before moving on to formal calculations and analytical procedures.

Statistical Hypotheses

9.1.1 Null Hypothesis (H₀)

The Null Hypothesis (H₀) serves as the baseline or reference point in hypothesis testing. It represents the assumption that there is no effect, no difference, or no relationship in the population. H₀ provides a standard against which the observed data is compared, allowing researchers to determine whether any observed difference is likely due to random variation rather than a true effect [2].

Key Points:

  • Acts as a benchmark for testing statistical evidence.
  • Assumed true initially and is tested for possible rejection, not proven.
  • Denoted as H₀ in all statistical analyses.

Examples:

  1. Drug Effect on Blood Pressure:
    The null hypothesis states that a new drug has no effect on blood pressure:
    \[ H₀: \mu_\text{treatment} = \mu_\text{control} \]

  2. Comparison of Teaching Methods:
    The null hypothesis states there is no difference in average test scores between two teaching methods:
    \[ H₀: \mu_1 = \mu_2 \]

In practice, H₀ provides a conservative assumption. Only if the sample data provides strong enough evidence against H₀ do we consider rejecting it in favor of the Alternative Hypothesis (H₁). This ensures decisions are data-driven and objective, minimizing the risk of concluding an effect exists when it does not [1].

9.1.2 Alternative Hypothesis (H₁)

The Alternative Hypothesis (H₁ or Ha) represents the statement that contradicts the Null Hypothesis (H₀). It reflects the effect, difference, or relationship the researcher expects to detect in the population [1]–[3].

Key Points:

  • Indicates the presence of a real effect or difference.
  • Denoted as H₁ or Ha.
  • Can be two-tailed (detecting a difference in either direction) or one-tailed (detecting a difference in a specific direction).

Examples:

  1. Drug Effect on Blood Pressure:
    The alternative hypothesis states that a new drug reduces or changes blood pressure:
    \[ H₁: \mu_\text{treatment} \neq \mu_\text{control} \]

  2. Comparison of Teaching Methods:
    The alternative hypothesis states that one teaching method improves test scores compared to the other:
    \[ H₁: \mu_1 > \mu_2 \]

In hypothesis testing, H₁ is accepted only if the sample evidence is strong enough to reject H₀. This ensures that conclusions about population effects are supported by data, reducing the risk of drawing incorrect inferences [1].

9.1.3 Type I/II Errors

In hypothesis testing, errors can occur when making decisions based on sample data. The two main types of errors are Type I Error (α) and Type II Error (β). The table below summarizes the comparison with examples:

Error Type Definition Probability Example
Type I Error (α) Rejecting H₀ when it is actually true (false positive) α, commonly 0.05 Concluding a new drug lowers blood pressure when in reality it does not.
Type II Error (β) Failing to reject H₀ when it is actually false (false negative) β; Power = 1 − β Concluding a new drug has no effect on blood pressure when it actually does.

Notes:

  • Type I Error (α) is controlled by setting the significance level before conducting the test.
  • Type II Error (β) depends on the sample size, effect size, and variability. Increasing sample size reduces β and increases the power of the test.
  • Understanding both errors is crucial for making informed statistical decisions and balancing the risk of false positives and false negatives in research.

9.2 Hypothesis Test Methods

In statistics, hypothesis testing methods are used to determine whether the evidence from a sample is strong enough to reject the null hypothesis (H₀) in favor of the alternative hypothesis (H₁). The choice of test depends on the type of data, sample size, and population characteristics.

9.2.1 T-Test

The T-Test is used to compare the mean of a sample to a known value or to compare means between two groups when the population standard deviation is unknown and the sample size is relatively small.

Types of T-Test:

  1. One-sample T-Test: Compare sample mean to a known value.
  2. Independent two-sample T-Test: Compare means of two independent groups.
  3. Paired T-Test: Compare means of paired observations (e.g., before-after measurements).

Example:
- Testing whether the average test score of students differs from 75.
\[ H₀: \mu = 75 \quad vs \quad H₁: \mu \neq 75 \]

9.2.2 Z-Test

The Z-Test is used to compare means when the population standard deviation is known or the sample size is large (n ≥ 30). It assumes that the data is approximately normally distributed.

Types of Z-Test:

  1. One-sample Z-Test: Compare a sample mean to a known population mean.
  2. Two-sample Z-Test: Compare means of two independent populations with known standard deviations.

Example:

  • Testing whether a new teaching method changes the average score, assuming the population standard deviation is known:
    \[ H₀: \mu_\text{new} = \mu_\text{old} \quad vs \quad H₁: \mu_\text{new} \neq \mu_\text{old} \]

9.2.3 Chi-Square Test

The Chi-Square Test (χ² Test) is used for categorical data to examine whether the observed frequency distribution differs from the expected distribution.

Types of Chi-Square Test:

  1. Goodness-of-Fit Test: Tests if a single categorical variable follows a hypothesized distribution.
  2. Test of Independence: Tests whether two categorical variables are independent.

Example:

  • Testing whether gender (male/female) is independent of preference for online learning (yes/no):

\[ H₀: \text{Gender and preference are independent} \\ H₁: \text{Gender and preference are not independent} \]

9.3 Statistical Decision Making

Statistical Decision Making involves using the results of hypothesis tests to make informed decisions about the population. After performing a T-Test, Z-Test, or Chi-Square Test, we interpret the p-value and decide whether to reject or fail to reject the null hypothesis (H₀). This process allows us to draw conclusions while considering the risk of errors. Steps in Statistical Decision Making:

  1. Set significance level (α):

    • Common choices: 0.05 (5%), 0.01 (1%), or 0.10 (10%).
    • This determines the threshold for rejecting H₀.

  2. Perform the hypothesis test:

    • Calculate test statistic \((T, Z, χ²)\) based on sample data.
    • Compute the p-value.

  3. Compare p-value with α:

    • If p-value ≤ α: Reject H₀ → evidence supports H₁.
    • If p-value > α: Fail to reject H₀ → insufficient evidence to support H₁.

  4. Consider Type I and Type II Errors:

    • Type I Error (α): Rejecting H₀ when it is true.
    • Type II Error (β): Failing to reject H₀ when it is false.
    • Balance between α and β is important for decision reliability.

9.3.1 T-Test

  • Suppose we test whether the average score of a sample of students is different from 80.
  • T-Test yields: t = 1.82, p-value = 0.10
  • Significance level: α = 0.05

Decision:

  • Since p-value (0.10) > α (0.05), we fail to reject H₀.
  • Interpretation: There is insufficient evidence to conclude that the average score differs from 80.

9.3.2 Chi-Square Test

  • Suppose we test whether gender and preference for online learning are independent.
  • Chi-Square Test yields: χ² = 4.23, p-value = 0.04
  • Significance level: α = 0.05

Decision:

  • Since p-value (0.04) < α (0.05), we reject H₀.
  • Interpretation: There is evidence that gender and preference are not independent.

References

[1]
Jim Frost, Statistical inference overview, 2023, Available. https://statisticsbyjim.com/hypothesis-testing/statistical-inference/
[2]
Khan Academy, Statistical inference, 2023, Available. https://www.khanacademy.org/math/statistics-probability/significance-tests-one-sample
[3]
GeeksforGeeks, Statistical inference in statistics, 2023, Available. https://www.geeksforgeeks.org/statistical-inference/