Inferential Statistics

Inferential Statistics is a branch of statistics that allows us to make predictions, decisions, or generalizations about a population based on data collected from a sample. Unlike descriptive statistics, which summarizes data, inferential statistics uses probability theory to test hypotheses, construct confidence intervals, and estimate population parameters. It plays a key role in research, business, science, and policymaking by enabling informed conclusions from limited information, helping us infer trends and patterns in large datasets from smaller, manageable samples.

1.0What is Inferential Statistics?

Inferential Statistics is a branch of statistics that helps us draw conclusions or make predictions about a population based on a sample. It uses data from a small group (sample) to infer the characteristics of a larger group (population). This is essential when it's impractical or impossible to survey an entire population.

2.0Inferential Statistics Meaning

In simple terms, inferential statistics involves analyzing sample data and using it to make generalizations, decisions, or forecasts about a broader population. It’s the core of statistical inference—the process of drawing evidence-based conclusions.

3.0Descriptive & Inferential Statistics

4.0What Is Meant by Inferential Statistics?

Inferential statistics uses probability theory to estimate population parameters (like mean, variance) from sample statistics. It also helps test hypotheses and determine the strength of conclusions based on sample data.

5.0Types of Inferential Statistics?

There are four major types of inferential statistical methods:

1. Hypothesis Testing

Used to accept or reject assumptions about a population.

Common tests: Z-test, t-test, chi-square test

2. Confidence Intervals

Offer a likely span of values for a population parameter.

3. Regression Analysis

Measures the relationship between variables (e.g., linear regression).

4. ANOVA (Analysis of Variance)

Analyzes the means of multiple groups (three or more) for statistical significance.

6.0Inferential Statistics Formula

While there is no single formula, inferential statistics often involves formulas like:

1. Z-Score

$Z=\frac{\bar{X}-\mu }{\frac{\sigma }{\sqrt{n}}}$

2. Confidence Interval for Mean (σ known)

$\bar{X}\pm Z_{\alpha /2}\cdot \frac{\sigma }{\sqrt{n}}$

3. t-test statistic

$t=\frac{{\bar{X}}_1-{\bar{X}}_2}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}$

7.0Solved Examples on Inferential Statistics

Example 1: A sample of 40 students has a mean test score of 70 with a standard deviation of 8. Construct a 95% confidence interval for the population mean.

Solution:

Given:

• $\bar{x}=70,\sigma =8,n=40,Z_{0.025}=1.96$

Formula:

$CI=\bar{x}\pm Z\cdot \frac{\sigma }{\sqrt{n}}$

$=70\pm 1.96\cdot \frac{8}{\sqrt{40}}$

$=70\pm 1.96\cdot 1.26$

$=70\pm 2.47$

Answer: (67.53, 72.47)

Example 2: A company claims that the average battery life of its smartphones is 15 hours. A sample of 50 phones has a mean life of 14.5 hours with a standard deviation of 1.2 hours. Test the claim at 5% significance.

Solution:

• $H_0:\mu =15,H_1:\mu \ne 15$
• $\bar{x}=14.5,\sigma =1.2,n=50$

$Z=\frac{14.5-15}{\frac{1.2}{\sqrt{50}}}$

$=\frac{-0.5}{0.1697}\approx -2.945$

At $\alpha =0.05$, critical value = ±1.96

Since −2.945 < −1.96, reject H_0

Conclusion: There is sufficient evidence to reject the company’s claim.

Example 3: A sample of 100 bulbs has an average lifespan of 1200 hours and standard deviation of 100 hours. What is the standard error of the mean?

Solution:

$SE=\frac{\sigma }{\sqrt{n}}=\frac{100}{\sqrt{100}}=\frac{100}{10}=10$ 

Answer: Standard error = 10 hours

Example 4: Estimating Population Proportion

Q. In a survey, 120 out of 200 people said they prefer online classes. Construct a 90% confidence interval for the population proportion.

Solution:

$\hat{p}=\frac{120}{200}=0.6,Z_{0.05}=1.645$

$SE=\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}=\sqrt{\frac{0.6\cdot 0.4}{200}}=\sqrt{0.0012}\approx 0.0346$

$CI=0.6\pm 1.645\cdot 0.0346\approx 0.6\pm 0.057\Rightarrow (0.543,0.657)$

Answer: Confidence interval = (54.3%, 65.7%)

Example 5: A dietitian believes that a new diet lowers cholesterol more than the old one. The cholesterol levels (in mg/dL) of 10 patients after the new diet have a mean of 180 with a sample standard deviation of 10. Test the claim at 1% significance if the population mean under the old diet was 190.

Solution:

• $H_0:\mu =190,H_1:\mu <190$
• $\bar{x}=180,s=10,n=10,df=9$
• $t=\frac{180-190}{\frac{10}{\sqrt{10}}}=\frac{-10}{3.16}\approx -3.16$
• $t_{0.01,9}\approx -2.821$

Since -3.16 < -2.821, reject H_0

Conclusion: There is strong evidence that the new diet is more effective.

Example 6: What is the formula used in Inferential Statistics?

Ans: There’s no single formula, but common ones include:

• Z-score:

$Z=\frac{\bar{x}-\mu }{\sigma /\sqrt{n}}$

• Confidence Interval:

$\bar{x}\pm Z\cdot \frac{\sigma }{\sqrt{n}}$

• t-test formula, ANOVA formulas, etc.

8.0What Is the Meaning of Inference in Statistics?

Inference in statistics refers to the conclusion or decision made about a population based on sample data. This includes estimating parameters, testing hypotheses, and predicting outcomes using statistical methods.

9.0What distinguishes inferential statistics from descriptive statistics?

10.0Statistical Inference in Real Life

• Medicine: Predict effectiveness of drugs
• Business: Estimate market trends and consumer behavior
• Education: Predict performance of students across institutions
• Engineering: Quality assurance and defect prediction