Chapter Summary

This chapter built the logic of statistical inference — using the information in a sample to draw, and defend, conclusions about the whole population it came from — one of the core ideas of the CBSE Class 12 Applied Mathematics syllabus. Its main results and formulae are gathered below.

Drawing statistical inferences

• A hypothesis is an educated guess about a population that must be tested before it can be trusted.
• Estimation is the process of making inferences about a population from the information obtained in a sample.
• A sampling distribution is the distribution of the possible values of a statistic for a fixed sample size drawn from the population.
• A confidence interval captures the amount of uncertainty attached to a sample estimate of a population parameter.
• Hypothesis testing is the procedure used to accept or reject a statistical hypothesis.

Sampling error and the Central Limit Theorem

• Even a well-drawn sample rarely matches the population exactly; the gap is the sampling error:

where $\bar{x}$ is the sample mean and $\mu$ the population mean.

• Central Limit Theorem (CLT): the sampling distribution tends to be normal (bell-curve shaped) when $n$ is large, no matter what the shape of the population is.

Degrees of freedom

• For a sample of size $N$, the degrees of freedom is

t-tests

• T-test for one sample:

• T-test for two independent groups — when the variances are assumed equal, the samples are pooled into $S_p$: $t = \dfrac{\bar{x}_1 - \bar{x}_2}{S_p\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}};$ …