**Brief Concept:**

A hypothesis is a statement about the value of a population developed for testing a theory. Hypothesis testing is the statistical assessment of a statement or idea regarding a population. In CFA level I, candidate is required to know procedures used to conduct tests of population means, population variances, differences in means, differences in variances and mean differences. Understanding of various tests involving z-test, t-test, chi-square test and F-test and appropriate usage of tests is required to be understood in for the CFA exam.

**Key learning out of LOSs as per CFA Curriculum:**

**Hypothesis testing** process requires a statement of a null and an alternative hypothesis, selection of appropriate test statistic, specification of significance level, decision rule, calculation of a sample statistic, decision regarding the hypothesis based on the rest and a decision based on the test results. - A
**null hypothesis** is the hypothesis that the researcher wants to reject. The **alternate hypothesis** is what is concluded if there is sufficient evidence to reject the null hypothesis. **A two-tailed test** results from a two sided alternative hypothesis (e.g. Ha: µ ≠ µ_{o}). A **one tailed test** results from a one-sided hypothesis (e.g. Ha: µ < µ_{o} or Ha: µ > µ_{o}). The decision rule depends on the alternative hypothesis and the distribution of test statistic.

**Type I error** is the rejection of null hypothesis when it is actually true. **Type II error** is the failure to reject the null hypothesis when it is actually false.

Decision | True Condition |

| Ho is true | Ho is false |

Not reject Ho | Correct Decision | Incorrect Decision Type II Error |

Reject Ho | Incorrect Decision Type I Error Significance level, α = P (Type I error) | Correct Decision Power of the test = 1 – P(Type II error) |

**Significance level** can be interpreted as the probability that a test statistic will reject the null hypothesis by chance when it is actually true. - The
**power of a test** is the probability of rejecting the null when it is false. Power of a test is defined as **1 – P (type II error).** - Hypothesis testing compares a computed test statistic to critical value at a stated level of significance. A hypothesis for a population parameter is rejected when the sample statistics lies outside a confidence interval around the hypothesized value for the chosen level of significance.
- When population variance is known, z-statistic is used for tests of the mean of a normally distributed population:
**z = x - µ**_{o}/ (α/ sq. rt (n)) - When population variance is unknown, t-statistics is used for tests of the mean of a normally distributed population:
**t **_{n-1} = x - µ_{o}/ (s/ sq. rt (n)) **Parametric tests** like t- test, F- test and chi – square tests make assumptions regarding distribution of population from which samples are drawn, while **non-parametric tests** either do not consider a particular population parameter or have few assumptions about the sampled population. - When
**two population variances** are assumed to be equal, the denominator is based on the variance of the pooled samples, but when sample variances are assumed to be unequal, the denominator is based on a combination of two sample variances. For **two independent samples** from two normally distributed populations, the difference in means can be tested with a t-statistic. - Test of a hypothesis about the population variance for a normally distributed population used a
**chi- square test statistic C**^{2} = (n-1) s^{2} / α^{2}. Degree of freedom in chi-square statistic is n-1. **Paired comparison test** is concerned with the mean of differences between the paired observations of two dependent, normally distributed samples. A t-statistic is used: **t = x - µ**_{d} / s_{d}. - Test comparing two variances based on independent samples from two normally distributed populations uses
**F – distributed test** statistic: **F = x**_{1}^{2} / x_{2}^{2}. Here x_{1}^{2} is the variance of first sample and x_{2}^{2} is the variance of second sample. Usually x_{2}^{2} < x_{1}^{2}

**Type of questions expected in the topic area:**

The typical questions expected from this topic area include

- Defining null hypothesis and alternative hypothesis for test cases
- Selecting appropriate test structure from F test, two tailed test, One tailed test and Chi square test
- Calculation of t- statistic and z-statistic
- Whether a null hypothesis can be rejected or not given a level of significance