In the last two posts we read about sampling and learnt about the various

**sampling types**and**sampling distributions**. Today we start a new topic – hypothesis testing wherein we will learn about the steps followed in hypothesis testing, one tailed test and two tailed test. In the next part we will look into type I and type II errors and so on. So let’s begin…..## Hypothesis testing

In quantitative analysis, hypothesis testing is conducted to statistically assess the validity of a statement. Hypothesis is stated in terms of the population parameter to be tested. For example we may hypothesise that the mean returns of a portfolio are positive. Through hypothesis testing we seek to validate the reasonableness of this hypothesis, i.e., should this statement be accepted or rejected.

**Null hypothesis and alternative hypothesis:**

Hypothesis testing is based on sample statistics and probability. To start with we state two types of hypothesis, null hypothesis (H

_{0}) and alternative hypothesis (H_{a}). The null hypothesis is the statement that you are actually trying to reject and the alternative hypothesis is the statement you want to accept. The alternative hypothesis can be accepted only if there is sufficient evidence to support it and therefore reject the null, failing which the null cannot be rejected and will have to be accepted.**One tailed test and two tailed test**

The H

_{0 }is usually stated as H_{0: }µ = µ_{0 }where µ is the population mean and µ_{0 }is the hypothesised value. The alternative hypothesis can be either stated as a one tailed test or a two tailed test. The test depends on what is being tested. If the test is to validate if the mean return of the portfolio is positive, then it can be stated as a one tailed test in the upper tail. If the test is to validate if the mean return of the portfolio is other than zero, the two tailed test needs to be employed.Two tailed tests are constructed as,

H

_{0: }µ = µ_{0 ; }H_{a: }µ ≠ µ_{0}Since alternative hypothesis allows for values both above and below the hypothesised parameter, a two tailed test uses two critical values or rejection points.

The decision rule for a two tailed test is given as,

Reject H

_{0 }if,Test statistic > upper critical value

Test statistic < lower critical value

For a two tailed test using a 95% confidence level and 5% level of significance the critical values are ± 1.96. Therefore if the test statistic lies below -1.96 or above +1.96 we will reject H

_{0. }If the test statistic falls within the two critical values we will fail to prove that the sample statistic is sufficiently different from the hypothesised value and therefore we cannot reject the null.A one tailed test can be either in the upper tail or lower tail. Therefore a one tailed test is constructed as,

Upper tail: H

_{0: }µ ≤ µ_{0 ; }H_{a: }µ > µ_{0}Lower tail: H

_{0: }µ ≥ µ_{0 ; }H_{a: }µ < µ_{0}Remember now that it is the alternative hypothesis that we actually want to accept. Therefore for the upper tail the alternative hypothesis is framed as population mean is greater than the hypothesised value and for the lower tail it is framed as the population mean is lesser than the hypothesised value. In order to gather evidence to support the alternative we have to disprove the null and thus reject it.

Thus if we want to state that the mean return of the portfolio is positive, the one tailed upper tail test will be framed as,

Upper tail: H

_{0: }µ ≤ 0_{ ; }H_{a: }µ > 0To be continued…..

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That’s all in this post …..Thanks for reading….… in the next part of the hypothesis testing series we will learn about

**type I and type II errors**….so stay posted….p.s. If you liked what you read don’t forget to use the social sharing buttons on the page :-)…..

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**For solved examples please refer to the CFA Institute Books and Schweser CFA notes. The problems can be easily solved using the CFA institute approved financial calculators. Please refer to the CFA exam policy and CFA calculator guide.**

[…] Please refer to CFA exam policy and CFA calculator Guide. You may also want to read: Hypothesis testing (Part 1) Sampling and estimation (Part 2) Sampling and estimation (Part 1) Common probability […]