Hypothesis testing allows us to quantify relationships between known samples and the unknown population from which they were taken. What do I mean by that? Let's say I am interested in the returns from investing in a particular stock. The daily return might be calculated as the ratio of today's closing price over yesterday's closing price. Whether I was to take a *sample* of 30 returns (over 31 days) or 60 returns (over 61 days), I still *couldn't know* the *population* mean return, but I *could hypothesize* about it... so I do.

**.**

*regardless of its truth*Now, imagine an all-seeing and all-knowing supernatural bystander watching the same events unfurl... they *could know* the population mean... and even though they wouldn't be obliged to share the exact value with me, let's say that they'd at least tell me if my hypothesis was true or false; that is to say: if I hypothesized that the population mean was 4.5% and it actually was 4.5% then the hypothesis would be true, otherwise if would be false (it could be 4.2% or 4.3% or even -5% or 63%; the point is we don't know).

If we take our two test results and combine them with the two possible truth values, it produces this 2X2 matrix of outcomes.

Accept | Reject | |
---|---|---|

True | Correct | Type 1 Error |

False | Type 2 Error | Correct |

- True/False: does the actual population mean match the hypothesized mean?
- Reject/Accept: does our statistic fall outside/inside the confidence interval?
- Correct/Incorrect: did we accept a true (or reject a false) hypothesis or did we commit an error?

Let's ask the bystander if our hypothesis was indeed true or if it was false.

- Yes, it's true:
- At 90% we accepted it
- At 80% we rejected it (Type 1 Error)

- No, it's false:
- At 90% we accepted it (Type 2 Error)
- At 80% we rejected it.

**why**we were correct to reject the hypothesis; all that matters is that we did.

This video tries to explain it but I am not confident the author is correct. I'd prefer to side with authors of the CFA curriculum who say - in short - "it's complicated".

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