Confidence Intervals and Null Hypothesis Tests for Return Means
Summary
The document explains how a sample mean can be tested against a hypothesized population mean using confidence intervals. Its example uses daily S&P 500 returns and a null hypothesis that the population mean is zero. At the 90% confidence level, the sample mean falls outside the stated interval, so the null is rejected; at 95%, it falls inside, so the evidence is insufficient to reject it. The interval widens at the higher confidence level, making that test more conservative.
The response relates this decision to statistical significance: under the assumptions of a sufficiently large sample, approximate normality, and a valid standard error, the observed mean would be relatively unusual under the null at the 10% level, but not at the 5% level. The example illustrates evidence against a hypothesis, not proof that the population mean is zero or nonzero. It offers no detailed explanation of z-scores or trading applications, and its conclusions depend on the sampling and distribution assumptions.
Key ideas
- A confidence interval for a mean widens as the confidence level increases.
- A sample mean outside the null-centered interval provides evidence against the null hypothesis at that test level.
- Failing to reject a null hypothesis does not establish that it is true.
- The interpretation relies on assumptions about sample size, distribution, and standard error.
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# How is hypothesis testing work in population sampiling? # How is hypothesis testing work in population sampiling? I am learning the basics of quant trading from quantconnect's tutorial Confidence Interval and Hypothesis Testing. I understood the first part of the article but I dont understand "Hypothesis Testing" section. From the article > Now we can talk about hypothesis testing. Hypothesis test is essentially test your inference based on a sample. Let's use our dataset, the daily return of S&P 500 us our population. Assume that we don't know the mean of this population. I guess that the mean of this population is 0. Is my guess correct? I need to test this hypothesis with my sample. Let's start from observing our sample: The null hypothesis(mean=0) and alternative hypothesis(mean not equal 0) are ``` H0:μ¯=0 H0:μ¯≠0 ``` and confidence interval used are ``` 90% confidence interval the mean is between (-0.00039756352254768874, 0.00039756352254768874) 95% confidence interval the mean is between (-0.00047513689280089639, 0.00047513689280089639) ``` They reject null hypothsesis for 90% confidence because real mean 0.000463 is outside the confidence mean range > Our mean of the sample is out of the 90% confidence interval. This means on a 90% confidence level, we can claim that the mean of our population is not 0. Then in 95%, real mean is inside the confidence intervals range and so we accept the null hypothesis. Shouldn't this the other way around because the real mean is inside the range of 95% confidence interval so mean of the population should be not zero ? What is the rationale behind it? Then they talk about z-score method which also I am not able to grasp? How do this applied to popultion sampiling? How is this idea applied in trading(any example)? ## Answer by Chris (score 2) https://quant.stackexchange.com/a/46186 This is pretty standard fare for a Stats 101 course, so as to rationale, etc. you might benefit from picking up a textbook or otherwise do some reading on this. In brief though, hypothesis testing allows us to assess the likelihood sample estimates are different than theorized values in the absence of actual population values. In the cases above, with a null hypothesis of 0, sufficient sample size to assume a normal distribution, and sample standard error, you can assert you would only see a sample mean as extreme as +/-0.00039756352254768874 by chance 10% of the time. Hence, as your calculated sample mean is outside that range, you would reject the null that the population mean is equal to 0 and assert that your sample mean is staistically significant at the 10% level. However, you can't reject the null in the 95% case because your sample mean isn't extreme enough.
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