On a two-tailed statistical test with significance level Alpha of 5%, and assuming the distribution is normally distributed.

if the sample data is 3 standard deviations away, should we reject the null hypothesis ?

Question

On a two-tailed statistical test with significance level Alpha of 5%,  and assuming the distribution is normally distributed.

if the sample data is 3 standard deviations away, should we reject the null hypothesis ?

BlackTom AI · Accepted Answer

When evaluating a two-tailed test at Alpha = 0.05 with a normal distribution, we compare the observed test statistic to the critical z-values at the 2.5% tails on each side, which are approximately ±1.96.
Option 1: 'No, do not reject the null hypothesis' would be correct only if the observed statistic fell within the acceptance region between -1.96 and +1.96. Since the data point is 3 standard deviations away, this lies well outside the ±1.96 bounds, indicating the result is unlikely under the null.
Option 2: 'Yes, reject the null hypothesis' aligns with the standard decision rule for a two-tailed test at alpha = 0.05: an absolute z-score of 3 exceeds 1.96, so we have sufficient evidence to conclude the observed effect is unlikely under the null hypothesis.
In plain terms, a 3-sigma departure from the mean corresponds to a p-value around 0.003, which is well below the 0.05 threshold, reinforcing the decision to reject the null. Conversely, staying within ±1.96 would imply a p-value above 0.05 and would not warrant rejection under typical criteria.
Therefore, the data point being 3 standard deviations away supports rejecting the null hypothesis, not failing to reject it.

ERMCPS5350_003_2025_3 - INTRO TO QUANTITATIVE RISK MANAGEMENT 9. Quiz for Class 9

On a two-tailed statistical test with significance level Alpha of 5%, and assuming the distribution is normally distributed. if the sample data is 3 standard deviations away, should we reject the null hypothesis ?

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