Statistical significance is a probability measure used by statisticians to describe the likelihood of an occurrence due to chance. In the context of hypothesis testing statistical significance is predetermined by the tester and known as significance level indicated by the Greek letter '$\alpha$'.
The p value calculated for the test statistic is also a measure of statistical significance. The null hypothesis is confidently rejected for very small values of calculated statistical significance, the p-value. Statistical significance does not convey the meaning of "importance" as the word significance normally means.

## Statistical Significance Definition

Statistical significance is the probability that sample observations can be attributed to chance, and does not convey the real situation. This is indicated by the Greek letter '$\alpha$' and is also known as type I error. The type I error is the probability of rejecting a true null hypothesis.
Statistical significance or the significance level $\alpha$ is predetermined by the researcher considering sampling error, sample size etc.
The level of significance can also be viewed as the area of rejection region.
The null hypothesis is rejected if the calculated test value falls in the rejection region or the calculated p value(significance probability) is less than the significance level preset for the test.

## Statistical Significance Formula

The level of significance for a hypothesis test and the confidence level set for finding the interval estimate are related as follows:
Significance level $\alpha$ expressed as a percent = 100 - confidence level set the interval estimate of Significance level a expressed as a decimal = 1 - confidence level expresses as a decimal.

For example: if a is set at 0.05, the confidence level for interval estimate of population parameter = 1 - 0.05 = 0.95 or 95%.The p value in a hypothetical test is the smallest significance level at which the null hypothesis is rejected. This means the null hypothesis is rejected when the computed p value is less than the level of significance $\alpha$.

The p-value for the computed test value z* are calculated as follows:
 Right-tailed testp-value = P(Z > Z*)Area to the right of Z*(Area in the right tail) Left-tailed testp-value = P(Z < Z*)Area to the left of Z*(Area in the left tail) Two tailed testp-value = 2P(Z > |Z*|)Area in both the tails

## Calculating Statistical Significance

Let us look at examples for calculating p-values for a given set of information and the type of the test.

Example 1:
Sample mean $\bar{x}$ = 25, $\alpha$ = 0.5 Sample standard deviation s = 2  no of observations in the sample = 36.
Calculate the p values for right tailed test with null hypothesis H$_{0}$  : $\mu$ $\leq$ 24. Explain at what level significance $\alpha$, the null hypothesis will be rejected.

Solution :
Z test value is calculated using the formula:

Z = $\frac{\bar{x}-\mu }{\frac{s}{\sqrt{n}}}$

= $\frac{25-24}{\frac{2}{\sqrt{36}}}$

= $\frac{1}{\frac{1}{3}}$

= 3
p-value corresponding to z = 3 is the area to the right of z = 3.
p-value for the test  = 0.5 - 0.4987 = 0.0013

You may note the p-value found is less than the $\alpha$ levels, 0.1,  0.05  0.01 and 0.005.
Hence the null hypothesis is rejected for all these significance levels.

Example 2: Sample mean $\bar{x}$ = 15, $\mu$ = 16, $\sigma$$^{2}$ = 16 and n = 16.
Calculate the p value for the left tailed test with null hypothesis  $H_{0}$ : $\mu$ $\geq$ 16 . Explain at what $\alpha$ level the null hypothesis will be rejected.

Z test value is calculated using formula:

Z = $\frac{\bar{x}-\mu }{\frac{\sigma }{\sqrt{n}}}$

= $\frac{15-16}{\frac{4}{4}}$

= -1

p-value corresponding to z = -1, is the area to the left of z = -1.
p-value for the test = 0.5 - 0.3413 = 0.1587
The p-value found is greater than all $\alpha$ levels less than 0.1
Hence the null hypothesis is not rejected for any of the common $\alpha$ levels chosen.