P-value under 5% but power under 80% Im trying to understand how to accurately report results on AB tests
If my type 1 error is 5% and type 2 is 20% , can we have instances where the p-value is < 0.05 and power is < 0.8?
If so, how ? What does it mean ?
Is something statistical significant when p-value is under 0.05 or power over 80% (assuming alpha and beta are defined to be 5% and 20%)?
EDIT: Im performing my power analysis during an experiment. Im trying to give updates on how a change in a button colour is performing. And in that update I want to specify the power it has.

Possible situations abound. Here a few:
Suppose you are testing $H_0: \mu = 50,$ against $H_a: \mu \ne 50$ at significance level $\alpha = 0.01 = 1\%,$ using $n = 10$ observations from a normal population
with $\sigma \approx 4.$
Then a power and sample size procedure (here from a recent release of Minitab)
gives power values about 70%, 75%, 80% for alternative values
$\mu_a = 54.82, 55.06,$ and $55.33,$ respectively.
Power and Sample Size 

1-Sample t Test

Testing mean = null (versus ≠ null)
Calculating power for mean = null + difference
α = 0.01  Assumed standard deviation = 4

Sample
  Size  Power  Difference
    10   0.70     4.81897
    10   0.75     5.05952
    10   0.80     5.32813

Corresponding points on the relevant power curve are shown below:


Note: A trivial example is to set the Rejection region
to be so far out into the tails of the null distribution (say, $|T|\ge 30)$ that $\alpha \approx 0.$ Then the power
(probability of rejection for the specified alternative)
could also be very small--in particular less than 80%.
