First, I would need to measure 1 gallon fill amount from multiple gas stations to make a normal distribution (Ex: 30 samples from 100 stations). Using this data set I can calculate the average and standard deviation the across the 100 stations. This gives me a normal distribution for me to make a hypothesis test. To make a hypothesis test I would first determine my level of significance to be 5% (Alpha = 5%). Using the excel function NORMAL.INV I would calculate the fill amount that corresponds to the lower 5% of my distribution curve. Since the cost advantage for the business owner is to charge me more for less fuel, I am setting only a lower critical value to include the full 5%. I would then measure 30 samples of 1 gallon of fuel from the gas pump in question and calculate the average (X_BAR_30). Now I can make a Hypothesis test:

If X_BAR_30 is greater than lower critical value, I would not reject gas pump and say it is accurate.

If X_BAR_30 is less than lower critical value, I would reject the gas pump and say it is not accurate.

To investigate if they were intentionally shorting customers vs out of calibration would be how I assign level of significance and frame my hypothesis test. If I am interested if they are intentionally shorting the customer, I would put the 5% level of significance at the lower critical value only. If I am interested if the pump were out of calibration that I would put half of the level of significance at the lower level and the other half at the upper level to determine my Lower Critical Value and Higher Critical Value.

My Hypothesis Test Reject region and do not reject region correspond to the critical values:

Intentionally shorting:

If X_Bar_30 is greater than Critical Value Lower, do not reject.

If X_Bar_30 is less than Critical Value Lower, reject.

Calibration

If X_Bar_30 is between Critical value Lower, Critical Value Higher; do not reject.

If Critical Value Lower > X_Bar_30 > Critical Value Higher; reject.