If the population mean payment time is 19.5 days, what is the  probability of observing a sample mean payment time of 65 invoices less  than or equal to 18.1077 days? 

The payment time case | Business & Finance homework help

The probability of observing a sample mean payment time of 65 invoices less than or equal to 18.1077 days is directly connected to the population mean payment time of 19.5 days. To calculate this probability, it is first necessary to calculate the standard deviation (SD) for the population. The SD measures the amount of variation or dispersion from the average, and can be calculated by taking the square root of variance (Var). Var is calculated by subtracting each data point from the population mean and then squaring that value before calculating a total sum over all values; dividing this total sum by N-1 yields Var, where N is equal to 65 in this case.

Once we have an estimate for SD, we can use it with our sample mean payment time of 18.1077 days and our known population mean payment time of 19.5 days to calculate Z-scores for both values; Z-score refers to how many standard deviations away from the mean a given value lies. If two scores are compared side-by-side, one can determine how much more or less likely one score is than another score based on its respective Z-score value in relation to those around it; if our sample mean score has a higher Z-score than other scores within its distribution, then it would be considered more unlikely when compared with other scores within that same distribution.

Finally, using these two calculated Z-scores as inputs into a normal distribution calculator will yield us an approximate probability associated with our observed sample mean being lower than or equal to 18.1077 days given our known population mean payments time of 19.5 days; this probability could range anywhere between 0% and 100%, depending on how far away these two means fall from one another relative their respective distributions across multiple observations—the greater their difference, the greater likelihood they are distinct distributions rather than belonging together within one common group, thereby yielding a smaller overall probability associated with them being related in any meaningful way whatsoever outside our established parameters here today.