Human Resource Consulting $($HRC$)$ surveyed a random sample of $66$ Twin Cities construction companies to find information on the costs of their health care plans. One of the items being tracked is the annual deductible that employees must pay. The Minnesota Department of Labor reports that historically the mean deductible amount per employee is $$498$ with a standard deviation of $$85.($Round your z$-$value to $2$ decimal places and final answers to $4$ decimal places. Leave no cells blank$-$be certain to enter $"0"$ if required.$)$

Required:

a$.$ Compute the standard error of the sample mean for HRC$.$

Standard error

b$.$ What is the chance HRC finds a sample mean between $$477$ and $$527?$

Probabity

c$.$ Cakulate the tikelihood that the sample mean is between $$492$ and $5512$

D$.$ What is the probability the sample mean is greater than $550?$