Step 1 :Subtract 20 from each item in the given sample: [48, 50, 40, 37, 30, 39, 41] to get a new sample: [28, 30, 20, 17, 10, 19, 21].
Step 2 :Compute the mean of the new sample by adding all the numbers in the sample and dividing by the number of items in the sample. The mean is \(\frac{28+30+20+17+10+19+21}{7} = 20.71\).
Step 3 :Compute the standard deviation of the new sample. This is a measure of how spread out the numbers in the sample are. It is calculated by taking the square root of the variance, which is the average of the squared differences from the mean. The standard deviation is \(\sqrt{\frac{(28-20.71)^2+(30-20.71)^2+(20-20.71)^2+(17-20.71)^2+(10-20.71)^2+(19-20.71)^2+(21-20.71)^2}{7}} = 6.23\).
Step 4 :Final Answer: The mean of the new sample is \(\boxed{20.71}\). The standard deviation of the new sample is \(\boxed{6.23}\).