2.146 Rescale the data The mean and standard deviation of a sample may change if data are rescaled (for instance, temperature changed from Fahrenheit to Celsius). For a sample with mean xQ, adding a constant c to each observation changes the mean to xQ +

c, and the standard deviation s is unchanged. Multiplying each observation by c 7 0 changes the mean to c xQ and the standard deviation to cs.

a. Scores on a difficult exam have a mean of 57 and a standard deviation of 20. The teacher boosts all the scores by 20 points before awarding grades.

Report the mean and standard deviation of the boosted scores. Explain which rule you used and identify c.

b. Suppose that annual income for some group has a mean of $39,000 and a standard deviation of $15,000.

Values are converted to British pounds for presentation to a British audience. If one British pound equals

$2.00, report the mean and standard deviation in British currency. Explain which rule above you used and identify c.

c. Adding a constant and/or multiplying by a constant is called a linear transformation of the data. Do linear transformations change the shape of the distribution?

Explain your reasoning.