For a sample with mean ¯y, adding a constant c to each observation changes the mean to ¯y+c, and the standard deviation s is unchanged. Multiplying each observation by c changes the mean to c¯y and the standard deviation to |c|s.

(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.

(b) Suppose that annual income of Canadian lawyers has a mean of $100,000 and a standard deviation of $30,000.

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.

(c) Observations from a survey that asks about the number of miles traveled each day on mass transit are to be converted to kilometer units (1 mile = 1.6 kilometers).

Explain how to find the mean and standard deviation of the converted observations.

3.79.* Show that Σ(yi − ¯y) must equal 0 for any collection of observations y1, y2, . . . , yn.

3.80.* The Russian mathematician Tchebysheff proved that for any k > 1, the proportion of observations that fall more than k standard deviations from the mean can be no greater than 1/k2. This holds for any distribution, not just bell-shaped ones.

(a) Find the upper bound for the proportion of observations falling (i) more than 2 standard deviations from the mean, (ii) more than 3 standard deviations from the mean,

(iii) more than 10 standard deviations from the mean.

(b) Compare the upper bound for k = 2 to the approximate proportion falling more than 2 standard deviations from the mean in a bell-shaped distribution.Why is there a difference?

3.81.* The least squares property of the mean states that the data fall closer to ¯y than to any other number

c, in the sense that the sum of squares of deviations about the mean is smaller than the sum of squares of deviations about

c. That is,

If you have studied calculus, prove this property by treating f

(c) =
(yi − c)2 as a function of c and deriving the value of c that provides a minimum. (Hint: Take the derivative of f

(c) with respect to c and set it equal to zero.)

Transcribed Image Text:

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