2. 职称英语
  3. Each of the following linear equations defines y as a function of x for all inte

Each of the following linear equations defines y as a function of x for all inte

游客2024-01-12  7
问题 Each of the following linear equations defines y as a function of x for all integers x from 1 to 100. For which of the following equations is the standard deviation of the y-values corresponding to all the x-values the greatest?
选项 A、y=x/3
B、y = x/2+ 40
C、y = x
D、y = 2x + 50
E、y = 3x-20
答案 E
解析 Recall that the standard deviation of the numbers in a data set is a measure of the spread of the numbers about the mean of the numbers. The standard deviation is directly related to the distances between the mean and each of the numbers when the mean and the numbers are considered on a number line. Note that each of the answer choices is an equation of the form y = ax+b, where a and b are constants. For every value of x in a data set, the corresponding value of y is ax + b, and if m is the mean of the values of x, then am + b is the mean of the corresponding values of y.
In the question, the set of values of x consists of the integers from 1 to 100, and each answer choice gives a set of 100 values of y corresponding to the 100 values of x. For each value of x in the data set,
(1)the distance between x and the mean m is \x - m\, and
(2)the distance between the corresponding y-value, ax + b, and the mean, am + b, of the corresponding y-values is \ax + b — am — b\, which is equal to \ax — am\, or |a||x-m|.
Therefore the greater the absolute value of a in the equation y = ax+b, the greater the distance between each y-value and the mean of the y-values; hence, the greater the spread. Note that the value of b is irrelevant. Scanning the choices, you can see that the equation in which the absolute value of a is greatest is y = 3x - 20. Thus the correct answer is Choice E.
This explanation uses the following strategy.
Strategy 8: Search for a Mathematical Relationship
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