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

Question

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?

A) y = x/3
B) y = x/2+40
C) y = x
D) y = 2x + 50
E) y = 3x − 20

A) y = x/3
B) y = x/2+40
C) y = x
D) y = 2x + 50
E) y = 3x − 20

IanStewart · Accepted Answer

When we add a constant number to every value in a set, we don't change how spread out that set is. So we don't change the sets standard deviation. So if you create a set of one hundred values by plugging x = 1, 2, 3, ... 98, 99, 100 into the equation

y = 3x + 10,000

you'll get the same standard deviation that you get when you plug those x-values into

y = 3x

because the sets only differ because we're adding the constant 10,000 to every value in the first set.

So we can ignore the numbers we're adding on at the end in each answer choice; we're just comparing these:

y = (1/3)x
y = (1/2)x
y = x
y = 2x
y = 3x

If you multiply all the values in a set by some constant, you stretch distances out (or shrink them, if we multiply by something between 0 and 1). So if we plug in 1, 2, 3, ... 100 into y = 3x, the distances between values will be 3 times as big as they are when we plug those numbers into y = x. So we'll get a larger standard deviation (in fact, one that is precisely 3 times larger). So E is the right answer here.

SMQazi · Answer

SD is nothing but the spread of numbers from the mean. Simply plug x=100 and x=1 in each of the equations to get the relative spread. Constant values don't change anything, so keep em or remove em, doesn't matter. Greatest spread is the answer, so.

X=1,100

a. y= 1/3 and 100/3 ->.33 and 33.33 --> spread of 33
b. y= 1/2+40 and 100/2+40 -> 40.5 and 90 --> spread of 49.5
c. y= 1 and 100 --> spread of 99
d. y= 2*1 + 50 and 2*100 + 50 -> 52 and 200 --> spread of 148
e. y= 3*1 - 20 and 3*100 - 20 -> -17 and 280 --> spread of 297

hence answer is E.

luisdicampo · Answer

Deconstructing the Question

We are comparing the standard deviation of the  y -values generated by each linear equation as  x  takes all integer values from  1  to  100 .

For a linear transformation of the form  y=ax+b , the standard deviation changes by a factor of  |a| . The constant term  b  only shifts all values and does not affect spread.

So we only need to compare the absolute value of the coefficient of  x .

Step-by-step

In choice A,

y=\frac{x}{3}

the coefficient of  x  is

\frac{1}{3}

In choice B,

y=\frac{x}{2}+40

the coefficient of  x  is

\frac{1}{2}

The  +40  does not affect the standard deviation.

In choice C,

y=x

the coefficient is

1

In choice D,

y=2x+50

the coefficient is

2

The  +50  does not affect the standard deviation.

In choice E,

y=3x-20

the coefficient is

3

The  -20  does not affect the standard deviation.

Now compare the absolute values:

\frac{1}{3}, \frac{1}{2}, 1, 2, 3

The greatest is  3 , so  choice E gives the greatest standard deviation.

Answer: E

KarishmaB · Answer

A higher SD means the numbers are more spread out on the number line.

Consider y = x. Here y is a set of numbers from 1 to 100. It has a certain SD, say S. 
When we divided every term of this list of 100 numbers by 2 or 3, the numbers come close together on the number line. So for options (A) and (B), the numbers will be much closer together and their SD will be smaller than S. Recall that adding a fixed constant to each number of the list does not change the SD. Hence the +40 in option (B) has no impact on its SD.

When we multiply each term of this list by 2, the numbers spread out on the number line and when we multiply by 3, the numbers spread out even further. Hence SD of option (D) will be more than S and that of option (E) will be even higher. Again, adding or subtracting a constant from each term has no impact on SD.

Hence option (E) will have the greatest SD.

Answer (E)

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

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

Prep Toolkit

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