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Practice with Statistics
Consider the following measurements: {1, 2, 3, 4, 8, 15, 16, 22, 24, 24, 57}
#1-3
Calculate the three measures of central tendency.
#4-5
Calculate two measures of dispersion.
#6-8
Calculate three measures of position.
Consider a set of normally distributed data where x = 312 and the standard deviation equals 14.
#9
Determine the mode.
#10
Determine the percent of data that lies above the data point 312.
#11
Determine the percent of data that lies between 298 and 326.
#12
Determine the percent of data that lies between 284 and 340.
#13
Determine the percent of data that lies above 354.
#14
Determine the percent of data between 305 and 333.
#15
Determine the percent of data above 291.
SOLUTIONS
Consider the following measurements: {1, 2, 3, 4, 8, 15, 16, 22, 24, 24, 57}
#1
The mean equals the sum of the data points divided by the number of data points as below.
x=
#2
1 + 2 + 3 + 4 + 8 + 15 + 16 + 22 + 24 + 24 + 57 176
=
= 16 .
11
11
The mode equals the data point that occurs most frequently in the data set. The frequency distribution
below shows that 24 is the mode of the data set.
x 1 2 3 4 8 15 16 22 24 57
f 1 1 1 1 1 1 1 1 2 1
#3
The median is the middle number (or the average of the two middle numbers) of the data set when it is
arranged in order from least to greatest.
1, 2, 3, 4, 8, 15, 16, 22, 24, 24, 57
The median is 15.
#4
The range of a data set is one measure of dispersion. The range is the difference of the highest and
lowest values of the data set: 57 - 1 = 56.
#5
The standard deviation (s.d.) is the square root of the ratio of the sum of the deviations squared and the
number of data points minus one, i.e., s.d . =
(
Σ⎡ X − X
⎢⎣
s.d . =
n −1
#6
(
Σ⎡ X − X
⎢⎣
n −1
) ⎤⎥⎦
2
.
) ⎤⎥⎦
2
s.d . =
225 + 196 + 169 + 144 + 64 + 1 + 0 + 36 + 64 + 64 + 1681
11 − 1
s.d . =
2644
= 264.4 ≈ 16.26
10
The second quartile is the median, which is 15.
1, 2, 3, 4, 8,
15, 16, 22, 24, 24, 57
#7
The first quartile is the median of the data below the median, which is 3.
#8
1, 2,
3, 4, 8
The third quartile is the median of the data above the median, which is 24.
16, 22,
24,
24, 57
Alternately, to answer #6 or #7 or #8, find the Z-score for any data point. For example find the Z-score
x−x
22 − 16
of 22: Z =
∴ Z 22 ≈
≈ 0.369 .
s.d .
16.26
#9
Determine the mode.
The mode of a normally distributed data set equals the median and the mean. The mean is given as 312,
so the mode is 312.
#10
Determine the percent of data that lies above the data point 312.
If data is normally distributed, half the data falls above the mean. Thus, 50% of the data falls above 312.
#11
Determine the percent of data that lies between 298 and 326.
298 has a z-score of -1. Moreover, 326 has a z-score of 1. Consequently, the percent of data that lies
between 298 and 326 equals the percent of data between -1 and 1 standard deviations. By the empirical
rule, 68% of the data falls within one standard deviation of the mean.
#12
Determine the percent of data that lies between 284 and 340.
284 has a z-score of -2, and 340 has a z-score of 2. Consequently, the percent of data that lies
between 284 and 340 equals the percent of data between -2 and 2 standard deviations. By the
empirical rule, 95% of the data falls within two standard deviations of the mean.
#13
Determine the percent of data that lies above 354.
354 has a z-score of 3. The empirical rule states that 99.7% of the data falls within three standard
deviations of the mean (between -3 and 3), which indicates that 49.85% of the data falls between the
mean and a positive three z-score. 50% - 49.85 = 0.15%, so 0.15% of the data falls above 354 for the
given data set.
#14
Determine the percent of data between 305 and 333.
305 has a z-score of -0.5. The table shows that 19.1% of data falls between -0.5 and mean (305 and
312). 333 has a z-score of 1.5. The table shows that 43.3% of data falls between mean and 1.5 (312 and
333). Consequently, 62.4% of the data falls between 305 and 333 (19.1% + 62.4% = 62.4%).
#15
Determine the percent of data above 291.
291 has a -1.5 z-score. The table shows that 43.3% of data falls between -1.5 and mean (between 291
and 312). Thus, 93.3% of data falls above 291 because 43.3% + 50% = 93.3%.
Practice with Functions
#1
If E = {−2, 2} and F = {−3, 0} , consider E × F in \ 2 . Calculate the greatest distance between any
two elements in E × F .
#2
For a science project, Gregor stands in his driveway at the top of each hour after sunrise until sunset and
records the length of his shadow. Use descriptive notation to identify the domain and range of this
function.
#3
Let X = Y = \ . Let X be the domain and Y the co-domain. Consider the graph of f below. Is the
function bijective?
f
#4
Sketch a graph of the function P : x 6 ( x + 3)( x + 1)( x − 1)( x − 3) . Show the correct x- intercepts,
2
near x-axis behavior, and end-behavior.
#5
The repulsive force, f, between the north poles of two magnets is inversely proportional to the square of
the distance, d, between them. Function f which gives the force in pounds between two magnets d
320
inches apart: f : d 6 2 . What does the horizontal asymptote of function f say about the repulsive
d
force between two magnets?
#6
The fraction of Carbon-14 remaining in the bone material of a dead animal is given by the
function Q : t 6 e −0.00012t where t represents years elapsed since the animal died. What fraction of
Carbon-14 remains in the bone of an animal that died 5,000 years ago?
#7
Astronomers use the distance modulus of a star as a measure of distance from Earth. The function
M : d 6 5 ⋅ log10 ( d ) − 5 maps d, a star's distance from Earth in parsecs to M, the star's distance
modulus. If a star is 10,000 parsecs from Earth, what is its distance modulus?
#8
⎡ 20 x
If V = ⎢
⎢x
⎣
4⎤
1 ⎥⎥ , find the values of x such that det (V ) = 54 .
2⎦
#9
Consider the function f : X 6 A ⋅ X where A = [5 − 2
#10
Consider the function f : x 6
⎡ 2⎤
3] . Map X = ⎢⎢3 ⎥⎥ .
⎢⎣ 4 ⎥⎦
4x + 5
. Name all the asymptotes of f ?
2x − 1
SOLUTIONS
#1
If E = {−2, 2} and F = {−3, 0} , consider E × F in \ 2 . Calculate the greatest distance between any
two elements in E × F .
E × F = {( −2, −3) , ( −2, 0 ) , ( 2, −3) , ( 2, 0 )}
d=
#2
( 2 − ( −2 ) ) + ( 0 − ( −3) )
2
2
= 16 + 9 = 5
For a science project, Gregor stands in his driveway at the top of each hour after sunrise until sunset and
records the length of his shadow. Use descriptive notation to identify the domain and range of this
function.
domain = {the hours of the day} , range = {Gregor's shadow length measurements}
#3
Let X = Y = \ . Let X be the domain and Y the co-domain. Consider the graph of f below. Is the
function bijective?
f
The range of f includes every y, so f is surjective. Nevertheless, the graph does
not pass the horizontal line test, so f is not injective. Consequently, f is not
bijective.
#4
Sketch a graph of the function P : x 6 ( x + 3)( x + 1)( x − 1)( x − 3) . Show the correct x- intercepts,
2
near x-axis behavior, and end-behavior.
#5
The repulsive force, f, between the north poles of two magnets is inversely proportional to the square of
the distance, x, between them. Function f which gives the force in pounds between two magnets x
320
inches apart: f : d 6 2 . What does the horizontal asymptote of function f say about the repulsive
x
force between two magnets?
Since the degree of the numerator (zero) is smaller than the degree of the denominator (two), the rational
function will approach the asymptote y = 0 as x grows very large, which means the repulsive force of
two magnets approaches zero pounds as the distance between the magnets grows large.
#6
The fraction of Carbon-14 remaining in the bone material of a dead animal is given by the
function Q : t 6 e −0.00012t where t represents years elapsed since the animal died. What fraction of
Carbon-14 remains in the bone of an animal that died 5,000 years ago?
Q :1, 000, 000 6 e
#7
−0.00012( 5,000 )
6 e −0.6 6 0.5488
Astronomers use the distance modulus of a star as a measure of distance from Earth. The function
M : d 6 5 ⋅ log10 ( d ) − 5 maps d, a star's distance from Earth in parsecs to M, the star's distance
modulus. If a star is 10,000 parsecs from Earth, what is its distance modulus?
M :10, 000 6 5 ⋅ log10 (10, 000 ) − 5 6 5 ⋅ 4 − 5 6 15
#8
#9
⎡ 20 x
If V = ⎢
⎢x
⎣
4⎤
1 ⎥⎥ , find the values of x such that det (V ) = 54 .
2⎦
20 x
4
1 = 54
x
2
1
20 x ⋅ − x ⋅ 4 = 54
2
10 x − 4 x = 54
6 x = 54
x=9
Consider the function f : X 6 A ⋅ X where A = [5 − 2
⎡ 2⎤
f : ⎢⎢3 ⎥⎥ 6 [5 − 2
⎢⎣ 4 ⎥⎦
#10
Consider the function f : x 6
y=2
x =1 2
⎡2⎤
3] . Map X = ⎢⎢3 ⎥⎥ .
⎢⎣ 4 ⎥⎦
⎡2⎤
3] ⋅ ⎢⎢3 ⎥⎥ 6 5 ( 2 ) + −2 ( 3) + 3 ( 4 ) 6 16
⎢⎣ 4 ⎥⎦
4x + 5
. Name all the asymptotes of f ?
2x − 1