ALGEBRA
CONTENT ACADEMY
Statistics
SOL A.9, A.10, A.11
SOL AII.2, AII.9, AII.10, AII.11, AII.12
February 18, 2015 & March 4, 2015
2014 Algebra 1 SPBQ data – Statistics
Formulas
•
What formulas do students need
to know related to Statistics?
•
What formulas are provided on
the formula sheet related to
Statistics?
2012 - Suggested Practice for SOL A.9
Students need additional practice finding values within a given standard deviation
of the mean.
A teacher gave a quiz. The following stem-and-leaf plot shows the scores of the students in her class.
Quiz Scores
STEM
LEAF
4
0
5
00
6
000
7
0000
8
000
9
00
10
0
The mean score of this data set is 70 and the standard deviation
(rounded to the nearest tenth) is 15.8.
Which scores are within one standard deviation of the mean?
Key: 6|0 equals 60
Common Errors? Misconceptions?
2012 - Suggested Practice for SOL A.9
Students need additional practice identifying an interval in which an element lies.
A data set has a mean of 16.5 and a standard deviation of 3. The element x has a
z-score of 1.5. In which interval does the element lie?
10.5 ≤ x < 13.5
13.5 ≤ x < 16.5
16.5 ≤ x < 19.5
19.5 ≤ x < 22.5
22.5≤ x < 25.5
Common Errors? Misconceptions?
2012 - Suggested Practice for SOL A.9
Students need additional practice finding an element of a data set given the mean,
standard deviation, and z-score.
A data set has a mean of 34 and a standard deviation of 4.5. An element in the data
set has a z-score of -1.2.
a)
Without doing a calculation, state whether this element is less than, equal to, or
greater than 34.
b)
Determine the element of the data set.
Common Errors? Misconceptions?
2013 - Suggested Practice for SOL A.9
Students need additional practice performing calculations with statistical
information.
a)
A data set has a mean of 55 and a standard deviation of 3.5. The z-score for a
data point is -1.2. What is the data point?
b.
A data set has a standard deviation of 3. The element 16 is an element of a data
set, with a z-score of 2.4. What is the mean of the data set?
Common Errors? Misconceptions?
2013 - Suggested Practice for SOL A.9
Students need additional practice performing calculations with statistical
information.
The number of minutes book club students read on Monday night is displayed by
the graph. The mean number of minutes for this data set is 21.18, and the standard
deviation of the data set is 6.5. The z-score for the data point representing the
number of minutes Tim read is 1.25. In which interval does this data point lie?
Common Errors? Misconceptions?
2014 - Suggested Practice for SOL A.9
Students need additional practice solving problems involving standard deviation.
A data set is shown.
If the standard deviation of the data set is approximately 1.25, how many of these elements
are within one standard deviation of the mean?
Common Errors? Misconceptions?
2014 - Suggested Practice for SOL A.9
Students need additional practice solving problems involving standard deviation.
1.
A data set has a mean of 45. An element of this data set has a value of 50 and a
z-score of 0.75. What is the standard deviation for this data set, rounded to the
nearest hundredth?
2.
Use two of the three numbers shown in the list to complete this sentence. A
data set could have a variance of
and a standard deviation of
.
9
40.5
81
Common Errors? Misconceptions?
2012 - Suggested Practice for SOL A.10
Students need additional practice analyzing changes to a data set when a data point is
added or removed, and analyzing two box-and-whisker plots to draw a conclusion about
the distribution of the data.
This box-and-whisker plot represents nine pieces of data.
No number is repeated.
The number 23 is removed from the data set and a new box-and-whisker plot is drawn. Compared to the values in the original
box-and-whisker plot, describe the changes to each of these values (increases, decreases or stays the same):
a)
the lower extreme
b)
the lower quartile
c)
the median
d)
the upper quartile
e)
the upper extreme
Need to see it?
Sample Data Sets
Before: 10, 12, 14, 15, 18, 20, 21, 22, 23
After: 10, 12, 14, 15, 18, 20, 21, 22
Common Errors? Misconceptions?
2012 - Suggested Practice for SOL A.10
Each of these box-and-whisker plots represents a data set with 10 distinct elements.
Which statements about these plots appear to be true?
a)
There are more elements in the lower quartile of plot B than plot A, because the left whisker of plot B is longer than the left
whisker of plot A.
b) Since both data sets have 10 distinct elements, the box of plot A and the box of plot B contain the same number of
elements.
c)
There are fewer elements in the upper quartile of plot B than plot A, because the right whisker of plot B is shorter than the
right whisker of plot A.
d) The interquartile range of plot A is greater than the interquartile range of plot B.
e)
The range of both plots are equal.
Common Errors? Misconceptions?
2013 - Suggested Practice for SOL A.10
Students need additional practice interpreting data plotted in box-and-whisker
plots.
Each of these box-and-whisker plots contain 15 unique elements.
1.
Write a statement comparing the range of both plots.
2.
Which box-and-whisker plot has the greater interquartile range?
2.
Which data set has more elements with a value of 11 or greater?
Common Errors? Misconceptions?
2013 - Suggested Practice for SOL A.10
•
•
Plot A represents the total number of songs downloaded by each of 15 students in Mr. Archer’s class during
October. Each student in Mr. Archer’s class downloaded a different number of songs from the others.
Plot B represents the total number of songs downloaded by each of 20 students in Mrs. Baker’s class during
October. Each student In Mrs. Baker’s class downloaded a different number of songs from the others.
During the month of October, what is the difference between the number of students who downloaded more
than 6 songs in Mrs. Baker’s class and the number of students who downloaded more than 6 songs in Mr.
Archer’s class?
Common Errors? Misconceptions?
2013 - Suggested Practice for SOL A.10
This box-and-whisker plot summarizes the number of pieces of pizza each of ten volunteers served at a
concession stand one night.
Another volunteer served 16 pieces of pizza that night, and 16 is added to the original data set. A new box-andwhisker plot is drawn. Which two statements comparing the new box-and-whisker plot to the original box-andwhisker plot must be true?
The interquartile range of the box-and-whisker plot increases.
The range of the box-and-whisker plot increases.
The value of the upper extreme increases.
The value of the median increases.
Common Errors? Misconceptions?
2014 - Suggested Practice for SOL A.10
Students need additional practice identifying and comparing the ranges,
interquartile ranges, and medians of box-and-whisker plots.
1. Which two plots appear to have the same value for the range?
2. Which two plots appear to have the same value for the interquartile range?
3. Which two plots appear to have the same value for the median?
Common Errors? Misconceptions?
2014 - Suggested Practice for SOL A.10
Which statement appears to be true regarding the box-and-whisker plots shown?
a. The interquartile range of the data for plot A is greater than the interquartile range of the data
for plot B.
b. The upper extreme of the data for plot A is greater than the upper extreme of the data for plot C.
c. The range of the data in plot A is the same as the range of the data in plot C.
d. The median of the data in plot A is greater than the median of the data in plot B.
Common Errors? Misconceptions?
2012 - Suggested Practice for SOL A.11
Students need additional practice making predictions using the linear or quadratic
curve of best fit.
Determine the quadratic curve of best fit for the data. Then estimate what the value
of y will be when x = -4.
Common Errors? Misconceptions?
2013 - Suggested Practice for SOL A.11
Students need additional practice making predictions using the linear or quadratic
curve of best fit.
This set of ordered pairs shows a relationship between x and y.
a)
What is the equation for the quadratic curve of best fit for this set of data?
b)
Predict the value of y when x = 8.
Common Errors? Misconceptions?
2013 - Suggested Practice for SOL A.11
This table shows the value, v, of an account at the end of m months. There was an
initial deposit of $50 and no other deposits were made.
m,
time in months
0
1
3
5
7
9
v,
value in dollars
50
129
299
485
687
905
If the value of the account continues to increase in the same way, predict the value of the account at
the end of 13 months. Use the quadratic curve of best fit to make the prediction.
Common Errors? Misconceptions?
2013 - Suggested Practice for SOL A.11
The data in the table shows the average United States farm size, in acres, for the years 2000-2007.
Average Farm Size
Year
Average Acres
Per Farm
2000
2001
2002
2003
2004
2005
2006
2007
434
437
436
441
443
444
446
449
Using the line of best fit for the data shown in the table, what is the best prediction of the average
farm size in the year 2014?
a. 437 acres
b. 441 acres
c. 447 acres
d. 463 acres
Common Errors? Misconceptions?
2014 - Suggested Practice for SOL A.11
Students need additional practice determining the linear or quadratic curve of best fit.
This set of ordered pairs shows a relationship between x and y.
Which equation best represents this relationship?
a.
b.
c.
Extension: Using the curve of best fit, what is
the value of y, rounded to the nearest whole
number, when the value of x is 8?
d.
Common Errors? Misconceptions?
2014 - Suggested Practice for SOL A.11
Students need additional practice determining the linear or quadratic curve of best
fit and making predictions.
This set of ordered pairs shows a relationship between x and y.
Using the line of best fit, which is closest to the output when the input is 5?
a.
b.
c.
d.
Common Errors? Misconceptions?
2014 - Suggested Practice for SOL A.11
Which equation best models
the relationship shown on the
grid?
a.
b.
c.
d.
Common Errors? Misconceptions?
2014 Algebra 2 SPBQ data – Statistics
Formulas
•
What formulas do students need
to know related to Statistics?
•
What formulas are provided on
the formula sheet related to
Statistics?
2012 - Suggested Practice for SOL AII.2
Students need additional practice finding the nth term of a sequence when a written
description of the sequence is given.
What is the seventh term of the geometric sequence with a first term of 729 and a
common ratio of ?
Common Errors? Misconceptions?
2013 - Suggested Practice for SOL AII.2
Students need additional practice finding the sum of a geometric series, particularly
when the common ratio is negative.
Find the sum of this series.
Common Errors? Misconceptions?
2014 - Suggested Practice for SOL AII.2
Students need additional practice finding the sum of a geometric series, particularly
when the common ratio is negative.
Find the sum of the infinite geometric series:
a)
b)
Common Errors? Misconceptions?
2012 - Suggested Practice for SOL AII.9
Students need additional practice identifying the equation for the curve of best fit
and making predictions using the curve of best fit.
A data set is displayed in this table. Using the exponential curve of best, what is the
value of y, rounded to the nearest hundredth, when x = 5?
x
-3
-2
-1
0
y
3.375
2.25
1.5
1
Common Errors? Misconceptions?
2013 - Suggested Practice for SOL AII.9
Students need additional practice finding the exponential curve of best fit for a set
of data and making predictions using this curve.
The table provides the value of an account over time that earned annual compound interest. There
was an initial deposit of $1,500 into the account, and no other deposits were made.
Value of Account Over Time
Time in years, x
0
5
10
15
20
25
30
Value in dollars
1,500.0
0
1,914.4
2
2,443.3
4
3,118.3
9
3,979.9
5
5,079.5
3
6,482.9
1
Assuming the account continues to grow in the same way, use the exponential curve of best fit to
find the value of the account at the end of 40 years, rounded to the nearest dollar.
Common Errors? Misconceptions?
2014 - Suggested Practice for SOL AII.9
Students need additional practice finding the curve of best fit for a set of data and
making predictions using this curve.
a)
Which type of equation would best model the data in this table?
x
30
60
90
120
150
180
A Exponential
b)
B Linear
y
2
4
8
16
32
64
C Logarithmic D Quadratic
Using the equation of best fit from the data in the table, what would be the
value of y if x = 300?
Common Errors? Misconceptions?
2012 - Suggested Practice for SOL AII.10
Students need additional practice identifying a variation equation that models a
situation, and solving problems involving variation.
Common Errors? Misconceptions?
2012 - Suggested Practice for SOL AII.10
Which equations represent this situation?
A car's stopping distance, d, varies directly with the speed it travels, s, and
inversely with the friction value of the road surface, f.
Common Errors? Misconceptions?
2012 - Suggested Practice for SOL AII.10
The amount of time required to stack boxes varies directly with the number of boxes
and inversely with the number of people who are stacking them. If 2 people can
stack 60 boxes in 10 minutes, how many minutes will be required for 6 people to
stack 120 boxes?
Common Errors? Misconceptions?
2013 - Suggested Practice for SOL AII.10
Students need additional practice finding the constant of proportionality and
solving real-world problems involving a combination of direct and inverse
variations.
a. If y varies inversely with the square of x, what is the constant of proportionality
when y = 10 and x = 5?
b. Body mass index (BMI) is directly proportional to a person’s weight in pounds
and inversely proportional to the square of a person’s height in inches. A person
with a BMI of 23.91 has a weight of 135 pounds and a height of 63 inches.
Rounded to the nearest hundredth, what is the BMI of a person with a weight of
145 pounds and a height of 65 inches?
Common Errors? Misconceptions?
2014 - Suggested Practice for SOL AII.10
Students need additional practice finding the constant of proportionality involving
a combination of direct and inverse variations.
If y varies directly with the square of x and inversely with the cube root of t, what is
the constant of proportionality if x = 4, y = 3, and t = 8 ?
Common Errors? Misconceptions?
2014 - Suggested Practice for SOL AII.10
Students need additional practice finding the constant of proportionality and solving realworld problems involving a combination of direct and inverse variations.
Assume that wind resistance varies jointly as an object’s surface area and velocity. If a ball
with a surface area of 25 square feet traveling at a velocity of 40 miles per hour experiences
a wind resistance of 225 Newtons, what velocity must a ball with 40 square feet of surface
area have in order to experience a wind resistance of 270 Newtons?
Common Errors? Misconceptions?
2012 - Suggested Practice for SOL AII.11
Students need additional practice using mean and standard deviation to find the
area under a normal curve and apply properties of the normal distribution to solve
problems.
The running times for a group of 200 runners to complete a one mile run are normally
distributed with a mean of 6.5 minutes and a standard deviation of 1.5 minutes.
a)
Approximately how many of the runners have a
time greater than 8 minutes?
b)
What percentage, rounded to the nearest tenth, of
these runners can complete this run in less than
3.5 minutes?
Common Errors? Misconceptions?
2013 - Suggested Practice for SOL AII.11
Students need additional practice using properties of the normal distribution curve
to find the probability of an event, the percent of data that falls within a specified
interval, and the number of expected values that fall within a specified interval.
A population of adult males had their heights measured. The heights were normally
distributed. Approximately what percentage of the heights, rounded to the nearest
whole number, are within one standard deviation of the mean?
a. 34 %
c. 68%
b. 95%
d. 99.7%
Common Errors? Misconceptions?
2013 - Suggested Practice for SOL AII.11
At a company, the data set containing the ages of applicants for a particular job was
normally distributed. The mean age of the applicants was 30 years old, and the
standard deviation of the data set was 3.5 years. Which is closest to the percent of
applicants that were 21 years old or younger?
a. 0.5 %
b. 2.0 %
c. 2.57 %
d. 9.0 %
For additional assistance: See Technical Assistance Document for AII.11
Common Errors? Misconceptions?
2013 - Suggested Practice for SOL AII.11
A normally distributed data set of 500 values has a mean of 35 and a standard
deviation of 7. Which is closest to the probability that a value in the data set will fall
between 42 and 46?
a. 0.04
b. 0.10
c. 10
d. 50
For additional assistance: See Technical Assistance Document for AII.11
Common Errors? Misconceptions?
2013 - Suggested Practice for SOL AII.11
A normally distributed data set of 600 values has a mean of 18.5 and a standard
deviation of 3.25.
1. What is the approximate number of values in the data set expected to be 22 or
greater?
2. What is the approximate number of values in the data set expected to be 16 or
fewer?
3. Which is closest to the expected number of values in the data set that lie between
21 and 27?
a. 6
b. 22 c. 130 d. 467
Common Errors? Misconceptions?
2014 - Suggested Practice for SOL AII.11
Students need additional practice in recognizing the properties of a normal
distribution.
Which description of a normal distribution is most likely NOT true?
a)
b)
c)
d)
Approximately 99.7% of the data will fall within three standard deviations of the
mean.
Approximately 95% of the data will fall within two standard deviations of the
mean.
Approximately 68% of the data will fall within one standard deviation of the
mean.
Approximately 34% of the data will fall within one standard deviation of the
mean.
For additional assistance: See Technical Assistance Document for AII.11
Common Errors? Misconceptions?
2014 - Suggested Practice for SOL AII.11
Students need additional practice using properties of the normal distribution curve
to find the probability of an event, the percent of data that falls within a specified
interval, and the number of expected values that fall within a specified interval.
a. A normally distributed data set has a mean of 0 and a standard deviation of 0.75.
What percent of the data would be expected to be between -1.5 and 1.5?
b. The scores of a college history test were normally distributed with a mean of 75
and a standard deviation of 6. What is the probability of a student’s score being
an 80 or lower?
Common Errors? Misconceptions?
2014 - Suggested Practice for SOL AII.11
Bayside Elementary School is visiting a local amusement park. One of the
amusement park’s attractions requires that children must be at least 44 inches tall
to ride. The heights of children at Bayside Elementary are normally distributed with
a mean of 43 inches and a standard deviation of 3.4 inches. What is the probability
rounded to the nearest tenth that a child selected at random does NOT meet the
height requirement for the amusement park attraction?
Common Errors? Misconceptions?
2012 - Suggested Practice for SOL AII.12
Decide whether each of these can be answered using a
permutation or a combination and then determine the
answer.
a)
Twenty horses competed in a race. In how many
ways could the horses have finished in first place
through third place?
b)
A 10 person student council will be selected from 18
students at a school. How many possibilities are
there for this student council?
Common Errors? Misconceptions?