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Probability & Statistics Summer Packet
Summer 2016
Name:
Probability & Statistics
ACHS Summer 2015 Math Packet for Students Entering Probability & Statistics Page 2 of 16
Topic: Summation Notation http://www.mathsisfun.com/algebra/sigma-notation.html
Example 1:
Solution
3(1) + 3(2) + 3(3) +3(4) + 3(5) = 3 + 6 + 9 + 12 + 15 = 45
Example 2:
Solution
12 + 22 + 32 + 42 = 1 + 4 + 9 + 16 = 30
Directions: Evaluate each expression. Show work.
A. ∑=
−11
1)72(
k
k Answer: ______________
B. ∑
=
−9
1
2 )38(k
kk Answer: ______________
C. ∑
=
+−10
1
2 )824(k
kk Answer: ______________
D. Answer: ______________
ACHS Summer 2015 Math Packet for Students Entering Probability & Statistics Page 3 of 16
Topic: Slope of a Line (given two points) http://www.coolmath.com/algebra/08-lines/06-finding-slope-line-given-two-points-01
Slope Formula: m =
Example 1: Find the slope of the line that passes through (1, 2) and (3, 4).
Solution
2–4 –2= =1–3 –2 1
Example 2: Find the slope of the line that passes through (3, 6) and (1, 8).
Solution
8–6 2= =1–3 –2 –1
Directions: For the problems below, find the slope of the line between each of the two given points. Show work. Write your answer in simplest form.
A. (1, 5) and (7, 8) Answer: ______________ B. (–5, 9) and (5, 11) Answer: ______________ C. (12, 13) and (7, 13) Answer: ______________ D. (–8, 2) and (–8, 7) Answer: ______________
ACHS Summer 2015 Math Packet for Students Entering Probability & Statistics Page 4 of 16
Topic: Permutations https://www.mathsisfun.com/combinatorics/combinations-permutations.html
Permutation Formula (notation): nPr = n!(n –r)!
Example 1: Evaluate 3P7
Solution
7! 7! 7(6)(5)(4)(3)(2)(1)= = =(7–3)! 4! 4(3)(2)(1)
7(6)(5 ) = 210
Example 2: How many different ways can first, second, and third place be awarded to 10 people?
Solution
n = 10 people r = 3 (1st , 2nd , 3rd place)
10! 10! 10(9)(8)(7)(6)(5)(4)(3)(2)(1)= = =(10–3)! 7! 7(6)(5)(4)(3)(2)(1)
10(9)(8 ) = 72010x9x8 = 720
Directions: Evaluate the following expressions. Show work.
A. 12P3 Answer: _______________ B. 8P5 Answer: _______________ C. In how many ways can a president, a treasurer and a secretary be chosen from among
7 candidates? Answer: _______________
D. How many ways can 1st, 2nd, 3rd, and 4th place be awarded to 10 runners?
Answer: _______________
ACHS Summer 2015 Math Packet for Students Entering Probability & Statistics Page 5 of 16
Topic: Combinations https://www.mathsisfun.com/combinatorics/combinations-permutations.html
Combination Formula (notation): nCr = n!(n –r)!r!
Example 1: Evaluate 3C7
Solution
7! 7! 7(6)(5)(4!)= = =(7–3)!3! 4!3! 4!(3)(2)(1)
7(6)(5) 210= =3(2)(1) 6 35
Example 2: Five cousins at a family reunion decide that three of them will go to pick up a pizza. How many ways can they choose three people to go?
Solution
n = 5 r = 3
5! 5! 5(4)(3!) 5(4) 20= = = = =(5–3)!3! 2!3! 2(1)(3!) 2(1) 2 10
Directions: Evaluate the following expressions. Show work.
A. 12C3 Answer: _______________ B. 8C5 Answer: _______________ C. In how many ways can a president, a treasurer and a secretary be chosen from among
7 candidates? Answer: _______________
D. Problem: From a group of 40 people, a jury of 12 people is selected. In how many
different ways can a jury of 12 people be selected? Answer: _______________
ACHS Summer 2015 Math Packet for Students Entering Probability & Statistics Page 6 of 16
Topic: Venn Diagrams (2 pages) http://www.regentsprep.org/regents/math/algebra/AP2/LVenn.htm
Example 1: Out of forty students, 14 are taking English Composition and 29 are taking Chemistry. If five students are in both classes, how many students are in neither class?
Solution Two students are taking neither class.
Example 2: Out of forty students, 14 are taking English Composition and 29 are taking Chemistry. If five students are in both classes, how are in either class?
Solution
There are 38 students in at least one of the classes.
Directions: Create a Venn diagram to display the information then solve.
A. In a class of 87 students, 40 take Chorus, 53 take Band, and 16 take both Chorus and Band. How many students in the class are not enrolled in either Chorus or Band?
9 5 24
2 English Chemistry 9 5 24
2 English Chemistry
ACHS Summer 2015 Math Packet for Students Entering Probability & Statistics Page 7 of 16
B. In a school of 800 students, 185 students are taking Spanish, 160 students are on sports teams, and 72 students participate in both activities. How many students don’t participate in either activity?
C. A veterinarian surveys 46 of his patrons. He discovers that 17 have hamsters, 21 have
guinea pigs, and 19 have birds. Eight have hamsters and guinea pigs, and 5 people have a guinea pig and a bird. Seven have hamsters and a bird, and of these, 3 people have a guinea pig. How many patrons have none of these pets?
ACHS Summer 2015 Math Packet for Students Entering Probability & Statistics Page 8 of 16
Topic: Writing equations of lines (using slope & y-intercept) – 2 pages http://www.mathsisfun.com/equation_of_line.html
http://www.regentsprep.org/regents/math/algebra/AC1/EqLines.htm
Slope Intercept Form: y = mx + b
Example 1: Find the slope of the line that passes through (1, 2) and has a slope of 5.
Solution
Plug in the slope (m) and point (x, y), then solve for b:
y = mx + b 2 = 5(1) + b 2 = 5 + b –3 = b
Now take m = 5 and b = –3. Plug them in to write your equation in slope-intercept form:
y = 5x – 3
Example 2: Find the slope of the line that passes through (1, 4) and (3, 10).
Solution Find the slope first: 10–4 6= =33–1 2
Use the m = 3 and any point and plug it in to solve for b:
y = mx + b 4 = 3(1) + b 4 = 3 + b 1 = b
Now take m = 3 and b = 1 and write your equation in slope-intercept form:
y = 3x + 1
Directions: Write the equation of the line using the given information. Show work.
A. Passes through (2, 4); slope of 3 Answer: _______________
B. Passes through ( –5, 8); m = Answer: _______________
ACHS Summer 2015 Math Packet for Students Entering Probability & Statistics Page 9 of 16
C. Passes through (0, 0); m = –4 Answer: _______________
D. Passes through (5, 1) and (3, 0) Answer: _______________
E. Passes though (–2, 3) and (–2, –1) Answer: _______________
ACHS Summer 2015 Math Packet for Students Entering Probability & Statistics Page 10 of 16
Topic: Standard Deviation (2 pages) http://www.mathsisfun.com/data/standard-deviation.html
Standard Deviation Formula: 2( )x ms
NΣ= –
Example: What is the population standard deviation for the numbers: 75, 83, 96, 100, 121 and 125?
Solution 1. Find the mean:
Mean = (75 + 83 + 96 + 100 + 121 + 125) ÷ 6 = 600 ÷ 6 = 100 2. Find the variance:
To calculate the variance, subtract each number by the mean, square the results, add the results, then average the final result. (75 – 100)2 + (83 – 100)2 + (96 – 100)2 + (100 – 100)2 + (121 – 100)2 + (125 – 100)2= (–25)2 + (–17)2 + (–4)2 + (0)2 + (21)2 + (25)2 = 625 + 289 + 16 + 0 + 441 + 625 = 1,996
Variance = 1,996 ÷ 6 ≈ 332.66... 3. Find the standard deviation:
To calculate the standard deviation, take the square root of the variance. Standard deviation = 332.66... ≈ 18.2 rounded to 1 decimal place
Directions: Calculate the standard deviation for each problem. Show work.
A. Ten friends scored the following marks in their end-of-year math exam: 23%, 37%, 45%, 49%, 56%, 63%, 63%, 70%, 72% and 82%.
Answer: ______________
ACHS Summer 2015 Math Packet for Students Entering Probability & Statistics Page 11 of 16
B. A booklet has 12 pages with the following numbers of words: 271, 354, 296, 301, 333, 326, 285, 298, 327, 316, 287 and 314.
Answer: ______________
ACHS Summer 2015 Math Packet for Students Entering Probability & Statistics Page 12 of 16
Topic: Stem and Leaf Plots https://www.mathsisfun.com/data/stem-leaf-plots.html
Example: Create a Stem and Leaf plot for the following temperatures (in degrees). 63, 63, 70, 72, 74, 82, 86, 95, 101
Directions: Create a stem and leaf plot, with a key, for each set of data. A. B.
Data: 12, 13, 20, 21, 25, 25, 28, 34, 36, 39, 53, 54, 54, 54, 56, 62, 65, 66, 66, 67, 68, 80, 83, 85, 98
Data: 45, 10, 79, 33, 15, 30, 26, 49, 53, 11, 28, 54, 42, 77, 33, 11, 36, 84, 58, 27, 47, 21, 43, 31, 19, 37, 45, 23, 71,33
ACHS Summer 2015 Math Packet for Students Entering Probability & Statistics Page 13 of 16
Topic: Measures of Central Tendency http://www.mathsisfun.com/data/central-measures.html
Directions: Find the mean, median, mode, and range for each. Show work.
A. 11, 10, 12, 12, 9, 10, 14, 12, 9 Mean: _______________
Median: _____________ Mode: _______________ Range: ______________
B.
Mean: _______________
Median: _____________ Mode: _______________ Range: ______________
Stem Leaf 1 0 3 6 2 1 6 7 8 3 5 5 6 4 1 1 5 6 9 5 0 3 6 8
ACHS Summer 2015 Math Packet for Students Entering Probability & Statistics Page 14 of 16
Topics: Frequency Tables and Histograms http://openhighschoolcourses.org/mod/book/view.php?id=206&chapterid=353
Example: The following numbers were rolled on a die: 1, 2, 3, 1, 2, 6, 2, 5, 1, 2, 6, 3, 2 Create a frequency table for the data.
Solution
Number Rolled Tally Frequency
1 lll 3 2 llll 5 3 ll 2 5 l 1 6 ll 2
Example: Create a histogram of the frequency table.
Solution
Interval Frequency 4-8 8
9-13 3 14-18 10 19-23 5
A. The scores on a mathematics test were 70, 55, 61, 80, 85, 72, 65, 40, 74, 68, and 84.
Complete the accompanying table, and use the table to construct a frequency histogram for these scores.
Score Tally Frequency 40-49 50-59 60-69 70-79 80-89
ACHS Summer 2015 Math Packet for Students Entering Probability & Statistics Page 15 of 16
Topics: Mean, Median, and Mode of Histograms http://passyworldofmathematics.com/mean-median-mode-for-grouped-data/
Example: Find the mean, median, and mode of the histogram.
Solution
Mean: Find the midpoint of the heights of each bar.
Mean = 2(137.5)+5(142.5)+6(147.5)+ 8(152.5)+3(157.5)+1(162.5)2+5+6 + 8 +3+1
= 3727.525
= 149.1
Median: Find the middle. There are 25 entries. The 13th entry will be the median. This is at 147.5 (or between 145 – 150)
Mode: Find the most. There are 8 entries for 152.5 (or between 150 – 155)
Directions: Find the mean, median, and mode of the histogram. Show work.
Mean = __________ Median = _________ Mode = ___________
137.5 142.5 147.5 152.5 157.5 162.5
ACHS Summer 2015 Math Packet for Students Entering Probability & Statistics Page 16 of 16
Topics: Quartiles and Box-and-Whisker Plots http://www.regentsprep.org/regents/math/algebra/AD3/boxwhisk.htm
Example: Find the quartiles for the numbers 5, 8, 4, 4, 6, 3, 8
Solution 1. Put the numbers in order: 3, 4, 4, 5, 6, 8, 8 2. Cut the list into quarters:
3. Identify the quartiles:
• Quartile 1 (Q1) = 4 • Quartile 2 (Q2), which is also the Median, = 5 • Quartile 3 (Q3) = 8
Directions: Find the quartiles (Q1, Q2, Q3) for each.
A. 8, 11, 20, 10, 2, 17, 15, 5, 16, 15, 25, 6 Q1 = __________, Q2 = __________, Q3 = __________
B. 13, 18, 6, 20, 25, 11, 9, 18, 3, 30, 16, 9, 8, 23, 26, 17 Q1 = __________, Q2 = __________, Q3 = __________
C. 4, 5, 6, 8, 9, 11, 13, 16, 16, 18, 20, 21, 25, 30, 31, 33, 36, 37, 40, 41
Q1 = __________, Q2 = __________, Q3 = __________
D.
Q1 = __________, Q2 = __________, Q3 = __________