ACHS Summer 2015 Math Packet for Students Entering Probability & Statistics Page 1 of 16

Probability & Statistics Summer Packet

Student Name: ______________________________________ Due Date: 9/7/2015

Summer Resource Contacts: kdaniel@roselleschools.org lvazquez@roselleschools.org

This packet is the FIRST TEST GRADE of Marking Period 1

Student Responsibilities

I will be able to improve my math performance by: • Completing the summer math packet • Reviewing math skills throughout the summer

___________________________________________ ______________ ___________________ Student Signature Current Grade Date Received

Parent Responsibilities

Parents will be able to ensure student success in math by: • Supporting the math goal of ACHS • Monitoring student completion of the summer packet

___________________________________________ ________________ ________________ Parent Signature Date Received Date Reviewed

GRADING 5 points per section for complete, correct answers with work where required.

Summation Notation = ______ pts. Standard Deviation = ______ pts. Slope of a Line = ______ pts. Stem and Leaf Plots = ______ pts. Permutations = ______ pts. Measures of Central Tendency = ______ pts. Combinations = ______ pts. Frequency Tables and Histograms = ______ pts. Venn Diagrams = ______ pts. Mean, Median, and Mode of Histograms = ______ pts. Writing equations of lines = ______ pts. Quartiles and Box-and-Whisker Plots = ______ pts.

TOTAL POINTS = __________ GRADE = ______%

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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(

kk Answer: ______________

B. ∑

=

−9

1

2 )38(k

kk Answer: ______________

C. ∑

=

+−10

1

2 )824(k

kk Answer: ______________

D. Answer: ______________

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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: ______________

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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: _______________

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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: _______________

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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

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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?

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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= =3

3–1 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: _______________

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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: _______________

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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: ______________

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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: ______________

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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

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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

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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

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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

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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 = __________