C.H. Flowers High School Mathematics DepartmentAP Statistics Summer preview Packet

2010 - 2011

Are You Ready for AP Statistics.

The fh-st unit of AP Statistics begins with the basics of descriptive statistics. Many of these concepts have beencovered in previous math and science courses. Over the summer you will review these topics and complete thispacket. Writing and reading comprehensiun are important components to the course. In this packet are three tasksthat require you to justify your reasons. Listed below are some Internet sites that contain information aboutdescriptive statistics.

]%e following packet is due at the first meeting of AP Statistics. This will be graded as several homework andassessment (quiz) grades.

Packet consists of the following:

II.III.IV.V.VI.

Measuring Central TendencyOrganizing DataConstructing Stem-and-Leaf Plots and Histoga’amsUsing the TI-83 CalculatorTasks 1, 2, and 3Combinations and Permutation

VII. S.12.4 Worksheet - Determining standaxd deviationVIII. S.12.7 Worksheet - Normal Distribution

IX. Preliminary Outline (Read pp.3 - 30)

NOTE: Prepare for a quiz on the Preliminary Chapter on the 1st day of class.

X. Chapter 1 Outlines (Read pp.37 - 100)

NOTE: go k~!~:/[ on howto use calculators (TI83+, 84+ & TI 89) in Statistics

Student Name:

Date Received:

Mr. B. OIowoyo, AP Statistics TeacherDepartment of Mathematics

Due the first day of class. NO EXCEPTIONS!!!!

ARE YOU READY FOR AP STATISTICS?

The first unit of AP Statistics begins with the basics ofdescriptive statistics. Many of these concepts have beencovered in previous math and science courses. Over the summeryou will review these topics and complete this packet. Writing andreading comprehension are important components to the courselIn this packet are three tasks that require you to justify yourreasons, Listed below are some Internet sites that containinformation about descriptive statistics.

The following packet is due at the first meeting of AP Statistics.This will be graded as the first homework assignment. There willtwo parts to the grade:

1. consideration of correctness of answer.2. completeness of work.

There will be a quiz on this materiall

I. Measuring Central TendencyFind the mean, median, and mode of the following collection.

15, 11, 19, 15, 14, 13, 17, 11, 12, 17, 15, 14, 15To begin, order the fourteen numbers.

11, 11, 12, 13, 14, 14, 14, 15, 15, 15, 17, 17, 19To find the mean, divide the sum of the numbers by 14.

mean= 2(11) +12+ 13+ 3(!4) +4(15) +2(17)+ 19 ~14A14

The mode is 15 because that is the number thatoccurs the most frequently.

The median is the average of thetwo middle numbers for even datalist~ Median=14+15 14.5

2

For odd data list, the median isthe middle number.

b. Find the quartiles of the collection in Example a. Then, sketch a box-and-whisker plot of thedata 14.5

11,11,12,13,14,14,14,15,15,15 15,17,17,19 ~Lower hal~ Upper half

The first quartile is 13 (the median of the lower half) ¯The second quartile is 14.5 (the median) 11The third quartile is 15 (the median of the upper half). 13 15A box-and-whisker plot for the data labels the endpoints of thedata and marks the quartilesl It is shown at the right.

The range is the difference of the highest and lowest data points. That is, the range is !9-11 = 8.

Exercises11 Find the mean, median, mode, and range of the following collection of scores on a test.

32, 72, 81, 95, 98, 58, 77, 75, 83, 97, 45, 89, 93, 57,82, 97, 52, 75, 79, 78, 99, 98, 54, 75, 85, 61, 55, 86

2. Find the first, second, and third quartiles of the collection of data in Exercise 1.3. Construc? a box-and-whisker plot of the collection of data in Exercise 1.4. Complete #1-3 for the following set of data. The weights (in pounds) of eleven children are asfollows: 39, 52, 40, 45, 46, 55, 48, 40, 43, 47, 44

II. Organizing DataAt a car dealership, the number of new cars sold in a week by each salesperson waS aS follows:

5,8,2,0,2,4,7,4,1,1,2,2,0,1,2,0,1,3,3,2.

a. Construct a frequency distributionfor this data.

b. Construct a line plot for this data

Number

6

32

0

Tall), Frequency XI 1o X X

t 1 X X XII zI! z X X X X XIIII I ~ X X X X X X X X

c. Construct a bar graph that shows the numberof salespeople who sold 0-8 cars.

N~tmber o~ ~ars Sold

Exercises1. Twenty-eight students in a class were asked how many cars their family owned. The resultswere as follows:

2,2,3,2ol, 2,2,4,3,2,0,1,0,1,1,2,2,3,2,3,3,5,1,1,3,0,1,2Construct a frequency distribution and a line plot for this datal

2. Each of the members of a recent high school graduatingclass was asked to name his/her favorite among thesesubjects: English, foreign language, history, mathematics,science. The results are shown in the table. Construct a bargraph that shows these results.

English 62Foreign Language 4OHistory 4OMathematics 18Science 33

III. Constructing Stem-and-Leaf Plots andHistograms

I. Construct a stem-and-leaf plot for the dataUnordered Data ste~

63, 52, 84, 83,23

51, 32, 58, 35, 445, 41, 65, 75, 559, 67, 25, 46 6

78

~-and-leaf Plot5

1561289357534

A stem-and-leaf plot orders data in increasing or decreasing order.

II. Histograms F1. Construct and label a horizontal number line rthat is scaled to contain all of the values of the q3variable of interest, u2. Construct and label a vertical axis so that the e

ngreatest frequency can be represented, c3. Construct the bars of equal width that are ~centered above each value. The heights of the ~bars represent the frequencies of the values. 25 35 45 55 65

Numbers75 85

Exercises1. Construct a stem-and-leaf plot for the data

15, 59, 66, 42, 48, 23, 70, 81, 35, 51, 68, 29, 77, 92,85, 16, 37, 59, 61, 76, 40, 25, 86, 11

2. Construct a histogram for the above data.

IV. Using the TI-83 Calculator1. Entering data into a list: STAT-~ 1 :Edit-* enter data in appropriate list2. To clear a list: highlight the list name and hit clear. The list is empty when you see thedashes.3. To sort a list in ascending order: STAT-~ 2:SortA(L1) ......4. To find the mean of a list: 2"d STAT-~ MATH -~3:mean(L1) ~Nam~o~5. To find the median of a list: 2nd STAT-~ MATH ~4:median(L1)6. To find the sum of a list: 2na STAT-~ MATH

Exercises1. Enter the data from Part I Exercise 1 in a list.21 Sort the data in ascending order and find the mode.3. Find the mean of the data.4. Find the median of the data.5. Find the sum of the data.

Task 1: Write two specific survey questions that you would ask voters in the next senatorialelection in your state. Choose the type of question and response (yes/no, scale of 1 to 5,numerical responses, etc.) that would be most appropriate for the issues involved. Whatrelationships would be expected when the responses are analyzed?

Task 2: Find a set of numbers that will satisfy the following conditions:¯ The median of a set of 20 numbers is 24.¯ The range is 42.¯ To the nearest whole number the mean is 24.¯ No more than three numbers are the same.

Show your strategy.

Task 3: Two pain relievers, A and B, are being compared for relief of postsurgical pain.Twenty different strengths (doses in milligrams) of each drug were tested. Eight hundredpostsurgical patients were randomly divided into 40 different groups. Twenty groups weregiven drug A. Each group was given a different strength. Similarly, the other twentygroups were given different strengths of drug B. Strengths used ranged from 210 to 400milligrams. Thirty minutes after receiving the drug, each patient was asked to describe hisor her pain relief on a scale of 0 (no decreaSe in pain) to 100 (pain totally gone).

The strength of the drug given in milligrams and the average pain rating for each group areshown in the scatterplot below. Drug A is indicated with A’s and drug B with B’s.

(a) BaSed on the scatterplot, describe the effect of drug A and how it is related tostrength in milligrams.(b) Based on the scatterplot, describe the effect of drug B and how it is related tostrength in milligrams.(c) Which drug would you give and at what strength, if the goal is to get pain relief of atleast 50 at the lowest possible strength? Justify your answer based on the scatterplot.

VI. Combinations and PermutationsThis a review topic from Algebra II that we will use in probability.

An Important Countinq Principle The computation of theoretical probabilities is based uponan important counting principle called logical multiplication.¯ If a person has 3 different sweatshirts and 2 different pairs of jeans, then there are 3

x 2 = 6 possible outfits.¯ If there is a family of 6 children, and assuming both genders are equally likely at birth,

how many different gender arrangements are there? For each child, there are 2possibilities (B or G) and each birth is independent of the others, therefore 2~ = 64possible arrangements.

How many ways are there of arranging 5 children to stand in a line? Far the first space,there are 5 choices, for the next space, 4 choices, and so on until the last space whenthere is only one choice, the last child~ The total number of ways is

5 x 4 x 3 x 2 x 1 = 5! = 120. To calculate this on a TI83 press 5, go to MATH, scrollright to PROB, then down to #4 !

Permutations have ORDER.¯ In how many ways can we pick a 4 letter word from the word MATH? As explained

above, the answer would be 4! = 24 ways.¯ In how many ways can we pick a 7 letter word from the letters ENGLAND? If all the

letters were different, there would be 7! ways. However, there are 2 N’s and so there

would be half as many ways. Similarly, there are ~ ways are getting a 9

letter word from the letters JEFFERSON.¯ In how many ways can we arrange 4 letters from a total of 6 letters? Using the

6! 6!counting principle, we can see that it would be 6 x 5 x 4 x 3 2! (6- 4)! In general,

the number of different permutations of n items taken r items at a time is denoted by

"Pr =

¯ To evaluate 1°P3 (arranging 3 items from 10), press 10 MATH PROB ,Pr 3

Combinations ORDER DOES NOT MATTER¯ There is only 1 way to choose 3 letters from C A T because CAT is considered the same

choice as TAC. So, there are FEWER ways of arranging items than when order matters.¯ For example from the set of 5 elements, {a,b,c,d,e} there are 10 ways of choosing 3

letters: {abc} {abd} {abe} {acd} {ace} {ade} {bcd} {bce} {bde} {cde}¯ In general, the number of ways that r items can be chosen from n elements is

r!(n- r)!To evaluate ~C3 (choosing 3 items from 5), press 5 MATH PROB ,Cr 3

Counting Rule

1. If a sandwich shop has 3 different types of meat, 4 types of bread, and 3 different type ofcheese. How many types of sandwiches can you create if you must have meat, bread andcheese on each?

2. How many ways can you hang 3 pictures tn a row on a wall?

3, HOW many ways can you visit 5 exclusive shops when you are looking for the perfect presentfor Mrs. Gustafson?

4. If a person has 4 pairs of shoes and 6 pairs of socks, then how many shoe-sock combos arepossible?

5. If there is a family of four, how many different gender arrangements are there?

6. If you are taking a multiple choice test (a, b, c, & d) consisting of 10 questions, then howmany different arrangements of answers are there?

Permutations: Order matter; think "president"; key word: arrange

7. How many ways can we pick a 6 letter "word" from the word HOKIES?

8. How many ways can we pick a 10 letter "word" from the word STATISTICS?

9. How many ways can we pick 3 different officers from a club of 20 members?

Combinations: Order doesn’t matter; think "committee"; key word: select

11. How many ways can we select a committee of 3 people from a club of 20 members?

12. How many ways can we pick 4 winners for 8 prizes if each prize is the same?

P. Preliminary Chapter "What is Statistics?"

Define the following key vocabulary

double blind (also seepage 370)

population

sample

voluntary response sampling (also seepage 331)

census

observational study

experiment

individuals

variable

categorical variable

quantitative variable

distribution

mode

mean

lurking variables (alsoseepage239)

probability (also seepage 392, ¶ l, §2)

variation, varia bility, chance variafion (check a dictionary, also see page 575)

statistical inference

Calculator Tricks

mean

Read pp. 3 - 25; and 28 - 30

X ): 1-varStats or 1-varStats L1

1-VaP Stats|

see page 59, step 1 - to enter the datasee page 82 step 4 - to calculate the meannote: calculator defaults to list 1 unless you say otherwise; so if data is in L1, youdon’t need to type L1. If you put data in a different list, you must tell thecalculator which list you want it to analyze.

Exercises

pg 26 "Case Closed". Complete on a separate sheet of notebook paper. Please writeneatly. Justifications and explanations count more than a correct numerical answer.

I .I Displaying Distributions with Graphs. Read pp. 37 - 63, and 67

Define the following key vocabulary

review ~ese ~rms

categorical variable

quantitative variable

roundoff error

Questions for you to answer will appear with a Q:Q: What types of graphs are appropriate to use with categorical variables?

Q: When can you not use a pie chart?

Q: What types of graphs are appropriate to use with quantitative variables? [abbr. given to help)

A: SP (orSLP) HBBSP RFHD RCFH ( or ? )

Q: Concerning Figure 1.3, What is the difference between graph (b) and (c)?

Q: How do you split stems? Another way to spit stems is shown in Figure 1.4; how was that done?

trimming ( or truncating )

class width

Q: Why are stemplots preferred over histograms?

frequency

frequency table

Q: What are the differences between a histogram and a bar graph?

Q: What 4 characteristics are used when examining (or describing) a distribution?

s c s and o

mode

unimodal, bimodal, trimodal, etc ....

continue on next pa.~e, olease ........

1.1 cont~ued

symmetric

skewed (seepage 126for a symmetric and right-skewed graph)

Q: Draw a sketch of a left-skewed graph.

outlier

constructing an ogive (relative cumulative frequency graph) - this will be gone over in class, butplease read it and see if it makes sense. Focus on identifying the percentile ,from the graph.

note: ogives can be done with bars or connect-the-dots

Calculator Tricks

histograms - see page 59. Try it! Follow the steps.

sorting data (to_ helpput yourWithdataSteminPl°tS’L1 especially!) ~ort..R (Ll) Done

- now from the home screen: [STAT] [2:SortA( ] [2nd] [1 ] [ ) ] [ENTER]- now go back and look al your list!

try this: to sod your presidential data from page 59 (it you stored it like the directions said to)-from the home screen:

r OPS HRTH ~or~.R( LPREZ)Bone[STAT] [2:SortA(] [2nd] [STAT][ 7 ] or where ever it is onyour talc[)] [ENTER]

- now to put the data back in L1 (if it’s not still there), so you can see it

[STAT] [ 1 :Edit ] arrow up to L1 [ ^],now pasle your list: [2°d] [STAT[ [7] [ENTER]

Recovering a lost list - if you ever delete a list by accident (or on purpose) and you want it back:

[~ r~ F~T~ 5etUPEditor. Done

Exercises

None. All the readings and calculator exercises are enough.

1.2 Describing Distributions with Numbers.

mean

median

resistant

Q: Explain: The mean is NOT resistant to outliers and the median is resistant to outliers.

Read pp. 69-96, and 99-100

formula to find location of median

Q: Which measure of cenler is more appropriate to use if data is symmetric? Skewed?

range

percentile

quartiles ( Q1, Q2 and Q3)

five number summery

boxplot

interquartile range

1.5(IQR) rule

modified box plot

variance

standard deviation

properties of the standard deviation

Q: How do you chose which numerical summary to use when describing distributions?

Effects of linear transformations ( Xnew = a + bxold )

1.2 continued

Calculator Tricks - see page 81

- 1-vats stats: will find mean, median, 5 number summary, standard deviation (Sx).- boxplots: ]2nd] [y=]; [zoom] [9:ZoomStat]; be sure to clear all graphs from [y=].

Exercises

to be determined - check with teacher

Name Class_ Date

Reteaching 12-4 Standard Deviation

OBJECTIVE: Determining standard deviation MATERIALS: None

The standard deviation of a collection of numbers (xl, x2, x3,..., Xn),

6, = ~/(xl ~)2 + (x2 ~)2 +n "’" + "’" + (xn ~)2’where~isthemeau°fxl’x2"’"xn""

Find the standard deviation for 100, 158, 170, 192.100 + 158 + 170 + 192

155(100 -- 155)2 = 3025(158 -- 155)2 = 9(170 - 155)2 =225(192 155)2 - 1369

~ = ~/3025+9 +255+ 13694

-.--- Find the mean.

Subtract the mean ~om each value inthe data set. Square each difference.

Find the standard deviation.

Find the standard deviation of each set of values.

1. 6.5 7.0 9.0 8.0 7.5 2. 5.6 5.8 5.9 6.1

3. 201 203 208 210 211 4. 12 14 15 17 19

A family goes grocery shopping every week. In a month the costs of thegroceries are $72.42, $91.50, $58.99, and $69.02.

5. What is the mean?

6. What is the standard deviation?

7. Within how many standard deviations of the mean is a cost of $50.00?

8. Within how many standard deviations of the mean is a cost of $102.00?

The distances driven by eight different vehicles using 12 gal of gasoline were174 mi, 271 mi, 208 mi, 196 mi, 340 mi, 214 mi, 236 mi, and 385 mi.

9. Find the mean and the standard deviation for the distances traveled.

10. How many items in the data set fall within one standard deviation ofthe mean? Within two standard deviations?

Algebra....................................................2 Chapter 12 [~s’s~°1"2"-~ °~e’t~t~i~; ....... []

Name Class + Date

Reteaching 12-7 Normal Distributions

OBJECTIVE: Using a normal curve to describe MATERIALS: Nonedata distribution

¯ The standard deviation tells how each data value in the set differs fromthe mean.

° Because normal curves contain the same probability distribution, theycan easily be used to make predictions on a set of data.

The length of life of a particular battery is normally distributed with the meanequal to 500 h. The standard deviation is equal to 50 h. Out of 250 batteriestested, find the number of batteries that are still working after 550 h.

Step 1: Sketch the normal curve. Label the mean 500. Label the standarddeviations using intervals of 50.

34% 34%13.5% 13.5%

350 400 450 500 550 600 650

Step 2: A battery that is still going after 550 h falls more than one standarddeviation above the mean. Calculate the percent of batteries at leastone standard deviation above the mean.

13.5% + 2.5% = 16% ~ Add the percentages that fall more that one standarddeviation above the mean.

0.16(250) = 40 ~ Find 16% of the 250 original batteries.

About 40 of the 250 batteries are still working after 550 h.

Sketch and label the normal curve for the following data. Make a predictionbased on the curve.

1. A fight bulb lasts an average of 219 h. Out of 1000 bulbs, how many willnot last 79 h if the standard deviation is 70 h?

In a math class of 26 students, a series of 100 multiplication problemscan be completed in a mean time of 4 min. The standard deviation is1 rain. How many math students will still be working after 5 min?

3. A group of 71 frogs had a mean hopping distance of 66 in. and astandard deviation of 3 in. How many flogs will hop more than 72 in.?