cfacteam.weebly.comcfacteam.weebly.com/uploads/2/6/8/0/26807326/probability... · web viewthe...

1

Benchmark MACC.912.S-CP.2.6

Reporting Category Statistics and Probability

Standard Conditional Probability & the Rules of Probability


Benchmark Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model.

Item Types This benchmark will be assessed using MC and GR items.

Benchmark Clarification

Find the conditional probability of an event given frequency data by identifying the possible outcomes and determining their respective probabilities.

Content Limits Limit to real world contexts and age appropriate situations.

Stimulus Attribute Tables and charts may be used.Items may be set in either real-world or mathematical contexts.

Response Attributes

Gridded response items will have fractional answers.The odds of an event must be expressed as a fraction.Fractional answers must be in simplified form.

2

SAMPLE ITEM

The state results for the FCAT 2.0 are shown in the table below. A student is considered proficient if he obtains an Achievement Level of 3 or higher. What is the probability that an 8th grade student was proficient on the math test in 2012? What are the odds that the student was NOT proficient?

FCAT 2.0 Mathematics – Next Generation Sunshine State Standards

Grade Year

Number of

Students

FCAT 2.0 Mean

Developmental Scale

Score

Percentage of Students by

Achievement Level1 2 3 4 5

3 2011

202,719

201 19

25

31

16 9

2012

203,207

202 18

24

30

18 10

4 2011

198,969

214 19

23

28

20 10

4 2012

193,802

215 18

22

27

20 12

5 2011

198,520

221 19

25

28

18 10

2012

199,844

222 19

24

27

18 11

6 2011

197,668

227 22

24

26

18 9

6 2012

199,076

227 23

25

25

18 10

7 2011

194,484

236 20

24

28

18 10

2012

198,277

236 20

24

27

18 10

8 2011

195,479

243 22

22

30

16 10

8 2012

194,346

243 22

21

30

16 11

Answer, part 1:

3

30/100 + 16/100 + 11/100 = 57/100 or .57

Answer, part 2:

22/100 + 21/100 = 43/100

43/100 : 57/100 or 43 : 57


Reporting Category

Statistics and Probability


Benchmark Number

MACC.912.S-CP.2.7

Benchmark Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.

Also Assesses or Assessed by

MACC.912.S-CP.1.2

Item Types This benchmark will be assessed using MC items.Content Limits

Limit to real world contexts and age appropriate situations.


Students will determine probable outcomes of events using the addition rule.

Stimulus Attributes

Test items may include illustrations of the following: tables, lists, tree diagrams, and other forms.

Response Attributes

Responses may be in decimal or reduced fraction form.

Sample Item If P(A) = 0.24 and P(B) = 0.52 and A and B are independent, what is P(A or B)?

(a) 0.1248(b) 0.28

4

* (c) 0.6352(d) 0.76(e) The answer cannot be determined from the information given.

Solution: 0.24 + 0.52 – (0.24)(0.52)


Reporting Category


Domain Conditional Probability & the Rules of Probability

Benchmark Number

MACC.912.S-CP.2.8

Benchmark Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.


MACC.912-CP.2.6/2.7, MACC.K12.MP.7.1

Item Types This benchmark will be assessed using MC, GR, and FR items.Content Limits Items may include multiple events or repeats of one event.


Students will determine probabilities of single or multiple events using probability addition and multiplication formulas.

Stimulus Attributes

Items may be set in either real world or mathematical contexts. Items may be a single event, multiple events, or a combination.

Response Attributes

Responses will be in fraction or decimal form and should be reduced to simplest form. Responses may be comprised of more than one answer. (find probabilities of more than one item in a question)

Sample Item Questions 1 and 2 relate to the following: An event A will occur with probability 0.5. An event B will occur with probability 0.6. The probability

5

that both A and B will occur is 0.1.

1. The conditional probability of A, given B (a) is 0.5.(b) is 0.3.(c) is 0.2.

* (d) is 1/6.(e) cannot be determined from the information given.

2. We may conclude that(a) events A and B are independent.(b) events A and B are disjoint.

* (c) either A or B always occurs.(d) events A and B are complementary.(e) none of the above is correct.

3.SAMPLE ITEM

The Saffir-Simpson Hurricane Scale labels hurricanes with a 1-5 rating based on their intensity. The overall probability that a hurricane is rated 1-4 is 0.985. The probability of a hurricane hitting Florida is 0.32. The probability that a hurricane is rated a 5 and misses Florida is 0.01. Given this information, create and complete a two-way frequency table and calculate the probability that a hurricane hits Florida given that it has a category 5 rating.Answer:

Rating of 5

Rating of 1-4

Hits Florida 0.005 0.315 0.32Misses Florida 0.01 0.67 0.68

0.015 0.985 1

Defining A1 = hits FloridaA2 = misses FloridaB = has rating of 5

Then P(A1)P(B│A1)

P(A1│B) = ----------------------------------------- P(A1)P(B│A1) + P(A2)P(B│A2)

6

(.32)(.005/.32)P(A1│B) = ----------------------------------------- = .333

(.32)(.005/.32) + (.68)(.01/.68)Or

P(A1│B) = (.005/.015) = .333

Answer: The probability that a hurricane hits Florida given that it has a category 5 rating is .333


Reporting Category



Benchmark Number

MACC.912.S-CP.1.1

Benchmark Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).


MACC.K12.MP.4.1

Item Types This benchmark will be assessed using MC and FR items.

Content Limits



Students will determine all possible outcomes of events.

Stimulus Attributes


Response Attributes


7

Sample Item 3. What is the sample space of flipping a coin and rolling a die?

Answer: P(Coin and Dice ) = P(Coin U Dice) = {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}


Reporting Category



Benchmark Number

MACC.912.S-CP.1.2

Benchmark Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.


MACC.K12.MP.2.1


Content Limits



Students will determine probable outcomes of events using the multiplication rule.

Stimulus Attributes


Response Responses may be in decimal or reduced fraction form.

8

AttributesSample Item Toss two balanced coins. Let A = head on the first toss, and let B = both tosses

have the same outcome. Are events A and B independent? Explain your reasoning clearly.

Solution: Yes, A and B are independent since P(A and B) is (.5)(.5)-0=0.25 and P(A)P(B)= (.5)(.5)=0.25

9

SAMPLE ITEM 2 MC

Anthony attends a high school that has four periods in the school day. Last semester, his friend took the same classes and recorded how often the teachers were absent. The data are shown in the table below. There were 2 quarters in the semester and 45 teacher work days in each quarter. Based on this data, what is the probability that all four of Anthony’s teachers will be absent on any given day?

Period

Teacher Name Subject

1st Quarter

Absences

2nd Quarter

Absences

1Mrs.

Rodriguez Algebra 5 42 Ms. Williams English 1 23 Mr. Keene US History 0 10

4 Mrs. MedinaEarth

Science 3 1

A) 1/60750B) 26/45C) 26/90D) 1/56782

Answer:

(9/90)(3/90)(10/90)(4/90) = 1/60750

10


Reporting Category



Benchmark Number

MACC.912.S-CP.1.3

Benchmark Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.


MACC.K12.MP.7.1


Content Limits




Stimulus Attributes


Response Attributes


Sample Item Use the conditional probability formula to demonstrate how the probability of being dealt a queen given that the card is red, is the same as the probability of being dealt a queen.

Answer: P(queen|red)= P (queen)·P(red) / P (red)(4/52)·(26/52)/(26/52) = 4/52 = 1/13 P(queen) = 4/52 = 1/13

11


Reporting Category



Benchmark Number

MACC.912.S-CP.1.4

Benchmark Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.


MACC.K12.MP.2.1, MACC.K12.MP.4.1


Content Limits



Students will determine probable outcomes of events using tables, lists, or tree diagrams.

Stimulus Attributes


Response Attributes


Sample Item Here are the counts (in thousands) of earned degrees in the United States in a recent year, classified by level and by the sex of the degree recipient:

Bachelor’s Master’s Professional Doctorate TotalFemale 616 194 30 16 856Male 529 171 44 26 770Total 1145 365 74 42 1626

1. If you choose a degree recipient at random, what is the probability that the person you choose is a woman?

12

Answer : 856/1626

2. What is the conditional probability that you choose a woman, given that the person chosen received a professional degree?

Answer : 30/74

3. Are the events “choose a woman” and “choose a professional degree recipient” independent? How do you know?

Answer: No if they were independent then P(woman) = P(woman|professional degree), the answers to #1 and #2 would be the same.

13


Reporting Category



Benchmark Number

MACC.912.S-CP.1.5

Benchmark Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.


MACC.912.S-CP.1.3 & MACC.912.S-CP.1.4


Content Limits




Stimulus Attributes


Response Attributes


Sample Item Researchers are interested in the relationship between cigarette smoking and lung cancer. Suppose an adult male is randomly selected from a particular population. Assume that the following table shows some probabilities involving the compound event that the individual does or does not smoke and the person is or is not diagnosed with cancer:

Event Probability smokes and gets cancer 0.08smokes and does not get cancer 0.17does not smoke and gets cancer 0.04does not smoke and does not get cancer 0.71

Suppose further that the probability that the randomly selected individual is a smoker is 0.25.

14

(a) Find the probability that the individual gets cancer, given that he is a smoker.

Answer : 0.08/0.25 = 0.32

(b) Find the probability that the individual does not get cancer, given that he is a smoker.

Answer : 0.17/0.25 = 0.68

(c) Find the probability that the individual gets cancer, given that he does not smoke.

Answer : 0.04/0.75 = 0.053

(d) Find the probability that the individual does not get cancer, given that he does not smoke.

Answer : 0.71/0.75 = 0.947

15


Reporting Category



Benchmark Number

MACC.912.S-CP.2.9

Benchmark Use permutations and combinations to compute probabilities of compound events and solve problems.



Item Types This benchmark will be assessed using MC and FR items.Content Limits

Items shall be limited to no more than five different sets. Limit to real world contexts and age appropriate situations.


Students will determine probable outcomes of events using the permutation and combination formulas.

Stimulus Attributes


Response Attributes

MC items of permutations should include answers using combination formula.

Sample Item 1. Students are celebrating their academic success with ice cream sundaes. There are 4 toppings available for their ice cream. Students may choose 2 out of the 4 toppings for their ice cream. This includes an option for a double helping of one topping. How many two-topping combinations can students choose?

Answer : 4C2 + 4= 6 + 4 =10

2. The school is going to hold an Algebra competition. Each teacher is to randomly select 2 students to represent their class. If Ms. Brown’s class has 10 boys and 14 girls, what is the probability that one boy and one girl will be chosen to represent her class?

Answer : (10C1 · 14C1) / 24C2 = (10· 14) / 276 = 35/69

16

Benchmark MACC.912.S-IC.1.1

Reporting Category

Probability and Statistics

Standard Making Inferences & Justifying ConclusionsBenchmark Number

MACC.912.S-IC.1.1

Benchmark Understand statistics as a process for making inferences about population parameters based on a random sample from that population.

BenchmarkClarification

Students will calculate, summarize, and interpret data using measures of center including mean and median and measures of spread including range, standard deviation, and variance. Students will use these measures to make comparisons and draw conclusions about data sets.



Item Types This benchmark will be assessed using FR, GR, and MC items.

Content Limits Items may include calculations for mean, median, variance, and standard deviation. Items may include the choice of which measure of center is the best representation of a data set.


Students will calculate, summarize, and interpret data using measures of center and spread.

Stimulus Attributes

Items must be set in mathematical or real-world context.

Response Attributes

Responses will be rational numbers.

Sample Items Sample Item 1 MCThe average grade point average (GPA) for 25 top-ranked students at a local college is listed below. (Use the rounding rule for the mean).3.80 3.77 3.70 3.74 3.70 3.86 3.76 3.68 3.67 3.573.83 3.70 3.80 3.73 3.673.78 3.75 3.73 3.65 3.663.75 3.64 3.78 3.73 3.64

17

Find the mean of the GPAs. Sample Response 1A. 3.7236B. 3.764 *C. 3.724D. 3.887Sample Response 1 C

Item Context: Mathematics (Statistics)

Sample Item 2 MCThe average grade point average (GPA) for 25 top-ranked students at a local college is listed below. (Use the rounding rule for the mean).3.80 3.77 3.70 3.74 3.70 3.86 3.76 3.68 3.67 3.573.83 3.70 3.80 3.73 3.673.78 3.75 3.73 3.65 3.663.75 3.64 3.78 3.73 3.64Find the median of the GPAs. Sample Response 2A. 3.7*B. 3.73 C. 3.74D. 3.75Sample Response 2 B


18

Sample Item Sample Item 3 FRSummer income for 14 high school students is listed below:1,208 2,4001,337 3,0201,055 456 810 1,7651,208 9872,343 1,654 734 1,356

Find the standard deviation. (Use the rounding rule for standard deviation).

Sample Response 3 719.2Item Context Mathematics (Statistics)

Sample Item 4 FRSummer income for 14 high school students is listed below:1,208 2,4001,337 3,0201,055 456 810 1,7651,208 9872,343 1,654 734 1,356

Find the variance.

Sample Response 4 517249Item Context Mathematics (Statistics)

7 1 9 . 2

5 1 7 2 4 9

19


* Teacher may access in the classroom.

Reporting Category



MACC.912.S-IC.1.2

Benchmark Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model?


LACC.9.10.WHST.1.1, LACC.9.10.WHST.2.4

Item Types This benchmark will be assessed using FR or MC items.

Content Limits Items may include calculations for mean, median, variance, and standard deviation.


Students will calculate, summarize, and interpret data using measures of center and spread. Students will use these measures to make comparisons and draw conclusions about statistics and probability models.

Stimulus Attributes


Response Attributes

Responses may be written understanding of inferences conditions.* Teacher may access in the classroom.

Sample Item

Suppose you have a population in which 60% of the people approve of gambling.

You want to take many samples of size 10 from this population to observe how the sample proportion who approve of gambling varies in repeated samples.

*1. Describe the design of a simulation using the partial random digits table below to estimate the sample proportion who approve of gambling. Then carry out five trials of your simulation.

20

6 0 0 9 1 9 3 6 5 1 5 4 1 2 3 9 6 3 8 8 5 4 5 3 4 6 8 1 6

3 8 4 4 8 4 8 7 8 9 1 8 3 3 8 2 4 6 9 7 3 9 3 6 4 4 2 0 0 6

8 2 7 3 9 5 7 8 9 0 2 0 8 0 7 4 7 5 1 1 8 1 6 7 6 5 5 3 0 0

6 0 9 4 0 7 2 0 2 4 1 7 8 6 8 2 4 9 4 3 6 1 7 9 0 9 0 6 5 6

6 8 4 1 7 3 5 0 1 3 1 5 5 2 9 7 2 7 6 5 8 5 0 8 9 5 7 0 6 7

*2. The sampling distribution is the distribution of all possible simple random samples of size 10 from this population. What is the mean of this distribution?

Answer: p = 0.6 is the mean of the distribution, to find the sample mean average the results from question 1.

Free Response Item

3. If you used samples of size 20 instead of size 10, which sampling distribution would give you a better estimate of the true proportion (of people who approve of gambling)? Explain your answer.

Answer: Samples of size 20 would give better estimates because of decreased variability.

21


Reporting Category



MACC.912.S-IC.2.3

Benchmark Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

Also Assesses MACC.K12.MP.3.1


Content Limits

Items may include research questions from surveys, observational studies, or experiments. Items may include sampling methods such as simple random, systematic, stratified, or cluster.


Students will use and compare surveys, experiments, and observational studies and decide which questions each is designed to answer. Student will use qualitative and quantitative measures to collect dataStudents will choose the type of sampling from a variety of situations as well as understand the reasoning behind random sampling. Students will identify sources of bias, including sampling and non-sampling errors.

Stimulus Attributes

Items must be set in real-world context.

Response Attributes

Answers need to be situations that are possible.

Sample Item 1. After a class receives the same lesson from the same teacher the class is divided randomly into two equal sized groups. One group is given a multiple choice test and the other group is given an essay test. Then the average of the test scores from each group are compared.

What is the technique for gathering data in this situation?

A. surveyB. observational study*C. experimentD. survey/experiment

22

2. Group businesses in Orlando, Fl according to type: medical, shipping, retail, manufacturing, financial, construction, restaurant, hotel, tourism, other. Then select a random sample of 10 businesses from each business type to survey.

What type of sampling is being used in this situation?

A. simple random samplingB. cluster samplingC. systematic sampling*D. stratified sampling

3. Students in grades 9-12 at an area high school were asked whether or not they should have tougher classes to help improve their testing performance on state assessments?

What form of bias was used in this situation?

A. type of questionB. language*C. sampleD. misleading

23


Reporting Category



MACC.912.S-IC.2.4

Benchmark Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.



Item Types This benchmark will be assessed using GR and MC items.

Content Limits Items may include confidence level and/or confidence interval. Items may include intervals for the mean when the population standard deviation is known and when the population standard deviation is not known. Items may include confidence intervals for proportions also. Items may include finding sample size needed for specific results.


Students will create, explain and interpret confidence intervals of a mean or proportion by understanding point estimate, confidence levels and sample size when the population standard deviation is known and when the population standard deviation is not known. Students will also interpret and apply the maximum error of the estimate ("margin of error").

Stimulus Attributes


Response Attributes

Answers will include confidence levels in percentages, confidence intervals with a mean or proportion estimate or sample sizes in whole numbers.

Sample Items1. When people smoke, the nicotine they absorb is converted to cotinine, which can be measured. A sample of 40 smokers has a mean cotinine level of 172.5. Assuming the population standard deviation is known to be 119.5, what is the 90% confidence interval estimate of the mean cotinine level of all smokers?

A. 74.64 < x < 164.36*B. 141.42 < x < 203.58C. 166.49 < x < 178.51D. 141.42 < x < 178.51

24

Gridded Response

2. The music industry recently has adjusted to the increasing number of songs being downloaded instead of being purchased on CDs. So, it has become important to estimate the proportion of songs being currently downloaded. If we want to be 95% confident that the sample percentage is within 1% point of the true population percentage of songs obtained by downloading, how many randomly selected song purchases must be surveyed to determine the percentage downloaded?Sample Response

9 6 0 4

25

Benchmarks: MACC.912.S-IC.2.5, & MACC.912.S-IC.2.6

Body of Knowledge


Standard Making Inferences & Justifying Conclusions

Benchmark Number

MACC.912.S-IC.2.5 & MACC.912.S-IC.2.6

Benchmark MACC.912.S-IC.2.5: Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.

MACC.912.S-IC.2.6: Evaluate reports based on data.

Also assesses or assessed by

LACC.910.WHST.1.1, LACC.910.WHST.2.4, LACC.910.WHST.3.9, and MACC.K12.MP.3.1, MACC.912.S-IC.1.1

Item Types This benchmark will be assessed using and MC, GR and FR items.


Students will understand, create and analyze the null hypothesis and the alternative hypothesis (basics of hypothesis testing). Students will understand the p-value is the probability of getting a sample statistic in the direction of the alternative hypothesis when the null hypothesis is true. Students will understand p-value rules for decisions to reject or not to reject the null hypothesis. Students will decide whether a Type I error (occurs when you reject the null hypothesis when it is true) and a Type II error (occurs if you do not reject the null hypothesis when it is false) have occurred in the outcome of the hypothesis testing.

Content Limits Items may include writing a null and alternative hypothesis for data or interpreting a given null and alternative hypothesis.Items may include stating the hypothesis, identifying the claim, computing the test value, finding the p-value, making a decision on given information and/or summarizing the results.

Stimulus Attribute


Response Attributes

Items may be writing the null and alternative hypothesis or analyzing the given null and alternative hypothesis. Answers may include the p-value and/or a decision to be made based on the p-value.

26

Sample Item 1 FR

The average undergraduate cost for tuition, fees, room, and board for all accredited universities was $26,025. A random sample in 2009 of 40 of these accredited universities indicated that the mean tuition, fees, room, and board for the sample was $27,690 with a population standard deviation of $5492. What is the null and alternative hypothesis for the conjecture?

A. H : µ= $26,025˳ and : µ > $26,025Ң B. H : µ < $26,025˳ and : µ = $26,025ҢC. H : µ =$26,025˳ and : µ < $26,025Ң

* D. H :µ = $26,025˳ and : µ ≠ $26,025Ң

Sample Item 2 FR

Using the statistics from sample item 1 and a 0.05 level of significance, what is the p-value of this sample?

Sample Response

Sample Item 3 MCThe average undergraduate cost for tuition, fees, room, and board for all accredited universities was $26,025. A random sample in 2009 of 40 of these accredited universities indicated that the mean tuition, fees, room, and board for the sample was $27,690 with a population standard deviation of $5492. At a 0.05 level of significance, which of these statements is the correct conclusion for the hypothesis test using the p-value?

* A. Reject the null hypothesis with p-value of 0.0274 with a significance level of 0.05.

B. Do not reject the null hypothesis with p-value of 0.0274 with a significance level of 0.05.

C. Except the null hypothesis with p-value of 0.0274 with a significance level of 0.05.

D. Not enough information to formulate a conclusion.

Item Context Probability & Statistics

0. 0 2 7 4

27

Sample Item 4 MC

The hypothesis of a sample gives the following conclusion for proportion of car crashes occurring less than a mile from home:

The proportion of car crashes occurring less than a mile from home is 0.23.

What are the Type I and Type II errors which could happen with these results?

A. Type I error: Fail to reject the claim that the proportion of car crashes occurring less than a mile from home is 0.23 when the proportion is actually different from 0.23.Type II error: Reject the claim that the proportion of car crashes occurring less than a mile from home is 0.23 when that proportion is actually 0.23.

*B. Type I error: Reject the claim that the proportion of car crashes occurring less than a mile from home is 0.23 when that proportion is actually 0.23. Type II error: Fail to reject the claim that the proportion of car crashes occurring less than a mile from home is 0.23 when the proportion is actually different from 0.23.

C. Type I error: Reject the claim that the proportion of car crashes occurring less than a mile from home is 0.23 when that proportion is actually different from 0.23.Type II error: Fail to reject the claim that the proportion of car crashes occurring less than a mile from home is 0.23 when the proportion is actually 0.23.

D. Type I error: Fail to reject the claim that the proportion of car crashes occurring less than a mile from home is 0.23 when the proportion is actually 0.23.Type II error: Reject the claim that the proportion of car crashes occurring less than a mile from home is 0.23 when that proportion is actually different from 0.23.

Item Context Probability & Statistics

28

Benchmark MACC.912.S-ID.1.1

Reporting Category


Standard Interpreting Categorical & Quantitative Data

Benchmark Number

MACC.912.S-ID.1.1

Benchmark Represent data with plots on the real number line (dot plots, histograms, and box plots).


MACC.K12.MP.4.1

Item Types This benchmark will be assessed using FR or MC items.

Content Limits

Items will include raw data.


Students will create a data distribution in the form of: dot plots, histograms, or box plots.

Stimulus Attributes

Data must be rational numbers

Response Attributes

Graphs must be appropriately labeled and draw to scale.

Sample Item Here are the season home run statistics of two major league baseball players I in increasing order. Draw parallel box plots to compare the two players.

Here are numbers for the first player:16 19 24 25 25 25 33 33 33 3434 3737 40 42 45 45 46 46 49 73

Here are the numbers for the second player:13 20 24 26 27 29 30 32 34 3438 3939 40 40 44 44 44 44 45 47

Draw parallel box plots for the two players:

1 *

2

HomeRuns 15 30 45 60 75

HRcodes

29


Reporting Category



Benchmark Number

MACC.912.S-ID.1.2

Benchmark Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) or five number summaries of two or more different data sets.


MACC.912.S-ID.1.1, MACC.K12.MP.5.1

Item Types This benchmark will be assessed using FR and MC items.

Content Limits

Items may include calculating measures of center and spread.


Students will calculate measures of center and spread of multiple data sets and compare the results.

Stimulus Attributes


Response Attributes

Answers may include mean, median, IQR, and standard deviation.

Sample Item Sample Item 1 MCThe average grade point average (GPA) for 25 top-ranked students at a local college is listed below. 3.80 3.77 3.70 3.74 3.70 3.86 3.76 3.68 3.67 3.573.83 3.70 3.80 3.73 3.673.78 3.75 3.73 3.65 3.663.75 3.64 3.78 3.73 3.64Find the five-number summary for the GPAs.Sample Response 1A. 3.64, 3.675, 3.73, 3.775, 3.86*B. 3.57, 3.67, 3.73, 3.775, 3.86 C. 3.64, 3.67, 3.73, 3.775, 3.80D. 3.57, 3.67, 3.73, 3.775, 3.9Sample Response 1 BItem Context: Mathematics (Statistics)

30

2. Use graphs and numerical summaries to describe how the following three data sets are similar and how they are different.

A: 5, 7, 9, 11, 13, 15, 17 B: 5, 6, 7, 11, 15, 16, 17 C: 5, 5, 5, 11, 17, 17, 17

Answer: Similar- Each set has the same mean of 11 and the same median of 11 all three sets are symmetric. All three sets have the same minimum value of 5 and maximum value of 17.

Different – They all have different standard deviations (spread)A. 4.32, B. 5.07, C. 6.0.

3. Sample Item MC

Which statement describes the histogram appropriately?

*A. The histogram is skewed to the right (positively skewed).B. The histogram is skewed to the left (negatively skewed).C. The histogram is a normal distribution.D. The histogram is bimodal.Item Context: Mathematics (Statistics)

31


Reporting Category



Benchmark Number

MACC.912.S-ID.1.3

Benchmark Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).


MACC.K12.MP.4.1, MACC.912.S-ID.1.2

Item Types This benchmark will be assessed using FR or MC items.Content Limits

Items may include identifying outliers from a set of data in a graph such as a histogram or box-and-whisker plot. Items may also include what effect adding a higher or lower value have on a particular mean, median, mode or range.


Students will identify outliers, the possible reasons for them and what effect they have on the mean, median, mode and range of data.

Stimulus Attributes


Response Attributes

Answers will include numerical outliers, reasons for outliers, or effect of specific outlier on mean, median, mode or range of data.

Sample Item Summer income for 14 high school students is listed below:1,208 2,4001,337 3,0201,055 456 810 1,7651,208 9872,343 1,654 734 1,356

Which of these statements is true?A. 456 is an outlier.

*B. 3020 is an outlier.C. 456 and 3020 are outliers.D. There are no outliers.

32


Reporting Category



Benchmark Number

MACC.912.S-ID.1.4

Benchmark Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.


MACC.K12.MP.6.1

Item Types This benchmark will be assessed using FR, GR, and MC items.

Content Limits

Items may include finding area under a curve between z-scores, finding z-scores with given probability or finding specific data values with given probability. Items may include students analyzing graphs with z-sores, data values or probabilities.


Students will identify the properties of a normal distribution, find area under the standard normal distribution, find probabilities of a variable under a normal distribution, and find specific data values for given percentages using the standard normal distribution.

Stimulus Attributes


Response Attributes

Items may be in decimal or percentage form. Items should specify whether the answer expected is in decimal or percentage form. Z-scores may be negative.

Sample Item Sample Item 1 FR

What is the probability (in decimal form) with the given information using the standard normal distribution?

P (-2.46 < z < 1.74)

Sample Response

0 . 9 5 2 1

33

Sample Item 2 MCFor men aged between 18 and 24 years, serum cholesterol levels (in mg/100mL) have a mean of 178.1 and a standard deviation of 40.7 (based in data from the National Health Survey). What is the z-score corresponding to a male, aged 18-24 years, who has a serum cholesterol level of 259.0 mg/100mL? Sample Response 2A. -1.99*B. 1.99C. 6.36D. 80.9Sample Response 2 BItem Context: Mathematics (Statistics)

Sample Item 3 FRWomen’s heights have a mean of 63.6 inches and a standard deviation of 2.5 inches. What is the z-score corresponding to a woman with a height of 70 inches?

Sample Response 3

Sample Response 2.56


2 . 5 6

34


Reporting Category



Benchmark Number

MACC.912.S-ID.2.5

Benchmark Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.


MACC.K12.MP.4.1, LACC.910WHST.3.9

Item Types This benchmark will be assessed using FR and MC items.

Content Limits

Items will be presented in two-way frequency tables.


Students will calculate and interpret frequencies found in two-way tables.

Stimulus Attributes


Response Attributes

Answers may be in decimal or percentage form. Answer may require written statements.

Sample Item In 2006, an electronic replay system debuted in both men’s and women’s professional tennis. Each player is allowed two unsuccessful challenges per set. Here are some data on the results of challenges made during the first few months of the new system.

Successful UnsuccessfulMen 201 288Women 126 224

1. Calculate the marginal distribution (in percents) for each of the two variables.Answer: Gender- Men 489/839 = 58.28%, Women 350/839 =41.72%Challenge results- Successful : 327/839 = 38.97%, Unsuccessful- 512/839 = 61.03% 2. Write a sentence describing what each marginal distribution tells you.A higher percentage of men make challenges than women. 61% of all challenges are unsuccessful.

35

3. Write a few sentences describing the relationship between gender and challenge results.Of the men who made challenges about 41% were successful. Only 36% of the women who made challenges were successful.

36

Benchmark MA.912.S-ID.2.6 & MACC.912.S-ID.3.7

Reporting Category


Standard MACC.912.S-ID.2.6 & MACC.912.S-ID.3.7Benchmark MACC.912.S-ID.2.6

1. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.

b. Informally assess the fit of a function by plotting and analyzing residuals.

c. Fit a linear function for a scatter plot that suggests a linear association.

MACC.912.S-ID.3.72. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

Also Asses MACC.K12.MP.4.1, MACC.K12.MP.5.1, MACC.K12.MP.8.1



Students will graph data on a scatter plot, describe the association, and interpret the slope.

Content LimitsData points limited to one, two, or three digits

Stimulus Attributes

Tables and charts may be used.

Items may be set in either real-world or mathematical contexts.

Response Attributes

Equations may be presented in standard or slope-intercept form.* Teacher may assess in the classroom.

37

SAMPLE ITEM

Mrs. Santiago would like to know how well her Algebra students will perform on the state’s End-of-Course Assessment. She has decided to gather some data from the previous year in order to determine a least squares regression line. She plans to use the students’ scores on her classroom test to predict how well they will do on the End-of-Course Assessment. Given the data in the table below

I. Graph the data on a scatter plot and describe how the variables are related.

Answer: *Teacher may assess in the classroom.

II. Find the equation of the least squares regression line.

III. Interpret the slope (rate of change) and the intercept (constant term) of this linear model in the context of the data.

Classroom Test Score

Algebra EOC Scale Score

90 45080 42575 42540 35080 43085 440

II.

A) y = .11x + 14.73B) y = 14.73x + .11C) y = 271.17x + 1.98D) y = 1.98x + 271.17

Answer:

38

D)

Solution:

(n∑xy)-(∑x∑y) (6 * 192,175) – (450 * 2520)m = ---------------------- = ---------------------------------------- =

1.98 (n∑x2)-(∑x)2 (6 * 35350) – (450)2

(∑y∑x2)-(∑x∑xy) (2520 * 35350) – (450 * 192,175)b = ------------------------ = ----------------------------------------------

= 271.17 (n-∑x2)-(∑x)2 (6 * 35350) – (450)2

y = mx + b = 1.98x + 271.17

III. Slope: The Algebra EOC Scale Score will rise 1.98 points for every one point increase on the classroom test score.

Y-Intercept: When the classroom test score is 0, the Algebra EOC score will be 271.17

39

Benchmark MACC.912.S-ID.3.8 & MACC.912.S-ID.3.9Reporting Category



Benchmark Number

MACC.912.S-ID.3.8

Benchmark MACC.912.S-ID.3.8: Compute (using technology) and interpret the correlation coefficient of a linear fit.

MACC.912.S-ID.3.9: Distinguish between correlation and causation.

Build on students’ work with linear relationships in eighth grade and introduce the correlation coefficient. The focus here is on the computation and interpretation of the correlation coefficient as a measure of how well the data fit the relationship. The important distinction between a statistical relationship and a cause-and-effect relationship arises in S.ID.9.


MACC.K12.MP.2.1

Item Types This benchmark will be assessed using FR, GR, or MC items.

Content Limits

Items may include calculating the correlation coefficient of a set of paired data and the interpretation of the relationship between the two variables.


Students will calculate the correlation coefficient of a set of paired data (x,y) to determine the strength and direction of the relationship between the variables. Student will also use scatter plots to aid their interpretation of the data.

Stimulus Attributes


Response Attributes

Answers will include the correlation coefficient and/or the interpretation of a given correlation coefficient.

Sample Item Sample Item 1 FRThe table below lists the numbers of murders and the population sizes (in hundreds of thousands) for large cities in America during a recent year.

What is the correlation coefficient of the data?

Murders 258

264

402

253

111

648

288

654

256

60

590

Population

4 6 9 6 3 29 15 38 20 6 81

40

Sample Response 1

Sample Response 1 0.727There is a strong positive relationship between the number of murders and the population size of large American cities.

0 . 7 2 7

41

Benchmark MACC.912.S-MD.1.1

*Teacher may assess in the classroom.


Standard Using Probability to Make Decisions

Benchmark Number MACC.912.S-MD.1.1

Benchmark Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.


MACC.K12.MP.4.1

Item Types This benchmark will be assessed using FR or MC items


Students will graph a random variable distribution on a scatter plot.

Content Limits Items may include probability distributions or raw data. Items may include discrete random variables, discrete uniform variables, binomial or exponential data.

Stimulus Attribute Items must be set in mathematical or real-world context.

Response Attributes Items may be in fraction or decimal form. *Teacher may assess in the classroom.

42

*Teacher may assess in the classroom.

Sample Item 1 FR

The probabilities that a randomly selected customer purchases 1, 2, 3, 4, or 5 items at a convenience store are 0.32, 0.12, 0.23, 0.18, and 0.15, respectively.

(a) Identify the random variable of interest. X = ________. Then construct a probability distribution (table), and draw a probability distribution histogram.

Solution:X= number of items purchased.

43




Benchmark Number MACC.912.S-MD.1.2

Benchmark Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.


MACC.K12.MP.5.1

Item Types This benchmark will be assessed using MC and GR items


Students will use the formula to calculate the expected value (mean) and variance of a random variable.



Response Attributes Items may be in fraction or decimal form.

Sample Item 1 MC

X 4 5 6 8 10P(X) 0.4 0.3 0.1 0.15 0.05

A) What is the mean of the Probability Distribution?

Solution: 4(0.4) + 5(0.3) +6(0.1) + 8(0.15) + 10(0.05) = 5.4

44

What is the variance of the Probability Distribution?

A. 0.35 B. 1.71

* C. 2.94D. 5.4

Solution: (4-5.4)²·(0.4) + (5-5.4)²·(0.3) + (6-5.4)²·(0.1) + (8-5.4)²·(0.15) + (10-5.4)²·(0.05) = 2.94

Item Context Probability and Statistics

45


*Teacher will assess in the classroom.



Benchmark Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes.


MACC.K12.MP.4.1



Students will use the formula to calculate the expected value (mean) of a random variable.




Sample Item *Teacher will assess in the classroom.

46


*Teacher will assess in the classroom.



Benchmark Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households?


MACC.K12.MP.4.1







Sample Item *Teacher will assess in the classroom.

47

Benchmark MACC.912.S-MD.2.5 & MACC.912.S-MD.2.6



Benchmark MACC.912.S-MD.2.5 : Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.

a. Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fast-food restaurant.

b. Evaluate and compare strategies on the basis of expected values. For example, compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident.

MACC.912.S-MD.2.6: Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).


MACC.K12.MP.4.1

Item Types This benchmark will be assessed using GR, FR or MC items.






Sample Item 11. A box contains ten $1 bills, five $2 bills, three $5 bills, one $10 bill, and one $100 bill. A person is charged $20 to select one bill.

a) Calculate the expected value and interpret this value in the context of this problem.b) Is the game fair?

Solution:

48

a) $1(10/20) + $2(5/20) + $5(3/20) +$10(1/20) +$100(1/20) = $7.25 is the expected amount a person would collect.b) No the game is not fair because the expected winnings are less than the amount charged to play.

49




Benchmark Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).


MACC.912.S-ID.1.4, MACC.K12.MP.1.1

Item Types This benchmark will be assessed using FR, GR, or MC itemsBenchmark Clarification

Students will analyze decisions and strategies using the probability formulas, and/or z-scores to solve real world statistics problems.

Content Limits Items may include finding z-scores, confidence intervals, or p-values from hypothesis test.

Stimulus Attribute Items must be set in mathematical or real-world context.Response Attributes Items may be in fraction or decimal form.

Sample Item Sample Item 1 GR

Engineers must consider the breadths of male heads when designing motorcycle helmets. Men have head breadths that are normally distributed with a mean of 6.0 inches and a standard deviation of 1.0 inch (survey data from Gordon Churchill). 100 men are randomly selected. What is the probability that the men have a mean head breadth less than 6.2?

Sample Response

0 . 9 7 7 2

cfacteam.weebly.comcfacteam.weebly.com/uploads/2/6/8/0/26807326/probability... · web viewthe...

Documents