189746423-laura-hasley-statistics-chi-squared-goodness-of-fit-test-lesson-plan - Copy

Teacher Candidate: ALAIN PRESILLAS

Subject: Statistics

Central Focus: (Content of the lesson) Chi-square Goodness of

Fit Test

Grade Level(s): 8 TO 12 (RESEARCH SUBJECTS)

Date: FEB 6 AND 11

Course for which the Lesson is developed :

Classroom Teacher:

Time allotted: Two MEETINGS-110 minutes each day, Second

Meeting will be Activity Test

Theoretical Principles and/or Research–Based Best Practices in this Lesson

Why are the learning tasks for this lesson appropriate for your students? Cite references

The learning tasks for this lesson are appropriate for my students because it gives them repetition in learning how to perform the test at hand. Through repetition, students can learn the steps for the test and how to conduct it when appropriate. This aligns most with

Atkinson’s and Shiffrin’s information-processing theory. From page 37 of

Integrating Education Technology into Teaching , the theory states that “learning is encoding information into memory, similar to the way a computer stores information.” So, by encoding the steps of the chi-square goodness of fit test into my students’ minds, the process will always be there, ready to use whenever needed.

Common Errors, Developmental Approximations, Misconceptions, Partial Understandings, or Misunderstandings for this

Lesson

A common error students could have would be accidentally using the chi-square test of independence. The two tests are similar, but definitely have distinguishing factors of when to use each one. I would address this with the class by emphasizing what to look for when deciding to use the chi-square goodness of fit test and creating a chart as a guide to use.

Implemented August 2013

Adapted from Layzell, D., (2013). Lesson Plan Model. Illinois State University; Leland Stanford Junior University (2012) ed-Teacher Performance Assessment; Tomlinson, C. (2004) How to differentiate

in mixed ability classrooms; Worldclass Instructional Design and Assessment (2012) WIDA 2012 Amplified ELD Standards.

Standard(s)/Benchmark(s) to be met in the Lesson: (ILS, Common Core, or

Professional Learning Standards)



CC. 9-12. S. IC. 1 Understand and evaluate random processes underlying statistical experiments.

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



CC. 9-12. S. IC. 2 Understand and evaluate random processes underlying statistical experiments.

Decide if a specified model is consistent with results from a given data-generating process.



CC. 9-12. S. IC. 6 Make inferences and justly conclusions from sample surveys, experiments, and observational studies. Evaluate reports based on data.

Learning Objective(s):



My 12 th

grade Statistics students will be able to identify when to use a chi-square goodness of fit test.



My 12 th

grade Statistics students will be able to perform a chi-square goodness of fit test when appropriate.



My 12 th

grade Statistics students will be able to deduce what the results of the test mean.



My 12 th

grade Statistics students will be able to distinguish the difference between the chi-square goodness of fit test and the chisquare test for independence.

Assessment Tool(s) and Procedures:



On day two, a fun review game will

 be played to assess the students’ retention of the test steps.

After day two in class, the kids will be assigned homework that will assess their grasp of the test. They can either take a worksheet home and complete it or they can conduct a live chi-square goodness of fit test with their family, record the results, and bring those into class.

Procedures: List in sequence the actions taken by teacher and students throughout the lesson.

A.

Engage Students: (Diagnostic/Pre-Assessment may be included here.)

In order to gauge what the students already know about the chi-square goodness of fit test, I’ll create a KWL chart on the white board that we’ll fill in as a class. I’ll see if they can use their prior knowledge of the chi-square test for independence to infer what the chi-square goodness of fit test is used for.

B.

Communicate the Purpose of the Lesson to Students (Objectives and Assessment):

The purpose of the lesson today is to learn a new type of inference test. It’s called the chi-square goodness of fit test.

We’ll learn how and when to use it and what the results mean in the grand scheme of things.

C.

Instructional Sequence: a.

Day One

1.

When the students come in and get settled, I’ll ask, “Has anyone heard of the chi-square good of fit test before?”

Depending on how many people have or have not heard of the test, I’ll ask either “What do you know about the test?” (if




kids have heard of it before) or “What do you think the test is used for?” (if kids haven’t heard of the test before).

2.

After questioning the kids on about the test, I’ll create a KWL chart on the board so we as a class can write down what is known and what they’d like to know.

3.

After the KWL chart, I’ll give notes about the characteristics of chi-squared and about the distribution on the Elmo. a.

The notes would include: i.

Chi-square is always positive ii.

Chi-square measures a relative difference between observed and expected counts, rather than an absolute difference iii.

Larger values of chi-square indicate larger differences between observed and expected counts iv.

Larger chi-square values lead to smaller p-values v.

Chi-square distribution is skewed right vi.

Precise shape of the distribution depends on the degrees of freedom (as the degrees increase, the skew decreases)

4.

I’ll project this scenario through the Elmo: Acme Toy Company prints baseball cards. The company claims that 30% of the cards are rookies, 60% veterans, and 10% are All-Stars. The cards are sold in packages of 100. Suppose a randomlyselected package of cards has 50 rookies, 45 veterans, and 5 All-Stars. Is this consistent with Acme's claim? Use a 0.05 level of significance.

5.

Using this data, I’ll explain the nine steps for the chi-square goodness of fit test. As we walk through the steps of the test,

I’ll write them on the white board as the kids put it in their notebooks

1.

Name the test: Chi-square goodness of first test

2.

They have to state what the null hypothesis is. I’ll ask, “Who can tell me what a null hypothesis is?” After someone answers that, I’ll say, “So if that’s the definition of a null hypothesis, who can tell me what the null hypothesis would be for this test?”

3.

The alternative hypothesis. I’ll follow a similar strategy that I used for asking about the null hypothesis.

4.

The significance level. I’ll ask, “Who can tell me what the significance level is?” The students should remember this term from prior inference tests that have already been taught.

5.

Conditions for the test. These conditions must be met so the test can be carried out properly. First, a random sample has to be used and second, the expected count has to be greater than or equal to five. To get the expected count, you have to find the sum of the relative differences. I’ll explain to the students that to find the expected count, you have to take the total number in the random sample and multiply it by the percentages given. For the baseball card example, you would take 100(.3) to find that the expected number of rookie cards to be found in the package would be 30. The same process would be carried out for the other two types of cards.

6.

Sketch the distribution. From the notes we just took, I’ll ask, “What’s the general shape of a chi-squared distribution?” When someone answers, “It’s right-skewed,” I’ll proceed to show them how to mark where chi-




squared would fall on the chart.

7.

Find the value for the test statistic, chi-square, and the degrees of freedom. Finding the value of chi-square was discussed earlier in the notes about it and to find degrees of freedom, the students have to do the number of categories minus one. For instance, with the baseball card example, it’d be three minus one because they are three different types of baseball cards that can be in the package. Both of these values can also be found using a TI-83 or

TI-84 calculator.

8.

Find the p-value. This process is done with their calculators

9.

Interpret the p-value

6.

After I’m done explaining the steps, I’ll hook up the Elmo to my laptop and use graphing software to create the chi-squared distribution more accurately. This will allow the kids to visualize what they just calculated and they can see how the results look.

7.

After graphing the results, I’ll bring out a large bag of M&Ms and project this graph through the Elmo.

8.

Each student will get a cup full of M&Ms and each pair will proceed to conduct the test.

I’ll explain how each partner set must conduct the test before eating any M&Ms.

9.

With the person they are sitting next to, the kids will carry out a chi-square goodness of fit test to see if the proportion of colors in their sample of M&Ms matches the proportions provided.

10.

I’ll walk around to make sure each person is counting their own M&Ms and that the kids are each writing out the steps for the test in their notebooks.

11.

Once everyone has completed their tests, we’ll come back together as a class and discuss their findings. b.

Day Two

1.

As the class comes in, I’ll review with the steps of the chi-squared goodness of fit test and the activity that was done the day before.

2.

I’ll ask, “Who can tell me the basic purpose of the chi-square test for independence” (a test previously taught). After that,

I’ll ask for someone to give me the difference between the two chi-square tests.

3.

Once the basic differences are distinguished between the tests, I’ll start making a table on the board. One column will be labeled chi-square goodness of fit test and the other will be labeled chi-square test for independence. As a class, we’ll note distinguishing characteristics between the two tests.

4.

Once the chart is done, I’ll spend the rest of class doing a review game with the students. I’ll put the class into groups of four, and I’ll present various sets of data. With each set, I’d ask questions like, “Which chi-square test would you use to find,” “What would the null hypothesis be for this test,” and “What are the conditions for this test?” These questions will help the students review not only the difference between the two chi-square tests but the steps involved for each.

5.

Within each group, they’ll be a small white board where the kids can write down their answers. After an appropriate




amount of time, I’ll ask to see the white boards and decide which teams get a point. At the end, the team with the most points will get some of the remaining M&Ms from yesterday’s activity.

6.

At the end of class, I’ll assign the homework. Kids can either do a worksheet with three problems on it. The kids would have to distinguish which test to use and conduct a full test for each problem based on the data provided. Or they can go home and conduct a test similar to the M&M activity with their family at home. They’d only have to do one and they could choose what they’d test. If they chose to do this, the steps for the test conducted and proof that the family was involved would have to be shown.

D.

Monitoring Student Engagement and Learning: (What will the teacher do to ensure student engagement?)

To ensure the student engagement, I’ll walk around the room as the kids conduct the test by with their partners. I’ll make sure both students are working together to perform the test within each partnership and that they both understand the steps to the test.

E.

Student Interactions: ( How will you structure opportunities for students to work with partners or in groups? What criteria will you use when forming groups?)

For the activity right after taking notes, I’ll have the students work with the person they are sitting next to. The seats in the classroom are already set up into columns two desks wide.

F.

Closure: Purpose of the Lesson is clearly restated by students and/or teacher (Objectives and Assessment)

To check the understanding of my students, on day two, I’ll play a fun review game to see where the kids are. While

I’ll have the class divided into teams, I’ll closely observe how students act within each group to see if any individual is having trouble.

Lesson Plan Details

Instructional Materials: Elmo, white board, , Expo markers, M&Ms, cups, personal notes to go off of when teaching, questions for review game, worksheet for homework, calculators

Use of Technology: I’ll use the Elmo for projecting the information needed to conduct the chi-square goodness of fit test. Also, I’ll use a graphing program to create the chi-square distribution of the example done in class and use the Elmo again to project the results.

Also, each student will have a TI-83 or TI-84 to use when conducting the test.

Safety in the Physical Environment:

I’ll tell the kids to be careful with their cups of M&Ms and to try not to spill any on the floor.

However, if a spill does occur, I’ll have a broom and dustpan ready to sweep up a large amount of M&Ms in order to avoid wasting time picking them up by hand or someone possibly slipping on them. Also, I’ll ask if anyone has any type of allergy to chocolate or colored dyes before starting the M&M activity. While the kids are seniors in high school, I will quickly go over the process of not eating the M&Ms too fast due to the risk of choking and how to proceed if anyone, per chance, started to choke. Finally, to avoid emotional harm for not having someone to work with, I’ll use the seating arrangement to make groups for the M&M activity and review game.




List the Academic Language used in the

Lesson.

Academic Language

Explain how the Academic Language is scaffolded in the Lesson using Sensory,

Describe the Evidence that Students know and use Academic Language

Include Tier 1, 2, and 3 language.



Chi-square



Inference test



Test statistic



Null/alternative hypothesis



Significance level



Observed count



Expected count



Right-skew



P-value



Degrees of freedom

Graphic and/or Interactive supports.

Each student has some basic understandings behind inference tests and their components because other types of tests have already been taught to the students. The graphic support of the chisquare distribution will build on each kid’s knowledge of what a prior test’s distribution is and they will be able to see how the p-value and chi-square are presented in the graph. appropriately:

The review game will not only assess if the students know how to conduct the test but also the language involved. When I ask questions like, “What are the degrees of freedom for this test,” or “What is the value of chi-square for this example,” I can tell which students know what I’m talking about based on their answers.

Assessment

(Identify the type(s) of assessment used in this lesson. Explain how it provides evidence that students will meet the objective(s). At least one type of assessment is required in a lesson plan.)

Diagnostic (Pre-): (Formal or Informal

: The KWL chart used would be informal because there’s no set grade based on it. It’s a way to map out topics and details about chi-square that the class discusses before learning about the new test.

Formative: (Formal or Informal ): In order to monitor students during class, I’ll do it through informal observation. While I lecture,

I’ll look around the room to make sure kids are paying attention and taking notes. During the M&M activity, I’ll walk around to each pairing to see who is participating and who’s just fooling around. Finally, during the review game, I’ll listen to various teams to see who within the team is participating and who’s just sitting to the side letting everyone else do the work.

Reflective: (Formal or Informal: The reflective part of class would be at the end of day one and it would also be informal. After everyone has finished their M&M activity, we’d come back and discuss as a class their results. I’d see how many people rejected the null hypothesis and how many people did not reject it. Based on those numbers, we’d discuss as a class whether or not the chart that was used to provide the color proportion of M&M was a true one.

Summative: (Formal or Informal): The summative part would be formal. After learning about the chi-square goodness of fit test in various ways, the kids can choose which type of homework to do: the worksheet with three tests to conduct or doing one live test with their family. The next day, I’d collect the worksheets or the results of the live test performed, along with the proof, and grade them.




Differentiating Instruction

Identify the Element(s) of the Lesson that is Differentiated: Content Process Product

Explain how it is Differentiated for the whole class, groups of students with similar needs, individual students OR students with

IEPs or 504 plans.

The element I differentiated was the product. In order to assess the kids’ understanding, I allowed two options: either do a worksheet where you read three sets of data given to you and perform the right test or create a test on your own and conduct it with your family.

This allows kids to be more creative with their work if they want to be. Some students might have trouble conducting the test without actually performing it in real life, so this gives those types of students a chance to practice the steps in a more comfortable way for them.

Identify the Student Characteristic that you will use to Differentiate: Student Readiness Student Interest Student

Learning Profile

Explain how it is used to Differentiate for the whole class, groups of students with similar needs, individual students OR students

with IEPs or 504 plans.

The characteristic I used to differentiate was student interest. If there’s a student who really enjoys starbursts, they can conduct a similar test to the M&M one and possibly have more fun doing that than a worksheet. However, if people are more interested in just being told what to test and not caring about the topics, the worksheet with the data provided would be more up their alley.

Analyzing Teaching (Reflection):

To be completed after the lesson is taught

A.

Give evidence that the lesson was successful for students meeting the learning objective/s. (at least one example)

B.

If you could teach this lesson to the same group of students again, what are two or three things you would do differently to improve the learning of these students based on their varied developmental and academic needs and characteristics? Consider missed opportunities and other aspects of planning, instruction, and/or assessment.

Clearly state each change you would make .

Explain why and how you would change it.

Explain the difference it would make in

student learning. Give evidence from your experience in teaching the lesson.




Chi-Square Goodness of Fit Test vs. Chi-Square Test for Independence Homework

For each problem decide which test to use. Then, conduct the test using the appropriate steps

1.

A cereal company claims that the chances of getting a certain toy as a prize in a cereal box are as followed: toy car-20%, sticker-60%, doll-15%, all-star baseball card-5%. From a random sample of 200 boxes of cereal, 96 prizes were stickers, 34




were dolls, 63 were cars, and 7 were baseball cards. Is this consistent with the company’s claim? Use a significance level of

.01.

2.

A public opinion poll surveyed a simple random sample of 1000 voters. Respondents were classified by gender (male or female) and by voting preference (Republican, Democrat, or Independent). Are gender and voting preference related? Use a

.05 significance level

GENDER

FEMALE

DEMOCRATIC

200

VOTING PREFERENCE

REPUBLICAN INDEPENDENT

150 50




MALE 250 300 50

3.

The proportion of jolly rancher flavors in a bag is presented below. After randomly choosing a bag containing 125 candies, the numbers of each flavor in the bag are also presented below. Does the sample taken suggest that the actual flavor distribution is different than the proportions presented?

Blue

Jolly Rancher Flavor

Cherry Grape Green Watermelon




Given

Proportions

Number of

Flavors in sample of

125

Raspberry

8%

16

20%

26

30%

32

Apple

25%

27

17%

24




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