## Lesson

Let's look more closely at data and the questions they can help to answer.

Exercise (PageIndex{1}): Pencils on A Plot

- Measure your pencil to the nearest (frac{1}{4})-inch. Then, plot your measurement on the class dot plot.
- What is the difference between the longest and shortest pencil lengths in the class?
- What is the most common pencil length?
- Find the difference in lengths between the most common length and the shortest pencil.

Exercise (PageIndex{2}): What's in the Data?

Ten sixth-grade students at a school were each asked five survey questions. Their answers to each question are shown here.

(egin{array}{lllllllllll}{ ext{data set A}}&{0}&{1}&{1}&{3}&{0}&{0}&{0}&{2}&{1}&{1}{ ext{data set B}}&{12}&{12}&{12}&{12}&{12}&{12}&{12}&{12}&{12}&{12}{ ext{data set C}}&{6}&{5}&{7}&{6}&{4}&{5}&{3}&{4}&{6}&{8}{ ext{data set D}}&{6}&{6}&{6}&{6}&{6}&{6}&{6}&{6}&{6}&{6}{ ext{data set E}}&{3}&{7}&{9}&{11}&{6}&{4}&{2}&{16}&{6}&{10}end{array})

- Here are the five survey questions. Match each question to a data set that could represent the students’ answers. Explain your reasoning.
- Question 1: Flip a coin 10 times. How many heads did you get?
- Question 2: How many books did you read in the last year?
- Question 3: What grade are you in?
- Question 4: How many dogs and cats do you have?
- Question 5: How many inches are in 1 foot?

- How are survey questions 3 and 5 different from the other questions?

Exercise (PageIndex{3}): What Makes a Statistical Question?

These three questions are examples of **statistical questions**:

- What is the most common color of the cars in the school parking lot?
- What percentage of students in the school have a cell phone?
- Which kind of literature—fiction or nonfiction—is more popular among students in the school?

These three questions are not examples of **statistical questions**:

- What color is the principal’s car?
- Does Elena have a cell phone?
- What kind of literature—fiction or nonfiction—does Diego prefer?

- Study the examples and non-examples. Discuss with your partner:
- How are the three statistical questions alike? What do they have in common?
- How are the three non-statistical questions alike? What do they have in common?
- How can you find answers to the statistical questions? How about answers to non-statistical questions?
- What makes a question a statistical question?

Pause here for a class discussion.

- Read each question. Think about the data you might collect to answer it and whether you expect to see
**variability**in the data. Complete each blank with “Yes” or “No.”- How many cups of water do my classmates drink each day?
- Is variability expected in the data? ______
- Is the question statistical? _____

- Where in town does our math teacher live?
- Is variability expected in the data? ______
- Is the question statistical? _____

- How many minutes does it take students in my class to get ready for school in the morning?
- Is variability expected in the data? ______
- Is the question statistical? _____

- How many minutes of recess do sixth-grade students have each day?
- Is variability expected in the data? ______
- Is the question statistical? _____

- Do all students in my class know what month it is?
- Is variability expected in the data? ______
- Is the question statistical? _____

- How many cups of water do my classmates drink each day?

Exercise (PageIndex{4}): Sifting for Statistical Questions

- Your teacher will give you and your partner a set of cards with questions. Sort them into three piles: Statistical Questions, Not Statistical Questions, and Unsure.
- Compare your sorting decisions with another group of students. Start by discussing the two piles that your group sorted into the Statistical Questions and Not Statistical Questions piles. Then, review the cards in the Unsure pile. Discuss the questions until both groups reach an agreement and have no cards left in the Unsure pile. If you get stuck, think about whether the question could be answered by collecting data and if there would be variability in that data.
- Record the letter names of the questions in each pile.
- Statistical questions:
- Non-statistical questions:

Are you ready for more?

Tyler and Han are discussing the question, “Which sixth-grade student lives the farthest from school?”

- Tyler says, “I don’t think the question is a statistical question. There is only one person who lives the farthest from school, so there would not be variability in the data we collect.”
- Han says: “I think it is a statistical question. To answer the question, we wouldn’t actually be asking everyone, 'Which student lives the farthest from school?' We would have to ask each student how far away from school they live, and we can expect their responses to have variability.”

Do you agree with either one of them? Explain your reasoning.

### Summary

We often collect data to answer questions about something. The data we collect may show **variability**, which means the data values are not all the same.

Some data sets have more variability than others. Here are two sets of figures.

Set A has more figures with the same shape, color, or size. Set B shows more figures with different shapes, colors, or sizes, so set B has greater variability than set A.

Both numerical and categorical data can show variability. Numerical sets can contain different numbers, and categorical sets can contain different categories or types.

When a question can only be answered by using data and we expect that data to have variability, we call it a **statistical question**. Here are some examples.

- Who is the most popular musical artist at your school?
- When do students in your class typically eat dinner?
- Which classroom in your school has the most books?

To answer the question about books, we may need to count all of the books in each classroom of a school. The data we collect would likely show variability because we would expect each classroom to have a different number of books.

In contrast, the question “How many books are in your classroom?” is *not* a statistical question. If we collect data to answer the question (for example, by asking everyone in the class to count books), the data can be expected to show the same value. Likewise, if we ask all of the students at a school where they go to school, that question is not a statistical question because the responses will all be the same.

### Glossary Entries

Definition: Categorical Data

A set of categorical data has values that are words instead of numbers.

For example, Han asks 5 friends to name their favorite color. Their answers are: blue, blue, green, blue, orange.

Definition: Dot Plot

A dot plot is a way to represent data on a number line. Each time a value appears in the data set, we put another dot above that number on the number line.

For example, in this dot plot there are three dots above the 9. This means that three different plants had a height of 9 cm.

Definition: Numerical Data

A set of numerical data has values that are numbers.

For example, Han lists the ages of people in his family: 7, 10, 12, 36, 40, 67.

Definition: Statistical Question

A statistical question can be answered by collecting data that has variability. Here are some examples of statistical questions:

- Who is the most popular musical artist at your school?
- When do students in your class typically eat dinner?
- Which classroom in your school has the most books?

Definition: Variability

Variability means having different values.

For example, data set B has more variability than data set A. Data set B has many different values, while data set A has more of the same values.

## Practice

Exercise (PageIndex{5})

Sixth-grade students were asked, “What grade are you in?” Explain why this is *not* a statistical question.

Exercise (PageIndex{6})

Lin and her friends went out for ice cream after school. The following questions came up during their trip. Select **all** the questions that are statistical questions.

- How far are we from the ice cream shop?
- What is the most popular ice cream flavor this week?
- What does a group of 4 people typically spend on ice cream at this shop?
- Do kids usually prefer to get a cup or a cone?
- How many toppings are there to choose from?

Exercise (PageIndex{7})

Here is a list of questions about the students and teachers at a school. Select **all** the questions that are statistical questions.

- What is the most popular lunch choice?
- What school do these students attend?
- How many math teachers are in the school?
- What is a common age for the teachers at the school?
- About how many hours of sleep do students generally get on a school night?
- How do students usually travel from home to school?

Exercise (PageIndex{8})

Here is a list of statistical questions. What data would you collect and analyze to answer each question? For numerical data, include the unit of measurement that you would use.

- What is a typical height of female athletes on a team in the most recent international sporting event?
- Are most adults in the school football fans?
- How long do drivers generally need to wait at a red light in Washington, DC?

Exercise (PageIndex{9})

Describe the scale you would use on the coordinate plane to plot each set of points. What value would you assign to each unit of the grid?

- ((1,-6), (-7,-8), (-3,7), (0,9))
- ((-20,-30), (-40,10), (20,-10), (5,-20))
- ((frac{-1}{3},-1), (frac{2}{3},-1frac{1}{3}),(frac{-4}{3},frac{2}{3}),(frac{1}{6},0))

(From Unit 7.3.3)

Exercise (PageIndex{10})

Noah’s water bottle contains more than 1 quart of water but less than (1frac{1}{2}) quarts. Let (w) be the amount of water in Noah’s bottle, in quarts. Select** all** the true statements.

- (w) could be (frac{3}{4}).
- (w) could be (1).
- (w>1)
- (w) could be (frac{4}{3}).
- (w) could be (frac{5}{4}).
- (w) could be (frac{5}{3}).
- (w>1.5)

(From Unit 7.2.2)

Exercise (PageIndex{11})

Order these numbers from least to greatest:

(|-17|qquad |-18|qquad -18qquad |19|qquad 20)

(From Unit 7.1.7)

## Statistical Questions (Grade 6)

These lessons help Grade 6 students learn how to recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers.

**Common Core: 6.SP.1**

### Suggested Learning Targets

- I can recognize that data has variability.
- I can recognize a statistical question (examples versus non-examples).

**6.SP.1 Statistical Vs. Non-Statistical Questions** < The following table gives some examples of statistical questions and non-statistical questions. Scroll down the page for more examples and solutions.

A statistical question is a question that should have different answers.

How to recognize a statistical question?

- A question is not a statistical question if it has an exact answer. For example &ldquoHow old are you?&rdquo
- A question is a statistical question if the answer is a percent, range, or an average. For example &ldquoHow old are the students in this room&rdquo

Identify which questions are statistical and which questions are not statistical.

• What is the favorite menu item for customers in the local restaurant?

• What time do most people eat their lunches?

• What did my dad eat for lunch today?

• What do 7th graders prefer to eat for lunch?

In a survey about practice time per week for high school athletes, 22% practice 1 hour, 40% practice 2 hours, 25% practice 3 hours, 10% practice 4 hours and 3% practice more than 4 hours.

Which one question is the most likely to have produced these results?

• What is the average practice time per week required by your sport?

• How much time do you spend doing homework during the week?

• Is practice time longer on Mondays than Tuesdays?

• Which sport practices the most?

Jessica conducted a survey using a representative sample of 50 customers from three local landscaping businesses in town. She found that 30% purchased maple trees, 24% purchased dogwoods, 20% purchased oaks, 16% purchased pines and 10% chose other types of trees.

Which statements about the survey that Jessica conducted are most likely to be true? Select all that apply.

• Jessica surveyed only the customers who purchased a tree.

• Jessica asked customers what type of tree they purchased.

• Jessica asked customers what type of plants they have in their yards.

• The sample consists of 50 customers from three local landscaping businesses in town.

• The population Jessica wants to know about consists of any customer of any landscaping business.

**What Is A Statistical Question?**

Definition: A statistical question has answers that will probably vary. Usually a statistical question will ask about a population of multiple people, events or things.

**Examples Of Statistical Questions**

- What time did the students in this class get up this morning?
- How many votes did the winning candidate for the Presidents of the Student Body receive in each of the past 20 years?
- What were the high temperatures in all of the Latin American capitals today?

**Examples Of Non-Statistical Questions**

- What time did I get up this morning?
- How many votes did the winning candidate for the Student Body receive this year?
- What was the high temperature in Mexico City today?

**Statistical And Non-Statistical Questions**

**Examples:**

Which of the following are statistical questions?

- How old are the people who have watched this video in 2013?
- Do dogs run faster than cats?
- Do wolves weigh more than dogs?
- Does your dog weigh more than that wolf?
- Does it rain more in Seattle than Singapore?
- What was the difference in rainfall between Singapore and Seattle in 2013?
- In general, will I use less gas driving at 55 mph than 70 mph?
- Do English professors get paid less than math professors?
- Does the most highly paid English professor at Harvard get paid more that the most highly paid math professor in MIT?

**Statistical Questions - Common Core Standard**

Students must know variability refers to the spread of data.

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.

## Solution

- Not statistical. This question is answered by counting the number of days in March. This produces a single number. This question is not answered by collecting data that vary.
- Not statistical. This question is answered by a single number. It is not answered by collecting data that vary.
- Statistical. This question would be answered by collecting data, and there would be variability in that data.
- Statistical. This question would be answered by collecting data, and there would be variability in that data.
- Not statistical. This question is answered by a single response. It is not answered by collecting data that vary.
- Not statistical. This question would be answered by counting the bricks. This produces a single number. This question is not answered by collecting data that vary.Â
- Non-statistical (there is oneÂ temperature).

## Solution

- Statistical question
- Not a statistical question
- Statistical question
- Statistical question
- Statistical question
- Statistical question

Each of the statistical questions would be answered by collecting data and there would be variability in the data.

Questions identified as not statistical questions are not answered based on data that vary.

The question "What is a typical number of holes for the buttons in the jar?" is a statistical question. To answer this question, students might compute the mean or the median (both measures of center that are used to describe a typical value). But in either case they would need to collect data on the number of holes by recording a value for each button. Because not all buttons have the same number of holes, there would be variability in the data that would be used to answer this question. That is what makes this a statistical question.

The question "How many buttons are in the jar?" is answered by counting the buttons. This produces a single value--it is not answered by collecting data that vary. It is not a statistical question.

The question "How large is the largest button in the jar?" isÂ a statistical question. The size of the largest button is a population characteristic and this question would be answered by collecting data on the sizes of all the buttons in the population. In this way, the question would be answered in a way that takes variability in the population into consideration.Â

The question "If Zeke grabbed a handful of buttons, what are the chances that all of the buttons in his hand are round?" is a statistical question because this is asking for a probability that would be estimated by having Zeke grab many handfuls of buttons. For each handful grabbed, whether or not all of the buttons were round would be recorded. This would result in categorical data (with values of "all round" and "not all round"), but again there would be variability in this data. These data could then be used to estimate the probability of interest to provide an answer to the question posed.

Like the first question, the last two questions (v and vi) are statistical questions because they would be answered by collecting data that vary. To answer the question about the typical number of holes, data on number of holes would be collected for each button in the jar. The question about how the buttons are distributed according to color would be answered by recording the color of each button in the jar and then summarizing these data in a table or a graph.

Some possible statistical questions are:

- What is a typical shape for buttons in the jar?
- What is the distribution of the diameters of the round buttons in this jar?

## Lesson 2 Summary

We often collect data to answer questions about something. The data we collect may show

**variability**, which means the data values are not all the same.Some data sets have more variability than others. Here are two sets of figures.

Set A has more figures with the same shape, color, or size. Set B shows more figures with different shapes, colors, or sizes, so set B has greater variability than set A.

Both numerical and categorical data can show variability. Numerical sets can contain different numbers, and categorical sets can contain different categories or types.

When a question can only be answered by using data and we expect that data to have variability, we call it a

**statistical question**. Here are some examples.- Who is the most popular musical artist at your school?
- When do students in your class typically eat dinner?
- Which classroom in your school has the most books?

To answer the question about books, we may need to count all of the books in each classroom of a school. The data we collect would likely show variability because we would expect each classroom to have a different number of books.

In contrast, the question “How many books are in your classroom?” is

*not*a statistical question. If we collect data to answer the question (for example, by asking everyone in the class to count books), the data can be expected to show the same value. Likewise, if we ask all of the students at a school where they go to school, that question is not a statistical question because the responses will all be the same.

## Lesson 2

Sixth-grade students were asked, “What grade are you in?” Explain why this is

*not*a statistical question.### Solution

Teachers with a valid work email address can click here to register or sign in for free access to Formatted Solution.

### Problem 2

Lin and her friends went out for ice cream after school. The following questions came up during their trip. Select

**all**the questions that are statistical questions.How far are we from the ice cream shop?

What is the most popular ice cream flavor this week?

What does a group of 4 people typically spend on ice cream at this shop?

Do kids usually prefer to get a cup or a cone?

How many toppings are there to choose from?

### Solution

Teachers with a valid work email address can click here to register or sign in for free access to Formatted Solution.

### Problem 3

Here is a list of questions about the students and teachers at a school. Select

**all**the questions that are statistical questions.What is the most popular lunch choice?

What school do these students attend?

How many math teachers are in the school?

What is a common age for the teachers at the school?

About how many hours of sleep do students generally get on a school night?

How do students usually travel from home to school?

### Solution

Teachers with a valid work email address can click here to register or sign in for free access to Formatted Solution.

### Problem 4

Here is a list of statistical questions. What data would you collect and analyze to answer each question? For numerical data, include the unit of measurement that you would use.

- What is a typical height of female athletes on a team in the most recent international sporting event?
- Are most adults in the school football fans?
- How long do drivers generally need to wait at a red light in Washington, DC?

### Solution

### Problem 5

Describe the scale you would use on the coordinate plane to plot each set of points. What value would you assign to each unit of the grid?

- ((1, ext-6)) , (( ext-7, ext-8)) , (( ext-3, 7)) , ((0, 9))
- (( ext-20, ext-30)) , (( ext-40, 10)) , ((20, ext-10)) , ((5, ext-20))
- ((frac < ext<->1><3>, ext-1), (frac<2><3>, ext-1 frac13), (frac < ext<->4><3>,frac23), (frac16, 0))

Expand Image

### Solution

### Problem 6

Noah’s water bottle contains more than 1 quart of water but less than (1 frac<1><2>) quarts. Let (w) be the amount of water in Noah’s bottle, in quarts. Select

**all**the true statements.### Solution

### Problem 7

Order these numbers from least to greatest:

### Solution

IM 6–8 Math was originally developed by Open Up Resources and authored by Illustrative Mathematics®, and is copyright 2017-2019 by Open Up Resources. It is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). OUR's 6–8 Math Curriculum is available at https://openupresources.org/math-curriculum/.

Adaptations and updates to IM 6–8 Math are copyright 2019 by Illustrative Mathematics, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Adaptations to add additional English language learner supports are copyright 2019 by Open Up Resources, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

The second set of English assessments (marked as set "B") are copyright 2019 by Open Up Resources, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Spanish translation of the "B" assessments are copyright 2020 by Illustrative Mathematics, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

The Illustrative Mathematics name and logo are not subject to the Creative Commons license and may not be used without the prior and express written consent of Illustrative Mathematics.

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## Contents

Examples exist of mathematically correct results derived by incorrect lines of reasoning. Such an argument, however true the conclusion appears to be, is mathematically invalid and is commonly known as a

*howler*. [1] The following is an example of a howler involving anomalous cancellation:Bogus proofs, calculations, or derivations constructed to produce a correct result in spite of incorrect logic or operations were termed "howlers" by Maxwell. [3] Outside the field of mathematics the term

*howler*has various meanings, generally less specific.The division-by-zero fallacy has many variants. The following example uses a disguised division by zero to "prove" that 2 = 1, but can be modified to prove that any number equals any other number.

The fallacy is in line 5: the progression from line 4 to line 5 involves division by

*a*−*b*, which is zero since*a*=*b*. Since division by zero is undefined, the argument is invalid.after which the antiderivatives may be cancelled yielding 0 = 1. The problem is that antiderivatives are only defined up to a constant and shifting them by 1 or indeed any number is allowed. The error really comes to light when we introduce arbitrary integration limits

*a*and*b*.Since the difference between two values of a constant function vanishes, the same definite integral appears on both sides of the equation.

Many functions do not have a unique inverse. For instance, while squaring a number gives a unique value, there are two possible square roots of a positive number. The square root is multivalued. One value can be chosen by convention as the principal value in the case of the square root the non-negative value is the principal value, but there is no guarantee that the square root given as the principal value of the square of a number will be equal to the original number (e.g. the principal square root of the square of −2 is 2). This remains true for nth roots.

### Positive and negative roots Edit

Care must be taken when taking the square root of both sides of an equality. Failing to do so results in a "proof" of [9] 5 = 4.

The fallacy is in the second to last line, where the square root of both sides is taken:

*a*2 =*b*2 only implies*a*=*b*if*a*and*b*have the same sign, which is not the case here. In this case, it implies that*a*= –*b*, so the equation should readAnother example illustrating the danger of taking the square root of both sides of an equation involves the following fundamental identity [10]

which holds as a consequence of the Pythagorean theorem. Then, by taking a square root,

Evaluating this when

*x*= π , we get thatThe error in each of these examples fundamentally lies in the fact that any equation of the form

and it is essential to check which of these solutions is relevant to the problem at hand. [11] In the above fallacy, the square root that allowed the second equation to be deduced from the first is valid only when cos

*x*is positive. In particular, when*x*is set to π , the second equation is rendered invalid.### Square roots of negative numbers Edit

Invalid proofs utilizing powers and roots are often of the following kind:

Alternatively, imaginary roots are obfuscated in the following:

The error here lies in the last equality, where we are ignoring the other fourth roots of 1, [note 2] which are −1,

*i*and −*i*(where*i*is the imaginary unit). Since we have squared our figure and then taken roots, we cannot always assume that all the roots will be correct. So the correct fourth roots are*i*and −*i*, which are the imaginary numbers defined to square to −1.### Complex exponents Edit

When a number is raised to a complex power, the result is not uniquely defined (see Failure of power and logarithm identities). If this property is not recognized, then errors such as the following can result:

The error here is that the rule of multiplying exponents as when going to the third line does not apply unmodified with complex exponents, even if when putting both sides to the power

*i*only the principal value is chosen. When treated as multivalued functions, both sides produce the same set of values, being <*e*2 π*n*|*n*∈ ℤ> .Many mathematical fallacies in geometry arise from using an additive equality involving oriented quantities (such as adding vectors along a given line or adding oriented angles in the plane) to a valid identity, but which fixes only the absolute value of (one of) these quantities. This quantity is then incorporated into the equation with the wrong orientation, so as to produce an absurd conclusion. This wrong orientation is usually suggested implicitly by supplying an imprecise diagram of the situation, where relative positions of points or lines are chosen in a way that is actually impossible under the hypotheses of the argument, but non-obviously so.

In general, such a fallacy is easy to expose by drawing a precise picture of the situation, in which some relative positions will be different from those in the provided diagram. In order to avoid such fallacies, a correct geometric argument using addition or subtraction of distances or angles should always prove that quantities are being incorporated with their correct orientation.

### Fallacy of the isosceles triangle Edit

The fallacy of the isosceles triangle, from (Maxwell 1959, Chapter II, § 1), purports to show that every triangle is isosceles, meaning that two sides of the triangle are congruent. This fallacy has been attributed to Lewis Carroll. [13]

Given a triangle △ABC, prove that AB = AC:

- Draw a line bisecting ∠A.
- Draw the perpendicular bisector of segment BC, which bisects BC at a point D.
- Let these two lines meet at a point O.
- Draw line OR perpendicular to AB, line OQ perpendicular to AC.
- Draw lines OB and OC.
- By AAS, △RAO ≅ △QAO (∠ORA = ∠OQA = 90° ∠RAO = ∠QAO AO = AO (common side)).
- By RHS, [note 3] △ROB ≅ △QOC (∠BRO = ∠CQO = 90° BO = OC (hypotenuse) RO = OQ (leg)).
- Thus, AR = AQ, RB = QC, and AB = AR + RB = AQ + QC = AC.

As a corollary, one can show that all triangles are equilateral, by showing that AB = BC and AC = BC in the same way.

The error in the proof is the assumption in the diagram that the point O is

*inside*the triangle. In fact, O always lies at the circumcircle of the △ABC (except for isosceles and equilateral triangles where AO and OD coincide). Furthermore, it can be shown that, if AB is longer than AC, then R will lie*within*AB, while Q will lie*outside*of AC, and vice versa (in fact, any diagram drawn with sufficiently accurate instruments will verify the above two facts). Because of this, AB is still AR + RB, but AC is actually AQ − QC and thus the lengths are not necessarily the same.There exist several fallacious proofs by induction in which one of the components, basis case or inductive step, is incorrect. Intuitively, proofs by induction work by arguing that if a statement is true in one case, it is true in the next case, and hence by repeatedly applying this, it can be shown to be true for all cases. The following "proof" shows that all horses are the same colour. [14] [note 4]

- Let us say that any group of
*N*horses is all of the same colour. - If we remove a horse from the group, we have a group of
*N*− 1 horses of the same colour. If we add another horse, we have another group of*N*horses. By our previous assumption, all the horses are of the same colour in this new group, since it is a group of*N*horses. - Thus we have constructed two groups of
*N*horses all of the same colour, with*N*− 1 horses in common. Since these two groups have some horses in common, the two groups must be of the same colour as each other. - Therefore, combining all the horses used, we have a group of
*N*+ 1 horses of the same colour. - Thus if any
*N*horses are all the same colour, any*N*+ 1 horses are the same colour. - This is clearly true for
*N*= 1 (i.e. one horse is a group where all the horses are the same colour). Thus, by induction,*N*horses are the same colour for any positive integer*N*. i.e. all horses are the same colour.

The fallacy in this proof arises in line 3. For

*N*= 1, the two groups of horses have*N*− 1 = 0 horses in common, and thus are not necessarily the same colour as each other, so the group of*N*+ 1 = 2 horses is not necessarily all of the same colour. The implication "every*N*horses are of the same colour, then*N*+ 1 horses are of the same colour" works for any*N*> 1, but fails to be true when*N*= 1. The basis case is correct, but the induction step has a fundamental flaw.

## The Sum of the Geometric Series 1 + 1/2 + 1/4 + · · ·

My name is Krishna. I'm now in Grade 12. When I was in Grade 11, I saw a question in the math club.

Actually, I have already asked a similar question like the one below, but this one is little different.

The question is,

What is the value of, 1 + (1/2) + (1/4) + (1/8) . ?

How can you find a definite value for an answer when it keeps on continuing?

Thank you.

Krishna

and this sequence of numbers (1, 3/2, 7/4, 15/8, . . . ) is converging to a limit. It is this limit which we call the "value" of the infinite sum.

How do we find this value?

If we assume it exists and just want to find what it is, let's call it S . Now

This same technique can be used to find the sum of any "geometric series", that it, a series where each term is some number r times the previous term. If the first term is a , then the series is

In using this technique, we have assumed that the infinite sum exists, then found the value. But we can also use it to tell whether the sum exists or not: if you look at the finite sum

As long as | r | < 1, the term r ^( n +1) will go to zero as n goes to infinity, so the finite sum S will approach a / (1- r ) as n goes to infinity. Thus the value of the infinite sum is a / (1- r ), and this also proves that the infinite sum exists, as long as | r | < 1.

## NYS Next Generation Mathematics Learning Standards for Grades 3-8: Post-Test Standards Designations Applicable to Test Administrations Beginning Spring 2023

**Public Comments:**The public comment period closed on 1/31/19. The New York State Education Department has reviewed the comments that were received regarding the draft post-test standards designations for Grades 3-8 Next Generation Mathematics assessments. Overall, the comments were very positive. Based on those comments, no changes were made to the draft designations. The draft post-test standards designations below for Grades 3-8**Next Generation Mathematics**assessments are now finalized. **Please note: These post-test standards designations will not be reflected in the Grades 3-8 state assessments until full implementation of the NYS Next Generation Mathematics Learning Standards, which begins in School Year 2022-2023.**These post-test standards designations are to be used as districts move into the Building Capacity stage of the Next Generation Learning Standards Roadmap and Implementation Timeline. The first goal of this stage is “Support local school district needs to integrate the Next Generation Mathematics Learning Standards into local curriculum.” It will be important for districts to consider the changes that have occurred with the post-test standards designations as they begin examining their current district curricular materials and resources and determining the changes needed to ensure alignment to the NYS Next Generation Mathematics Learning Standards.

**Please note that these post-test standards designations will not be reflected in the Grades 3-8 assessments until full implementation of the NYS Next Generation Mathematics Learning Standards, which begins in School Year 2022-2023.**In 2015, New York State (NYS) began a process of review and revision of its current mathematics standards adopted in January of 2011. Through numerous phases of public comment, virtual and face-to-face meetings with committees consisting of NYS educators (Special Education, Bilingual Education and English as a New Language teachers), parents, curriculum specialists, school administrators, college professors, and experts in cognitive research, the New York State Next Generation Mathematics Learning Standards (2017) were developed. These revised standards reflect the collaborative efforts and expertise of all constituents involved.

In order to support implementation of these revised standards, a process began in 2018 to designate which Standards for Mathematical Content from each grade, 3 – 8, will be assessed on that grade’s annual NYS assessment, and therefore must be taught prior to the date of the assessment, and which standards will be taught after the date of the NYS assessment each year. This process included three rounds of face-to-face meetings with committees consisting of NYS educators. Each committee started by independently reviewing the standards and drafting an initial recommendation. They then reviewed the recommendations of the groups that preceded them, discussed the merits of all options, and finished by providing their final recommendation that considered their initial ideas as well as those of previous groups. The final recommendations outlined in the document were informed by all of the ideas and discussions of each of the committees.

Throughout the process, there were three main principles that guided the thinking of each committee and the final recommendations two of which have to do with maintaining connections among standards, to every extent practicable. It is important to note that, at every grade level, all other standards were considered for possible designation as post-test, but were not ultimately moved because of these three guiding principles.

## Question: Problems 1. The measured data on the thickness of a meta.

Problems

1. The measured data on the thickness of a metal layer from a vapor deposition study in Å in an instrument is: 48530, 48980, 50210, 49860, 48650, 49560, 49270, 48850, 49320, 48680.

The instrument has a range of 100,000 with a full-scale accuracy of 1.5%.

a. Present the descriptive statistics for this data

b. Calculate the systematic uncertainty, random uncertainty, total uncertainty, random uncertainly of the mean, and total uncertainty of the mean from the instrument.2. The inside diameters of bearings used in an aircraft landing gear assembly are known to have a standard deviation of σ = 0.002 cm. A random sample of 15 bearings has an average inside diameter of 8.2535 cm.

a. Test the hypothesis that the mean inside bearing diameter is 8.25 cm. Use a two-sided alternative and α = 0.05.

b. Find the P-value for this test.

c. Construct a 95% two-sided confidence interval on the mean bearing diameter

d. Repeat (a,b,c) above for α = 0.1. Compare with results for α = 0.05.

e. Repeat (a,b,c) above for α = 0.01. Compare with results for α = 0.05.3. A machine is used to fill containers with a liquid product. Fill volume can be assumed to be normally distributed. A random sample of ten containers is selected, and the net contents (oz) are as follows: 12.03, 12.01, 12.04, 12.02, 12.05, 11.98, 11.96, 12.02, 12.05, 11.99.

a. Suppose that the manufacturer wants to be sure that the mean net contents exceeds 12 oz. What conclusions can be drawn from the data. Use a = 0.01.

b. Construct a 95% two-sided confidence interval on the mean fill volume.

c. Does the assumption of normality seem appropriate for the fill volume data?

d. Repeat (a,b,c) above for a = 0.05. Compare with results for a = 0.01.4. Two machines are used for filling glass bottles with a soft-drink beverage. The filling processes have known standard deviations s1 = 0.010 L, and s2 = 0.015 L, respectively. A random sample of n1 = 25 bottles from machine 1 and n2 = 20 bottles from machine 2, results in average net contents of x1= 2.04 L, and x2 = 2.07 L.

a. Test the hypothesis that both machines fill to the same net contents, using a = 0.05. What are your conclusions?

b. Find the P-value for this test.

c. Construct a 95% two-sided confidence interval on the mean fill volume.5. Two quality control technicians measured the surface finish of a metal part, and the data is tabulated. Assume that the measurements are

normally distributed.

a. Test the hypothesis that the mean surface finish measurements made by the two technicians are equal. Use a = 0.05, and assume equal variances

b. Assuming that the variances are equal, construct a 95% confidence interval on the mean difference in surface-finish measurements.

Problem 5

T1 T2

1.45 1.54

1.37 1.41

1.21 1.56

1.54 1.37

1.48 1.20

1.29 1.31

1.34 1.27

1.356. Two different hardening processes – (1) saltwater quenching and (2) oil quenching – are used on samples of a particular type of metal alloy. The results are tabulated. Assume that hardness is normally distributed.

a. Test the hypothesis that the mean hardness for the saltwater quenching process equals the mean hardness for the oil quenching process. Use a = 0.05, and assume equal variances.

b. Assuming that the variances are equal, construct a 95% confidence interval on the mean difference in mean hardness

c. Does the assumption of normality seem appropriate for this data?Problem 6

Saltwater Oil

145 152

150 150

153 147

148 155

141 140

156 146

146 158

154 152

139 151

148 1437. The results of an experiment to investigate the low-pressure vapor deposition of polysilicon. The reactor has several wafer positions,

and four of these positions are selected at random. The response variable is film thickness uniformity. Three replicates of the experiment were run, and data is presented.

a. Is there a difference in the wafer positions? Use the analysis of variance, and a = 0.05?

b. Estimate the variability due to wafer positions.

c. Estimate the random error component

d. Analyze the results from this experiment and comment on model adequacy.

Problem 7

Position Uniformity

1 2.76 5.67 4.49

2 1.43 1.70 2.19

3 2.34 1.97 1.47

4 0.94 1.36 1.658. An experiment to determine the effect of C2F6 flow rate on etch uniformity on Si wafer is presented in a table. Three flow rates are tested, and the resulting uniformity (in %) is observed for six test units at each flow rate.

a. Does C2F6 flow rate affect etch uniformity? Use the analysis of variance, and a = 0.05?

b. Construct a box plot of the etch uniformity data. Use this plot, together with ANOVA results, to determine which gas flow rate would be best in terms of etch uniformity (a small % is best).

c. Plot the residuals vs. predicted C2F6 flow. Interpret this plot.

d. Does the normality assumption seem reasonable in the problem?

Problem 8

Observations

Flow 1 2 3 4 5 6

125 2.7 2.6 4.6 3.2 3.0 3.8

160 4.6 4.9 5.0 4.2 3.6 4.2

200 4.6 2.9 3.4 3.5 4.1 5.1

## Watch the video: Simple Statistical Instruments. Level of Measurement. Grade 7 Quarter 4 (October 2021).