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“Formal sample size issues are important only in academic (enumerative) statistics. In analytic statistics, the answer to “How much data?” is “Enough to characterize the underlying situation.” Why is this the case? How is academic or research statistics fundamentally different from an analytic situation such as this?

Formal sample size issues are essential only in academic (enumerative) statistics, while in analytic statistics, the answer to “How much data?” is “Enough to characterize the underlying situation.” This is because, in enumerative statistics, the elements of the population are well-defined and constant in the study. Furthermore, according to Deming,...

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Compare univariate and bivariate analysis.

It is the task of descriptive statistics to identify key trends specific to the current data set. More specifically, if the researcher has a data set for analysis that the hypothesis suggests may be causally related, it is critical to understand the number of variables used for such analysis. For...

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Compare dependent and independent variables.

It is crucial to understand the difference between dependent and independent variables when it comes to statistical analysis. Although both metrics are only characteristics of a particular distribution of a single variable (“Age” or “Height”), an idea of what exactly is an independent variable for a test makes sense. The...

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Discuss logical and linear regression.

Regression is commonly referred to as a statistical causal analysis in which a potential relationship between variables is identified. In this case, it is evaluated to what extent one variable can influence the other. At the same time, the direction of this influence is determined. For example, regression analysis can...

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Statistical Evidence Is Not Necessarily the Same as Truth

Discuss why it is important to recognize that statistical evidence is not necessarily the same as truth. Why do you think people are so quick to accept “statistics” as true? From a scientific point of view, is spinning the truth acceptable? Is this practice OK from a critical approach, in...

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The Impact of Integration Technology in Teaching Discrete Mathematics

Investigate the impact of integration technology in teaching discrete mathematics among business information technology students at university. To answer this question, establish the relationship between technology and university discrete mathematics students’ performance. Find out the teaching strategies to be used to improve the integration of technology in teaching discrete mathematics.

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Mathematical Approach Analysis

Make a mathematical approach analysis. What were the overviewed simple problem-solving tasks focused on? Discuss reflecting on the task involving a composite figure. Evaluate the task involving a triangle. Analyze the task of finding the side length. Did the students who successfully completed the tasks show certain differences in their...

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A Confidence Interval in the News and Medical Journals

In everyday terms, a confidence interval is the range of values around a sample statistic (such as mean or proportion) within which clinicians can expect to get the same results if they repeat the study protocol or intervention, including measuring the same outcomes in the same ways. Find an example...

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Modeling the Relationship Between Advertisement Expenditures and Sales Revenues

Consider the dataset below and respond to the questions that follow.  Advertisement ($’000) Sales ($’000): 1068 4489, 1026 5611, 767 3290, 885 4113, 1156 4883, 1146 5425, 892 4414, 938 5506, 769 3346, 677 3673, 1184 6542, 1009 5088. Construct a scatter plot with this data. Do you observe a relationship between both...

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Linear Regression and Its Usefulness at Work

How can the linear regression be useful to you in your workplace or chosen major? Linear regression is used to predict the value of one variable from another variable. Since it is based on correlation, it cannot provide causation. In addition, the strength of the relationship between the two variables...

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“The Median Isn’t the Message” Article by Gould

Read the article “The Median Isn’t the Message” by Stephen Jay Gould. What are your immediate thoughts on this article? If the median isn’t the message, then what is Gould’s message? What is the main statistical concept addressed in this article? What are Gould’s personal feelings about statistics, as put...

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Additive and Multiplicative Models in Time Series Modeling

When do we use an additive model? When do we use a multiplicative model? Time series is particularly useful for tracking variables such as revenues, costs, and profits over time. Time series models help evaluate performance and make predictions. Time series decomposition seeks to separate the time series (Y) into...

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The Crucial Role of Mathematics in Both Business and Society

Analyze the crucial role of mathematics in both business and society. To answer this question, discuss how business math is useful in analyzing a company’s financial performance. Describe how to evaluate and improve the quality of the information in the face of uncertainty, present and clarify options, model available alternatives...

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There are the Five Process Standards from Principles and Standards for School Mathematics. In the article “No Tears Here! Third-Grade Problem Solvers” by Hartweg and Heisler, a problem-solving approach was presented that was implemented in three classrooms. Discuss (in detail) the similarities between the Standards for Mathematical Practice and the problem-solving approach implemented that was discussed in the article.

The presented problem-solving approach follows the Standards for Mathematical Practice. Firstly, both stress the importance of checking students’ comprehension of the task. Moreover, the described approach encourages students to be persistent and try various strategies. The students could use multiple tools and present their thinking processes in different forms. While...

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According to the article “No Tears Here! Third-Grade Problem Solvers” by Hartweg and Heisler since the students did not have much experience with problem-solving, what kinds of problems were chosen at first and why? After the first few rotations of the meeting to discuss various aspects of the problem-solving implementation in the classroom, what did the teachers and consultants discover about most of the problems they had chosen from the traditional textbook, and how did this affect future problems that were chosen?

At first, the teachers used simple problems so that children were motivated by their success and were more inclined to this type of task. Later, they presented more complex problems, similar to those the children would have to solve during state tests. The teachers discovered that most problems in the...

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In the article “No Tears Here! Third-Grade Problem Solvers” by Hartweg and Heisler, what three parts were the mathematics classes broken down into (using Van de Walle’s three teaching phases)? Describe (in detail) each phase and what took part in each phase.

The mathematics classes were divided into three phases: “before,” “during,” and “after.” The “before” phase was dedicated to introductory activities, such as asking anticipatory questions, raising the children’s interest and attention, presenting the problem and clarifying whether everything was clear. The “during” phase included working in pairs or small groups...

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In the article “No Tears Here! Third-Grade Problem Solvers” by Hartweg and Heisler, did teachers evaluate students’ responses during the problem-solving phase as right or wrong? Why or why not?

The key focus of this phase is to allow students to brainstorm ideas freely without the pressure of being judged. In this way, they developed valuable argumentation, discussion, critical assessment, and self-correction skills. Therefore, teachers did not assess their responses as right or wrong but listened to the way students...

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Read the article “No Tears Here! Third-Grade Problem Solvers” by Kim Hartweg and Marlys Heisler. Due to the third graders’ struggle to write the mathematical steps and justify their answers, the teachers developed and tested two models to aid students in this process. Describe these two models. Which one would you choose to use with your students and why?

The first developed model was based on a “What?/Why?” T-chart. In the “What?” column, children wrote down all the steps they took, and in the “Why?” column, the justification for them. The alternative model suggested by the teachers implied using the “math house” scheme. Its “attic” was used for rephrasing...

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Read the article “No Tears Here! Third-Grade Problem Solvers” by Kim Hartweg and Marlys Heisler. Give at least three of the given changes in students’ thinking after integrating open-response problem-solving into the mathematics curriculum.

The students have less anxiety about approaching challenging problem-solving tasks. On the contrary, they now enjoy them and do not hesitate to try new methods when previously learned ones do not work. They became more confident and persistent in solving mathematical problems.

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Read the article “No Tears Here! Third-Grade Problem Solvers” by Kim Hartweg and Marlys Heisler. Briefly discuss the results from the student attitude assessment. What implications do these results have on you as a future educator?

After the project’s implementation, the students have a more positive attitude towards mathematics and problem-solving tasks. They do not give up after initial failures and keep trying new approaches. Many students find challenging tasks exciting rather than fearsome. The important idea I learned from this project is that the teacher’s...

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How to Teach Multiplication of Whole Numbers

How would you make the multiplication of whole numbers a lesson? In the answer, use your knowledge of mathematics unit planning and lesson planning. Based on the basic concept of multiplying, the knowledge of number principles, and the understanding of multiplication and adding. Describe Multiplication as repeated addition, multiplication of...

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Constructing Regular Polygons With 6, 8, or 10 Sides

How to construct one of the regular polygons with 6, 8, or 10 sides? Select one of the regular polygons with 6, 8, or 10 sides, and determine how to construct it using just a compass and a straight edge (you should not need to use any length measurements). The...

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Explain the difference between descriptive and inferential statistical methods and give an example of how each could help you draw a conclusion in the real world.

Descriptive statistics depict a set, while logical statistics take a sample of people for a specific model and generalize it to the entire group. Descriptive statistics are collected, for example, when local government elections take place. The inferential statistics are used to study the selected percentage of people from the...

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You would like to determine whether eating before bed influences your sleep patterns. Describe the steps you would take to conduct a statistical study on this topic. What is our hypothesis on this issue? What type of data will you be looking for? What methods would you use to gather information? How would the results of the data influence decisions you might make about eating and sleeping?

The hypothesis is that eating before going to bed harms the period of night sleep. It is necessary to use an inferential method and study scientific articles and statistical data on how many people of the same age, gender, and health status experience negative consequences from eating before sleep. I...

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A company that sells tea and coffee claims that drinking two cups of green tea daily has been shown to increase mood and well-being. This claim is based on surveys asking customers to rate their mood on a scale of 1-10 after days they drink or do not drink different types of tea. Based on this information, answer the following questions: How would we know if this data is valid and reliable? What questions would you ask to find out more about the quality or the data? Why is it important to gather and report valid and reliable data?

To understand how much this information can be trusted, it is necessary to study the results of the conducted research. Some data summaries show how many people were interviewed, their data, and answers to the survey questions. I would clarify for what period people have studied tea and its effect...

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Identify two examples of real-world problems that you have observed in your personal, academic, or professional life that could benefit from data-driven solutions. Explain how you wouId use data/statistics and the steps you would take to analyze each problem.

An urgent question is whether it is worth getting vaccinated against Covid-19. I will study the statistics of the most reliable sources, for example, the New York Times. According to statistics from the New York Times, 60.4% of Americans aged 18 to 65 are fully vaccinated. Based on these data,...

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What is the most common post hoc test for the ANOVA, and how does it work?

One of the most popular tests is the LSD, which compares the difference between the means. In this test, pairwise comparisons are made between the means, and confidence intervals are constructed for them. Ultimately, this test measures the validity of all confidence intervals.

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What is a post hoc test, and why is it important?

Post hoc tests are conducted after the ANOVA to determine the origin of the differences if they were found. So, if ANOVA shows differences between groups, post hoc shows precisely what the differences are. This is important because it allows for an in-depth analysis of differences and identifies variables that...

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Provide detailed explanations for the following comment: The one-way ANOVA uses the variance to compare the differences in the means between three or more groups.

ANOVA is a test used to detect a significant difference between three or more groups on an assessed parameter. Consequently, each group has some average, and an ANOVA test is conducted between groups to determine the significance of these differences. That is, how confident we can say that the averages...

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Why is a Z score a standard score? Why can standard scores be used to compare scores from different distributions? Why is it useful to compare different distributions?

A Z score is a standard score because it compares various variables through distribution standardization. This facilitates the establishment of how far a data point is from the mean. As such, scores above the mean possess positive z scores, and those below have negative z scores. However, since standard scores...

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Reflect on the video What Is a Triangle?

The teacher showed in the video What is a Triangle has an idea of how the development stages work, and it is noticeable that she has a lot of experience in teaching. The teacher adapts to the behavior of the students and selects new approaches or methods of teaching them....

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Explain the difference between a monomial, a binomial, and a trinomial.

Polynomials are described as algebraic expressions consisting of coefficients and indeterminate or variables combined by mathematical operations, which include multiplication, subtraction, addition, and division but not by a variable. The word polynomial consists of two Greek words: ‘poly,’ which means ‘many,’ and ‘nominal,’ which means ‘expressions or terms,’ and when...

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State the rules for multiplying integers.

Multiplication rules for numbers with the same sign. If the numbers are positive, then their product is found in the same way as natural numbers. To multiply two negative numbers, you need to put a plus sign and multiply the modules. This means that when multiplying numbers of the same...

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Define each of the following types of distributions and provide at least one example of data that would likely create the type of distribution: Normal Distribution; Positive Distribution; Negative Distribution; Bimodal Distribution; Inverted U-shaped Distribution.

Normal Distribution Known as the “bell curve,” it is symmetrical distribution, with values gradually rising to the mean value, which is the highest value as well, and gradually falling after it. Positive Distribution The distribution where data are skewed to the left (to the smaller value): its maximum is smaller...

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Define each of the following measures and provide at least one example of the most appropriate instance when to utilize the type of measure also include the appropriate Greek symbol for each: Mean; Median; Mode; Range; Standard Deviation.

Mean Depicted as μ, it is the average value of the sample of data. It is computed by the formula μ = (x1+x2+…+xn)/n. It is called an arithmetic mean: there are other mean values too, but the arithmetic mean is the most common. An example is an average mark on...

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Discuss the properties of polynomials.

A polynomial is a mathematical expression that consists of several components, or terms, containing both numerical and literal symbols and connected by addition. An essential rule of a polynomial is that all its parts cannot be added, which means that the polynomial is already represented in the simplest form. Thus,...

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Explain how problems can be solved using rational equations.

An equation, both of whose sides are rational expressions — in other words, it only includes numbers, literal symbols, arithmetical operations, and integral powers — is also rational. Solving a mathematical problem frequently presupposes deriving, then solving a rational equation. The student must first assign X as the independent variable....

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What is the difference between norm-referenced and criterion-referenced tests?

Norm-referenced test pertains to normalized tests that are intended to contrast and grade test-takers in comparison to one another. For example, it would reflect how many fewer or more correct answers a test taker gave compared to other test takers. On the other hand, criterion-referenced tests outcomes are founded on...

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What is probabilistic equivalence? Why is it important?

Probabilistic equivalence is a concept in experimental design methodology, which implies that the researcher is perfectly aware of the odds that he or she will find come across a pretest difference between the two groups under investigation. However, it does not necessarily mean that the two groups being studied will...

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Why would anyone ever want more than two levels of an independent variable?

Independent variable is that element of experimental research, which is manipulated or changed by the research in order to cause changes in other elements being studied. Therefore, the independent variable is one of the core or major components of experimental research. It is the variable that is altered or manipulated...

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What is NHST in statistics? Describe the assumptions of the model.

The NHST model is a statistic, such as a t-test, for determining whether a given variable causes the effect being tested. NHST tests often make certain assumptions. The NHST model (z-test and t-test) makes assumptions regarding the distribution of the tested variable. One such assumption relates to the normal distribution...

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What is modal logic?

Modal logic is an extension of formal logic that allows reasoning about certainties and possibilities. In comparison with classical logic, modal one considers that things may not always be as they are at that moment. For example, one may use modal logic by saying, “although the sun is usually yellow,...

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What are De Morgan’s Laws?

De Morgan’s Laws are a pair of transformation logical rules and are an example of mathematical duality. They can be expressed as: the negation of a disjunction is the conjunction of the negations while the negation of a conjunction is the disjunction of the negations.

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We want to find the average height of adult men in America. We choose a SRS of 36 adult American men and perform a 95% confidence interval. Using this procedure, we find that their average height is 71 in. +/- 0.52 inches. What is the margin of error of this confidence interval?

The margin of error shows how accurate one expects the value of the unknown parameter to be. The margin of error of the given confidence interval is equal to ± 0.52 inches. This means that the value of ± 0.52 inches is the maximum amount by which the sample results...

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You sample 100 apples from your farm’s harvest of over 200,000 apples. The mean weight of the sample is 112 grams. Let’s assume that you know the weight of all apples produced by your farm is normally distributed and the standard deviation of the weight is 40 grams. What is the 99% confidence interval for the mean weight of apples of your farm? If we want to shorten the length of the confidence interval by half while keeping the same confidence level at 99%, how many apples should we sample to begin with?

In order to shorten the length of the confidence interval by half without changing the confidence level, the value of (σ / (number of samples0.5)) should be halved. Given that σ is the constant value, the number of samples needs to be increased, so that (new number of samples0.5) /...

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Suppose you buy a case of 36 cans of soda and carefully measure the volume of each. You average the results and determine that the sample mean is 12.1 oz. Let’s assume that you know that the canning facility outputs cans whose volumes are distributed normally with a standard deviation of 0.24 oz. What is the 80% confidence interval of the mean of the volume? Is it larger or smaller than the 90% confidence interval. Does this make sense?

Since z* is equal to 1.282 for the 80% confidence interval, the 80% confidence interval of the mean of the volume is 12.1 ± 1.282 × (0.24 / (36 0.5)) = 12.1 ± 0.05128. It can be stated that the calculated interval is smaller than the 90% confidence interval, with...

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The mean income of households in Utica is $50,000 with a standard deviation of $10,000. Can we use Table A, the Standard Normal table, to estimate the probability that a single randomly chosen Utica household will have an income of less than $52,500? Why or why not?

Given that income distribution in Utica is normal, it is possible to use Table A to estimate the probability that a single randomly chosen Utica household will have an income of less than $52,500. This is because the value of $52,500 can be standardized, and the cumulative proportion for the...

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The total amount of money spent by shoppers in one visit to Wainwright’s department store is normally distributed with a mean of $50 with a standard deviation of $10. Use the Standard Normal table to compute the following probability: What is the probability that a shopper chosen at random will spend less than $47 during their next visit to Wainwright’s?

In order to use Table A, a standardized amount of money spent by shoppers should be calculated. For a shopper who spends $47 during one visit to Wainwright’s, this standardized amount of money equals z = (47 – 50) / 10 = -0.3. The cumulative proportion for z is 0.3821....

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View the Khan Academy video on the Central Limit Theorem and comment on it.

The Central Limit Theorem is a profound concept in statistics that makes statistically significant inferences about the general population based on samples. For example, a variable has a normal distribution (or any other type of distribution). One repeatedly takes samples from this general population and calculates the mean of each...

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Give an example of a survey (real or invented) that is poorly designed because it contains one or more type of bias. State clearly what kinds of bias the survey has.

The name of the store and the following example of a survey were invented by the author. Kaufmann’s is a large department store that offers expensive clothing for high-income customers, both men and women. A limited collection of jewelry was launched for the first time, and a survey was designed...

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Find an example in your environment (in a newspaper article, a TV news show, a political campaign, etc.) in which two variables are correlated and the author/presenter/speaker is implying that one of these variables causes the other. Comment on the strength of the evidence that causation is really present. Can you think of another factor (a confounding or lurking variable) that might be the underlying cause of both variables?

In order to demonstrate the example when correlation does not imply causation, the article discussing the link between soft drinks and aggression in children was chosen. It is stated that heavy soda drinkers are much prone to violent behavior than other children. For this particular research, the intake of soft...

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Take a look at the career batting averages of four of the greatest baseball players of all time. When watching a game many people will say that a given player is “due” for a home run or a hit after a period of uncharacteristically low offense. Let’s say Willie Mays is in a slump – he has made an out the last 10 times at bat. What are his chances of getting a hit? Is he “due” for a hit? Why or why not?

The chance of a hit in ideal conditions is 0.5. The player can either hit the ball and make a hit or miss with equal probability. In this sense, it is wrong to say that he must make a hit, but he can make a hit with a chance of...

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Take a look at the career batting averages of four of the greatest baseball players of all time. In terms of the law of large numbers, what can we assume about how accurately these career averages reflect the actual abilities of the players?

The AVG parameter shows the success of the batting during a baseball game. Having discarded all other factors, the probability of hitting the ball with a bat is 0.5. From the law of large numbers, this parameter does not say anything about the likelihood of a successful strike during the...

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The probability of tossing a coin and getting heads is 0.5 or 50%. Try this several times by hand, or use the GeoGebra Simulation. Imagine the simulation below, and that you were going to flip the coin 100 more times. Will you get 50 more heads and 50 more tails? Why or why not?

In fact, according to Bernoulli’s law, the average result will tend to be 0.5, as the number of shots has increased, but the number of added goals and tails will not be the same. The intuitive answer that there will be 50 goals and 50 tails is wrong because it...

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The probability of tossing a coin and getting heads is 0.5 or 50%. Try this several times by hand, or use the GeoGebra Simulation. Imagine the simulation below, and that you were going to flip the coin just one more time. What is the probability you will get heads? What is the probability you will get tails?

It is a mistake to assume that the probability of heads or tails falling out during a throw depends on previous results. Despite the last 108 throws, the likelihood of a head fall for a symmetrical coin will still be 0.5. But it is worth distinguishing the probability of a...

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Suppose a large class at Utica College has just given an exam and I picked 20 students at random from the class and asked each person their score on the exam. Here are their replies: 100, 98, 94, 94, 93, 93, 92, 90, 86, 86, 85, 84, 84, 78, 77, 73, 72, 68, 55, 12. Calculate the median of this data set.

To find the median, the data should be arranged in order from least to greatest: 12, 55, 68, 72, 73, 77, 78, 84, 84, 85, 86, 86, 90, 92, 93, 93, 94, 94, 98, 100. Since there is an even number of items, the median is calculated as the arithmetical...

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Explain and evaluate the views of Pythagoras regarding the nature of substance.

Pythagoras was another philosopher interested in discovering the nature of the substance. In his opinion, mathematics (more specifically numbers) is the answer to the question. Since numbers cannot be mistaken, understanding them would provide knowledge about the nature of the substance. He is widely known for his studies of geometry...

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The manager of a company wants to investigate job satisfaction among its employees. One morning after a meeting, she talks to all 25 employees who attended. Does this sampling plan result in a random sample? What type of sample is it? Explain.

This is not a random sample since employees who did not attend the meeting did not have a chance to be selected. Instead, this is convenience sampling since subjects were chosen only because the manager could easily access them.

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A phone company obtains an alphabetical list of names of homeowners in a city. They select every 25th person from the list until a sample of 100 is obtained. They then call these 100 people to advertise their services. Does this sampling plan result in a random sample? What type of sample is it? Explain.

This sampling plan does not result in a random sample since individuals do not have an equal chance to be chosen. In fact, individuals that are at the end of the list may be not even reached if there are more than 2,500 people living in the city. Since the...

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Describe how you could use regression analysis to help make a decision in your current job, a past job, or a life situation. Include a description of the decision, what would be the independent and dependent variables, and how data could ideally be collected to calculate the regression equation.

I could think of one way of using a simple linear regression model to help me make better health choices. Some time ago, I stumbled upon a piece of research that showed that regular exercise correlated with better sleep quality. While it is now pretty established that physical activity improves...

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Describe how you could use confidence intervals to help make a decision in your current job, a past job, or a life situation. Include a description of the decision, how the interval would impact the decision, and how data could ideally be collected to determine the interval.

Everyday use of the concept of confidence intervals that I can think of is buying clothes. It is common knowledge that sizing is often arbitrary and depends on the company. Sometimes, however, I think it is neither negligence nor malicious intent. People come in many shapes and sizes, and it...

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