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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...

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...

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...

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...

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...

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,...

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.

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...

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...

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...

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....

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...

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...

Explain the difference between an expression and an equation.

A mathematical phrase that groups numbers, variables, and operators to display the meaning of something is called an expression. An equation can be described as a statement with two expressions equal to each other. An expression is a piece of a sentence that represents a single numeric value. In contrast,...

Briefly explain why the scale of measurement is important and relevant to statistics.

The scales of measurement enable to manipulate data more precisely: to compare them, to see differences and similarities between them. Different scales are helpful for different purposes: the nominal scale is useful when the task shows the properties of the objects in the sample, while an interval or ratio scale...

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...

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...

Explain arithmetic operations with whole numbers, integers, fractions, and decimals.

There are four basic arithmetic operations, notably addition, subtraction, multiplication, and division. With whole numbers that are also referred to as integers, those mean calculating “the number of individual items” based on adding them to one another or, contrariwise, dividing into groups. For example, in arithmetic operations with integer data,...

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,...

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....

The trajectories of fireworks are modeled by quadratic equations. The equations can be used to predict the maximum height of a firework and the number of seconds it will take from launch to explosion. Discuss the properties of quadratic equations and how they are applied as models in various situations.

As the left side of a quadratic equation contains both the first and second power of the same variable, it is apparently possible to factor. The two resulting expressions need to equate to zero and solving; the solutions will be the roots of the initial one. Graphically, a quadratic equation...

How does focusing on the process rather than the end product improve output?

Since a process entails a series of activities, focusing on the process by controlling each of the activities to provide the correct input would improve output. In contrast, focusing on the end-product does not improve output because it only gives a cumulative output of numerous activities, and thus, difficult to...

In quantitative research, explain 3 types of non-experimental research designs with some detail and examples if possible.

Types of non-experimental research designs: Descriptive: These designs explain a phenomenon. Examples include graphs, averages, and percentages. Relationships design: They describe the correlation between variables (two or more). Survey designs: They represent data gathering techniques essential in correlational and descriptive studies due to versatility, generalizability, and efficiency. Examples include cross-sectional...

Explain the Likert scale, questionnaire, checklist.

The Likert Scale This is a scaling method for two extremities, which measure either negative or negative answers to a statement. Questionnaire This is a research tool that consists of a series of questions and other requirements with the aim of collecting information from participants. Checklist This is a listing...

Choose and explain 5 threats to the validity of qualitative research.

Threats to the validity of qualitative research: Descriptive validity: This refers to what the researcher is unable to document while collecting information. Interpretation validity: The researcher requires capturing the study as interpreted by the person being researched. Researcher bias: Though every researcher is bound to be biased, the bias should...

Summarize what an ANOVA F-test can and cannot tell you about the population means.

An ANOVA F-test can tell about the extent of variation among population means. A large F-ratio implies that the variation of population means among groups is greater than the variation of population means within groups. In essence, ANOVA F-test can tell if the variability among population means due to treatment...

Explain main effects and interactions as related to two-way ANOVA.

In two-way ANOVA, the main effects indicate the unique effect of each independent variable on a dependent variable, controlling for other independent variables. Interactions are the combined effects of independent variables on the dependent variables. The interaction effect shows if independent variables act independently or synergistically to produce the apparent...

Explain the difference between capability and control.

A capability is the potential of a process to meet required standards while control comprises activities performed to ensure that a process is predictable and consistent with required standards.

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...

Find and state a definition of parametric and non-parametric methods that distinguishes between the two. In your own words, explain the difference between parametric and non-parametric methods.

Parametric and non-parametric statistical methods denote two general classifications of statistical procedures. Parametric tests are grounded in the assumptions about the distribution of the fundamental population, which provided the sample. The most widely spread parametric assumption implies that data are divided in an approximately normal manner. Non-parametric tests are not...

How do I interpret measures of central tendency as well as measures of dispersion when dealing with various variables that have continuous data in SPSS, and how do these measures affect my results?

Measures of central tendency and dispersion attempt to describe data that is currently available in the SPSS dataset, with the most commonly used ones being the mean, median, and mode for measures of central tendency, and the standard deviation, variance, and range for measures of dispersion. Owing to the fact...

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...

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...

What is error variance and how is it calculated?

Error variance is an important concept in experimental design methodology. It is used to describe the differences that exist between different variables in the study. In more specific terms, error variance is the variability that cannot be accounted for in experimental research using the systematic differences that the researcher is...

What is an F-ratio? Define all the technical terms in your answer.

F-ratio is a statistical concept in experimental research design, which helps to assess whether variance in two different samples is equal. It is also used to determine the variance within groups of sample data as well as the variance between two groups of data. The F-ration is mostly applied in...

What does p = .05 mean? What are some misconceptions about the meaning of p =.05? Why are they wrong? Should all research adhere to the p = .05 standard for significance? Why or why not?

The p-value in research describes the probability that the H0 is true at a given level of significance. Thus, p = .05 means that, assuming a correct H0, the probability that the data (dependent variable) are significant is 0.05. If p < 0.05 (at 95% confidence), the H0 is rejected....

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...

What do inferential statistics allow you to infer?

Inferential statistics allow a researcher to draw conclusions regarding the population based on the sample data. Thus, they enable the researcher to generalize the study’s findings to other populations. Inferential statistics determine the characteristics of the population based on the sample as well as the cause-and-effect relationship between the IV...

What is a prior probability?

Prior probability is the estimated probability of the state of affairs before a practical survey is conducted to acquire relevant details. This is usually a conclusion based on past events. For instance, Peter passed the test (A) last year; therefore, the probability P (Pc) he will pass test C is...

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,...

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.

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...

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) /...

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...

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...

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....

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...

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...

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...

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...

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...

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...

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...

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...

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...

Dataset below gives a picture of the first 15 values in a dataset that contains information on students within a particular university. Which are the variables (or ‘fields’) and which are the cases? Suggest two questions that we could possibly answer through an analysis of the data in dataset.

From the data given, the age of the students would be the variables since variables represent data that cannot be measured empirically. The cases would be the qualification entry requirements to the university since it is the subject matter of the analysis, and unlike the age of the students, it...

Tutor O-Rama claims that their services will raise student SAT math scores at least 50 points. The average score on the math portion of the SAT is μ=350 and σ=35. The 100 students who completed the tutoring program had an average score of 385 points. Is the students’ average score of 385 points significant at the 5% and 1% levels to support Tutor O-Rama’s claim of at least a 50-point increase in the SAT score? Is the Tutor O-Rama students’ average score of 385 points significantly different at the 5% and 1% levels from the average score of 350 points on the math portion of the SAT? What conclusion can you make, based on your results, about the effectiveness of Tutor O-Rama’s tutoring?

a. The average score of 385 points should be compared to the score of 400 points, considering that the tutor claims that their services will increase the average score of 350 by at least 50. z = ((x-μ)/√(σ2/N)) = ((385-400)/√(352/100)) = -4.285 The p-value for the z-score is 0.000009, which...

You want to investigate the workplace attitudes concerning new policies that were put into effect. You have funding and support to contact at most 100 people. Choose a design method and discuss the following: a. Describe the sample design method you will use and why. b. Specify the population and sample group. Will you include everyone who works for the company, certain departments, full or part-time employees, etc.? c. Discuss the bias, on the part of both the researcher and participants.

a. It is a good idea to choose stratified random sampling since it yields more accurate results, as compared to simple random sampling. Workers will be divided into groups by their position. Then, depending on the number of groups, a certain number of workers will be randomly chosen from each...

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.

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...

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...

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...

After several grueling months of data collection and statistical analysis, you finally were able to test the following hypothesis: Null hypothesis: There is no relationship between yearly changes in Gross Domestic Product (GDP) and yearly changes in the national deficit. Alternative hypothesis: There is a relationship between yearly changes in Gross Domestic Product (GDP) and yearly changes in the national deficit. Your results told you to reject the null hypothesis (Χ2(4) = 2.35, p = 0.012). However, what did the p-value tell you about the results of the hypothesis test?

Scholars emphasize the significance of the p-value in scientific research and statistics. P-value is the probability of receiving the result that is at least as extreme as the one mentioned in the sample data with the assumption that the null hypothesis is true. When the p-value is less than the...

a. Show how you can use numbers, tables and graphs to describe a single set of values. b. Show how you can use numbers, tables and graphs to describe an association between two sets of values.

Numbers, tables, and graphs are very useful when describing a single set of values or showing an association between two sets of values. A single set of values are used in everyday life. For instance, the teaching staff is fond of using single sets of values when evaluating the learners’...