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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,...

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...

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...

Define statistics.

Statistics deals with gathering, grouping, examining, and interpreting numerical values based on their measurable probability. Thus, large data can be analyzed using the measures of central tendency. The information collected can be categorized as quantitative and qualitative data.

Invent a chant or rap to learn the multiplication tables.

To help students remember the multiplication table better, one may use the following chant: Multiplying is so fun!Let’s revise it one by one:(the rhythm slightly changes, the beat becoming more pronounced):Two (beat) times two (beat) is (pause) four,Two (beat) times three (beat) is (pause) six.The teacher should continue until the...

Work with one other person to look at the NCTM website. Find another math concept or skill you think could be taught through music. Collaborate to develop a strategy, and test it on children to see if it works. Reflect on the success of your idea(s).

During the collaboration with another student, it was theorized that the concept of fractions could be taught with the help of music. The approach based on using rhythm as the method of explaining fractions was used on fourth graders. The students seemed to have acquired a better idea of fractions...

Try a strategy of teaching math using music with children at an appropriate age level. Reflect on what happened in terms of children’s engagement and learning.

The strategies implying the integration of music into the process of teaching math to students seem to be very promising as far as their effect on students’ critical thinking is concerned. Therefore, the application of the strategy that involves computing fluency was utilized in the classroom. A drum was used...

What are some authentic connections between math and music?

The connections between math and music might seem elusive at first, yet a closer analysis of the specified subjects will show that music can help students understand math better. For example, learners can use the music beat to understand the idea of division. The process of building fluency in third-to–sixth-graders...

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...

Your babysitter claims that she is underpaid given the current market. Her hourly wage is $12 per hour. You do some research and discover that the average wage in your area is $14 per hour with a standard deviation of 1.9. Calculate the Z score and use the table to find the standard normal probability. Based on your findings, should you give her a raise? Explain your reasoning as to why or why not.

The z-score is equal to: z = (12-14)/1.9 = -1.05 Given that the right-tailed p-value for this z-score is equal to 0.85, 85% of babysitters are paid more than my babysitter, which is why it is feasible to give her a rise.

Cough-a-Lot children’s cough syrup is supposed to contain 6 ounces of medicine per bottle. However, since the filling machine is not always precise, there can be variations from bottle to bottle. The amounts in the bottles are normally distributed with σ = 0.3 ounces. A quality assurance inspector measures 10 bottles and finds the following (in ounces): 5.95 6.10 5.98 6.01 6.25 5.85 5.91 6.05 5.88 5.91. Are the results enough evidence to conclude that the bottles are not filled adequately at the labeled amount of 6 ounces per bottle? a. State the hypothesis you will test. b. Calculate the test statistic. c. Find the P-value. d. What is the conclusion?

a. H0: the mean volume of all the bottles is 6 ounces. Ha: the mean volume of all the bottles is not equal to 6 ounces. b. The z-score is equal to: z = ((x-μ)/√(σ2/N)) = 5.989-(6/√(0.09/10)) = -0.115 c. The p-value is equal to 0.907.d. The null hypothesis should...

Drug A is the usual treatment for depression in graduate students. Pfizer has a new drug, Drug B, that it thinks may be more effective. You have been hired to design the test program. As part of your project briefing, you decide to explain the logic of statistical testing to the people who are going to be working for you. a. Write the research hypothesis and the null hypothesis. b. Then construct a table like the one below, displaying the outcomes that would constitute Type I and Type II errors. c. Write a paragraph explaining which error would be more severe, and why.

a. Ha: the effectiveness of drug B is greater than the effectiveness of drug A; H0: the effectiveness of drug B is equal to the effectiveness of drug A.b. Table 1 presents the outcomes of type I and type II errors. – – Reality Reality – – Drug B is...

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...

An education expert is researching teaching methods and wishes to interview teachers from a particular school district. She randomly selects 10 schools from the district and interviews all of the teachers at the selected schools. 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 because subgroups were randomly selected, and participants were not selected randomly. This is cluster sampling because the entire population was divided into internally heterogeneous groups (clusters), and several clusters were chosen for analysis using simple random sampling.

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...

Describe how you could use hypothesis testing 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 null and alternative hypotheses, and how data could ideally be collected to test the hypotheses.

When I waited tables a few years ago, I was often curious about the average number of restaurant visitors depending on the day of the week. The information was of value to me because I wanted to know what to expect from a shift, and some of them were really...

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’...