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A random variable is a variable (typically represented by x) that has a single numerical value, determined by chance, for each outcome of a procedure.

A probability distribution is a description that gives the probability for each value of the random variable. It is often expressed in the format of a graph, table, or formula.

A discrete random variable has either a finite number of values or a countable number of values, where “countable” refers to the fact that there might be infinitely many values, but they can be associated with a counting process, so that the number of values is 0 or 1 or 2 or 3, etc.

A continuous random variable has infinitely many values, and those values can be associated with measurements on a continuous scale without gaps or interruptions.

Examples: 1.) Discrete: x = the number of eggs that a hen lays in a day. This is a discrete random variable because its only possible values are 0, or 1, or 2, and so on. No hen can lay 2.343115 eggs, which would have been possible if the data had come from a continuous scale. 2.) Continuous: x = the amount of milk a cow produces in one day. This is a continuous random variable because it can have any value over a continuous span. During a single day, a cow might yield an amount of milk that can be any value between 0 and 5 gallons. It would be possible to get 4.123456 gallons, because the cow is not restricted to the discrete amounts of 0,1,2,3,4, or 5 gallons.

Requirements for a Probability Distribution: 1.) Σ P(x) = 1 where x assumes all possible values. 2.) 0 ≤ P(x) ≤ 1 for every individual value of x.

Mean for a probability distribution:

Variance for a probability distribution:

Variance for a probability distribution (shortcut):

Standard deviation for a probability distribution:

Identifying Unusual Results with the Range Rule of Thumb…...

.... . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Limitations and Common Misinterpretations of Hypothesis Testing . . . . . . . . . . 1 1 6 10 15 17 Stat 3011 Chapter 9 CHAPTER 9: HYPOTHESIS TESTS Motivating Example A diet pill company advertises that at least 75% of its customers lose 10 pounds or more within 2 weeks. You suspect the company of falsely advertising the beneﬁts of taking their pills. Suppose you take a sample of 100 product users and ﬁnd that only 5% have lost at least 10 pounds. Is this enough to prove your claim? What about if 72% had lost at least 10 pounds? Goal: 9.1 Elements of a Hypothesis Test 1. Assumptions 2. Hypotheses Each hypothesis test has two hypotheses about the population: Null Hypothesis (H0 ): Alternative Hypothesis (Ha ): 1 Stat 3011 Chapter 9 Diet Pill Example: Let p = true proportion of diet pill customers that lose at least 10 pounds. State the null and alternative hypotheses for the diet pill example. 3. Test Statistic Deﬁnition: Test Statistic A test statistic is a measure of how compatible the data is with the null hypothesis. The larger the test statistic, the less compatible the data is with the null hypothesis. Most test statistics we will see have the following form: What does a large value of |T | reﬂect? NOTE: 2 Stat 3011 Chapter 9 4. p-value The p-value helps us to interpret the test statistic. Deﬁnition: p-value Assume H0 is true. Then the p-value is the probability...

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...say whether I was able to learn how to be a better teacher and what the teacher did that I could possibly use in the future. While analyzing and going through the process of this assignment it is helping realize how to become a better teacher as well. I would also like to get more comfortable and experience on using this template of the paper. Memories Of A Teacher My teacher, Mr. G, used many different instructional techniques and approaches to his lessons. Mr. G had taught me math for three years in a row, so I think that I have a good grasp on his approaches to the lessons that he would teach. He would assign many homework assignments, as well as in-class assignments, which helped me and other students understand and get practice with the lesson that we were learning. I think that with math having a lot of homework is a good thing. In my mind, the only way to learn how to do math is plenty of practice. The more you practice, the easier it will be. Mr. G would also have the students do some math problems on the chalk board or smart board to show the class and go over the corrections with the whole class so that everyone would understand the problem. Playing “racing” games also helped and added fun to the class. With the “racing” games, the students would get into groups and have to take turns doing problems on the chalk board and see who could get the correct answer first. It added fun and a little friendly competition to the class. It also helped the students want to......

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...Diana Garza 1-16-12 Reflection The ideas Stein presents on problem saving and just math in general are that everyone has a different way of saving their own math problems. For explains when you’re doing a math problem you submit all kinds of different numbers into a data or formula till something works or maybe it’s impossible to come up with a solution. For math in general he talks about how math is so big and its due in large measure to the wide variety of situations how it can sit for a long time without being unexamined. Waiting for someone comes along to find a totally unexpected use for it. Just like has work he couldn’t figure it out and someone else found a use for it and now everyone uses it for their banking account. For myself this made me think about how math isn’t always going to have a solution. To any math problem I come across have to come with a clear mind and ready to understand it carefully. If I don’t understand or having hard time taking a small break will help a lot. The guidelines for problem solving will help me a lot to take it step by step instead of trying to do it all at once. Just like the introduction said the impossible takes forever. The things that surprised me are that I didn’t realize how much math can be used in music and how someone who was trying to find something else came to the discovery that he find toe. What may people were trying to find before Feynmsn....

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...STAT 302 – Statistical Methods Lecture 8 Dr. Avishek Chakraborty Visiting Assistant Professor Department of Statistics Texas A&M University Using sample data to draw a conclusion about a population • Statistical inference provides methods for drawing conclusions about a population from sample data. • Two key methods of statistical inference: o o Confidence intervals Hypothesis tests (a.k.a., tests of significance) Hypothesis Testing: Evaluating the effectiveness of new machinery at the Bloggs Chemical Plant • Before the installation of new machinery, long historical records revealed that the daily yield of fertilizer produced by the Bloggs Chemical Plant had a mean μ = 880 tons and a standard deviation σ = 21 tons. Some new machinery is being evaluated with the aim of increasing the daily mean yield without changing the population standard deviation σ. Hypothesis Testing: Evaluating the effectiveness of new machinery at the Bloggs Chemical Plant Null hypotheses • The claim tested by a statistical test is called the null hypothesis. The test is designed to assess the strength of the evidence against the null hypothesis. Usually the null hypothesis is a statement of “no effect” or “no difference”, that is, a statement of the status quo. Alternative hypotheses • The claim about the population that we are trying to find evidence for is the alternative hypothesis. The alternative hypothesis is one-sided if it states that a parameter is larger than or...

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...Top Math 147 - Fall 1997 - Test 3 Name: __________________________________ Section Number: __________________ Show ALL your work. Solutions with no work where it is necessary will receive NO credit. If you need extra paper raise your hand and ask one of the proctors for some. A normal table is provided at the end of the test. Good Luck. For questions 1-10 circle the answer which best completes the sentence or answers the question. (3 pts each) 1. A fair coin is tossed one hundred times and the number of heads is recorded. The same coin is then tossed 1000 times and the number of heads is recorded. We expect, (a) the difference between 50 and the number of heads in the first trial to be larger than the difference between 500 and the number of heads in the second trial. (b) to get exactly 500 heads in the second trial. c. the chance error expressed as a percentage of the number of tosses to be smaller in the first trial than in the second trial. c. all of the above statements. c. none of the above statements. 2. A box contains 99 zeros and 1 one. If we make draws from this box with replacement, a. the probability histogram for the sum of the draws ( when put in standard units) will follow the normal curve after a small number of draws. a. then the probability histogram for the numbers in the box is close to the normal curve if the number of draws is very large. a. we can use the binomial formula to compute the chance of......

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...I would like to find out whether a Man's Name affects his credit score. I recorded the Credit scores readings of 25 men Named Joe, 22 men named Jon and 19 men named Mark; My data appears in the table to the right. I want to know whether the differences in the average readings are significant. Basically whether the average reading of all men named Joes is different from the average reading of all Jon' or whether these averages differ from the average reading of all Marks. Joe Jon Mark 650 680 650 630 650 570 630 660 580 650 690 520 670 550 420 440 480 650 550 440 660 549 350 670 490 650 660 465 670 650 440 690 570 470 680 550 490 650 680 650 660 690 660 650 700 665 690 650 670 720 660 665 550 670 545 650 660 550 660 590 670 655 650 660 670 680 Using the ANOVA formulas, we have: • I = 3; • n1 = 26, n2 = 23, and n3 = 20, and x1,1 = 650, x1,2 = 680, x1,3 = 650, x2,1 = 630, etc.; • N = 25 + 22 + 19 = 69; • AV1 = 594, AV2 = 253.7, AV3 = 242.5 (at least approximately -- all figures except the degrees of freedom are approximate from here on); • AV = [26(594) + 23(626.52) + 20(626)]/66 = 642.03 • SSE = (650 - 594)2 + (680 – 626.52)2 + (650 - 626)2 + (630-594)2 + . . . (a total of 66 terms) =448591.7 • SSG = 26(594 – 642.03)2 + 23(626.52 –......

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...Manlapid, Charles Eugene Cervantes, Miguel Ruiz, Ramon De Guzman, Laurence Nepomuceno, Jeremy LOCAL AUTHORS Advantages of Going to a Shopping Mall Author: Juan (not indicated) http://shoppingwithjuan.com/advantages-shopping-mall.html All under one roof and special offer is the idea that jumps to mind when you think of a shopping mall. Today, shopping malls are the trend and if you ask people the benefits that they enjoyed going to a mall, they will give so many of them. However, today, shopping is funnier than ever before. You can eat as you shop and you can get the widest variety of services and products in a mall. If you would like a big experience, the mall is the way to go. You can give your children the best experience, the mall is the way to go. You can give your children the best experience by......

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...This article is about the study of topics, such as quantity and structure. For other uses, see Mathematics (disambiguation). "Math" redirects here. For other uses, see Math (disambiguation). Euclid (holding calipers), Greek mathematician, 3rd century BC, as imagined by Raphael in this detail from The School of Athens.[1] Mathematics is the study of topics such as quantity (numbers),[2] structure,[3] space,[2] and change.[4][5][6] There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics.[7][8] Mathematicians seek out patterns[9][10] and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity for as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become......

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...Mathematics Question Paper Q1. Find all solutions of the following equation on the interval [0, 2π ) cos(x) + 1 = sin(x) Q2. Using analytical means find all the zeros of the polynomial function P(x) = 2x3 + x2 - 13x + 6 By the Rational Zeros Theorem the rational zeros of P are of the form possible rational zero of P =factor of constant term =factor of 6/factor of 2 factor of leading coefficient Q3. A survey of 80 college students was taken to determine the musical styles they listened to. 42 students listen to rock, 34 to classical and 27 to jazz. 12 students listened to rock and jazz, 14 to rock and classical, and 10 to classical and jazz. 7 students listened to all three types of music. Of those surveyed how many listened to at least two of the musical styles? Q4. How many license plates are possible in a state if a license plate has three letters of the alphabet followed by 4 numbers of which the first number cannot be a zero? Q5. Determine the convergence of the following integral. 4 ∫ x/ (x ^2 -9) dx 0 Q6. Determine whether the following sequence is arithmetic or geometric, and then find the sum of the first ten terms of that sequence. 4, - 12, 36, - 108, ... Q7. Find the inverse of the following function. 7 - 8x f(x) = --------------- 2x + 5 Q8 . Find the equation of a line that passes through the point (-7,8)......

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...printout given above), Explain (0, (1, also provide the units of slope and y-intercept. Does (0, (1 make sense? d) Is there sufficient evidence to conclude that the model contributes information for predicting the Percentage of refund spent in 3-months? (State Hypothesis, and do the test.) e) Is there sufficient evidence to conclude, "As the family income increases than the Percentage of refund spent in 3-month decreases? (State Hypothesis, and do the test.) (Does it make sense to do this test? Explain) f) Calculate R-sq, what is the practical meaning of R-sq? g) Calculate the Standard error of Estimate, What is the practical meaning of S(? (Get the residual printouts – 5 points) In Minitab, Goto Stat>regression>regression, then follow the screen prints below to get the residual plots. [pic] [pic] And click ok h) State the regression Assumption 1 and test it using the residual plots. i) State the regression Assumption 2 and test it using the residual plots. j) State the regression Assumption 3 and test it using the residual plots. k) State the regression Assumption 4 and test it using the residual plots. l) Calculate R-sq(adjusted). m) Find 95% Confidence Interval for (0 n) Find 95% Confidence Interval for (1 o) Explain the relationship between Confidence Interval and Hypothesis testing. p) What is an Outlier? Are there any outliers? q) What is an...

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...can see how to split up the original equation into its factor pair, this is the quickest and allows you to solve the problem in one step. Week 9 capstone part 1 Has the content in this course allowed you to think of math as a useful tool? If so, how? What concepts investigated in this course can apply to your personal and professional life? In the course, I have learned about polynomials, rational expressions, radical equations, and quadratic equations. Quadratic equations seem to have the most real life applications -- in things such as ticket sales, bike repairs, and modeling. Rational expressions are also important, if I know how long it takes me to clean my sons room, and know how long it takes him to clean his own room. I can use rational expressions to determine how long it will take the two of us working together to clean his room. The Math lab site was useful in some ways, since it allowed me to check my answers to the problems immediately. However, especially in math 117, it was too sensitive to formatting of the equations and answers. I sometimes put an answer into the math lab that I knew was right, but it marked it wrong because of the math lab expecting slightly different formatting Week 9 capstone part 2 I really didn't use center for math excellence because i found that MML was more convenient for me. I think that MML reassures you that you’re doing the problem correctly. MML is extra support because it carefully walks you through the problem visually......

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...use 0.75. Plotting the Binomial Probabilities 1. Create plots for the three binomial distributions above. Select Graph > Scatter Plot and Simple then for graph 1 set Y equal to ‘one fourth’ and X to ‘success’ by clicking on the variable name and using the “select” button below the list of variables. Do this two more times and for graph 2 set Y equal to ‘one half’ and X to ‘success’, and for graph 3 set Y equal to ‘three fourths’ and X to ‘success’. Paste those three scatter plots below. Calculating Descriptive Statistics Open the class survey results that were entered into the MINITAB worksheet. 2. Calculate descriptive statistics for the variable where students flipped a coin 10 times. Pull up Stat > Basic Statistics > Display Descriptive Statistics and set Variables: to the coin. The output will show up in your Session Window. Type the mean and the standard deviation here. Mean: 4.600 Standard deviation: 1.429 Short Answer Writing Assignment – Both the calculated binomial probabilities and the descriptive statistics from the class database will be used to answer the following questions. 3. List the probability value for each possibility in the binomial experiment that was calculated in MINITAB with the probability of a success being ½. (Complete sentence not......

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...and solve problems in everyday life”. In my everyday life I have to keep the balance in my check book, pay bills, take care of kids, run my house, cook, clean etc. With cooking I am using math, measuring how much food to make for four people (I still haven’t mastered that one). With bills I am using math, how much each company gets, to how much money I have to spare (which these days is not much). In my everyday life I do use some form of a math. It might not be how I was taught, but I have learned to adapt to my surroundings and do math how I know it be used, the basic ways, none of that fancy stuff. For my weakest ability I would say I fall into “Confidence with Mathematics”. Math has never been one of my favorite subjects to learn. It is like my brain knows I have to learn it, but it puts up a wall and doesn’t allow the information to stay in there. The handout “The Case for Quantitative Literacy” states I should be at ease with applying quantitative methods, and comfortable with quantitative ideas. To be honest this class scares the crap out of me, and I am worried I won’t do well in this class. The handout also says confidence is the opposite of “Math Anxiety”, well I can assure you I have plenty of anxiety right now with this class. I have never been a confident person with math, I guess I doubt my abilities, because once I get over my fears and anxiety I do fine. I just have to mentally get myself there and usually it’s towards the end of the class. There are......

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...You can use StatCrunch to assist with the calculations. A link for StatCrunch can be found under Tools for Success in Course Home. Here is also a link: http://statcrunch.pearsoncmg.com/statcrunch/larson_les4e/dataset/index.html. You can also find additional help on both confidence intervals and StatCrunch in the Online Math Workshop under Tab: “MAT300 Archived Workshops”. Specifically you will be looking for Hypothesis Tests and Using Technology – Hypothesis Testing. Submit your answers in Excel, Word or pdf format. Submit your file through the M&M® project link in the weekly course content. Be sure to state clear hypotheses, test statistic values, critical value or p-value, decision (reject/fail to reject), and conclusion in English (what does reject/fail to reject the null mean in terms of your hypotheses). When doing calculations for the color proportions, keep at least 4-6 decimal places sample proportions, otherwise you will encounter large rounding errors. Masterfoods USA states that their color blends were selected by conducting consumer preference tests, which indicated the assortment of colors that pleased the greatest number of people and created the most attractive overall effect. On average, they claim the following percentages of colors for M&Ms® milk chocolate candies: 24% blue, 20% orange, 16% green, 14% yellow, 13% red and 13% brown. 3 pts. Test their claim that the true proportion of blue M&Ms® candies is 0.24 at the 0.05 significance......

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...STAT 346/446 - A computer is needed on which the R software environment can be installed (recent Mac, Windows, or Linux computers are sufficient).We will use the R for illustrating concepts. And students will need to use R to complete some of their projects. It can be downloaded at http://cran.r-project.org. Please come and see me when questions arise. Attendance is mandatory. Topics covered in STAT 346/446, EPBI 482 Chapter 5 – Properties of a Random Sample Order Statistics Distributions of some sample statistics Definitions of chi-square, t and F distributions Large sample methods Convergence in probability Convergence in law Continuity Theorem for mgfs Major Theorems WLLN CLT Continuity Theorem Corollaries Delta Method Chapter 7 – Point Estimation Method of Moments Maximum Likelihood Estimation Transformation Property of MLE Comparing statistical procedures Risk function Inadmissibility and admissibility Mean squared error Properties of Estimators Unbiasedness Consistency Mean-squared error consistency Sufficiency (CH 6) Definition Factorization Theorem Minimal SS Finding a SS in exponential families Search for the MVUE Rao-Blackwell Theorem Completeness Lehmann-Scheffe Location and scale invariance Location and scale parameters Cramer-Rao lower bound Chapter 9 - Interval Estimation Pivotal Method for finding a confidence interval Method for finding the “best” confidence interval Large sample confidence......

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