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E.g C = {4, 2, 1, 3} and D = {blue, white, red}are sets of natural numbers and set of colors respectively.

b) Element of Set: A number, letter, item or any other object contained in a Set is called Element of a Set. In a) above, elements of Set C are 1,2,3 and 4.

c) Order of a Set: are special binary relations. Suppose that P is a set and that ≤ is a relation on P, Then ≤ is a partial order if it is reflexive, antisymmetric and transitive.

d) Null Set: is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Common notations for the empty set include "{}", " ", and " ".

e) Finite Set: a finite set is a set that has a finite number of elements. For example,

is a finite set with five elements. The number of elements of a finite set is a natural number (non-negative integer),and is called the cardinality of the set. A set that is not finite is called infinite. For example, the set of all positive integers is infinite:

f) Proper Subset: A proper subset is a grouping of numbers in which all the numbers for two quantities have the same numbers, but are not equal.

g) Data: are values of qualitative or quantitative variables belonging to a set of items.

h) Statistics: Statistics is a branch of mathematics that deals with the collection, organization and interpretation of data.

i) Probability: is a measure or estimation of how likely it is that something will happen or that a statement is true. Probabilities are given a value between 0 (0% chance or will not happen) and 1 (100% chance or will happen). The higher the degree of probability, the more likely the event is to happen, or, in a longer series of samples, the greater the number of times such event is expected to happen.

j) Event: is a set of outcomes to which a…...

...Notes for Statistics 3011 University of Minnesota Fall 2012 Section 010 Instructor: Shanshan Ding Notes accompany the Third Edition of Statistics: The Art and Science of Learning From Data by Alan Agresti and Christine Franklin Contents CHAPTER 9: HYPOTHESIS TESTS 9.1 Elements of a Hypothesis Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Normal Hypothesis Test for Population Proportion p . . . . . . . . . . . . . . . . . . 9.3 The t-Test: Hypothesis Testing for Population Mean µ . . . . . . . . . . . . . . . . . 9.4 Possible Errors in Hypothesis Testing . . . . . . . . . . . . . . . . . . . . . . . . . . 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...

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...Lecture 1 Examples. STAT 102 In Exercises 1-13, identify which of these types of sampling is used: random, stratified, systematic, cluster, or convenience. 1. 2. 3. 4. 5. 6. When she wrote Marriage and Divorce: Legal and Psychological Issues, author Julia Kim based her conclusions on 4500 responses from 100,000 questionnaires distributed to women. A psychologist at the University of Saskatchewan surveys all students from each of 20 randomly selected classes. A sociologist at Grant MacEwen Community College selects 12 men and 12 women from each of 4 Statistics classes. Sony selects every 200th compact disc from assembly line and conducts a thorough test of quality. A gun registry lobbyist writes the name of each Member of Parliament on a separate card, shuffles the cards, and then draws 10 names. Due to a number of factors, a real estate broker classifies his clients as: upper-class Protestants, middle-class Protestants, lower-class Protestants, upper-class Catholics, etc. Over last years he had about 1200 clients from 15 different groups. Trying to analyze the tendency he randomly selected 3 clients from each group. A fashion expert polls online 50 males and 50 females about their brand of clothing. An Air Canada market researcher interviews all passengers on each of 10 randomly selected flights. A medical researcher from Acadia University interviews all leukemia patients in each of 20 randomly selected hospitals. A reporter for the Financial Post interviews every 25th...

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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...

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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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...x | y | x-xbar | y-ybar | (x-xbar)(y-ybar) | 14 | 2000 | -0.3 | -813.466667 | 244.0400001 | 19 | 9980 | 4.7 | 7166.533333 | 33682.70667 | 20 | 1000 | 5.7 | -1813.466667 | -10336.76 | 9 | 1000 | -5.3 | -1813.466667 | 9611.373335 | 15 | 1560 | 0.7 | -1253.466667 | -877.4266669 | 14 | 6559 | -0.3 | 3745.533333 | -1123.66 | 20 | 800 | 5.7 | -2013.466667 | -11476.76 | 5 | 908 | -9.3 | -1905.466667 | 17720.84 | 14 | 2800 | -0.3 | -13.466667 | 4.0400001 | 21 | 1800 | 6.7 | -1013.466667 | -6790.226669 | 4 | 1325 | -10.3 | -1488.466667 | 15331.20667 | 3 | 1000 | -11.3 | -1813.466667 | 20492.17334 | 16 | 1100 | 1.7 | -1713.466667 | -2912.893334 | 18 | 3261 | 3.7 | 447.533333 | 1655.873332 | 13 | 10282 | -1.3 | 7468.533333 | -9709.093333 | 10 | 8422 | -4.3 | 5608.533333 | -24116.69333 | 10 | 5500 | -4.3 | 2686.533333 | -11552.09333 | 9 | 1178 | -5.3 | -1635.466667 | 8667.973335 | 14 | 1254 | -0.3 | -1559.466667 | 467.8400001 | 6 | 8 | -8.3 | -2805.466667 | 23285.37334 | 12 | 3740 | -2.3 | 926.533333 | -2131.026666 | 35 | 6000 | 20.7 | 3186.533333 | 65961.23999 | 15 | 300 | 0.7 | -2513.466667 | -1759.426667 | 20 | 500 | 5.7 | -2313.466667 | -13186.76 | 8 | 3000 | -6.3 | 186.533333 | -1175.159998 | 16 | 1200 | 1.7 | -1613.466667 | -2742.893334 | 20 | 4241 | 5.7 | 1427.533333 | 8136.939998 | 14 | 1589 | -0.3 | -1224.466667 | 367.3400001 | 7 | 1497 | -7.3 | -1316.466667 | 9610.206669 | 28 | 600 | 13.7 | -2213.466667 | -30324.49334...

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...Lab 3 Due on Monday, June 23 FROM DATA TO DECISION Listed below are the ages of actresses and actors at the times that they won Oscars for the categories of Best Actress and Best Actor. The ages are listed in chronological order by column, so that corresponding locations in the two tables are from the same year. (Notes: In 1968 there was a tie in the Best Actress category, and the mean of the two ages is used; in 1932 there was a tie in the Best Actor category, and the mean of the two ages is used. These data are suggested by article “Ages of Oscar-winning Best Actors and Actress,” by Richard Brown and Gretchen Davis, Mathematics Teacher magazine. In that article, the year of birth of the award winner was subtracted from the year of the awards ceremony, but the ages in the tables below are based on the birth date of the winner and the date of the awards ceremony.) Analyzing the Results 1. Go to MyStatLab → Statcrunch → StatCrunch website → Open StatCrunch and will take you to the spreadsheet and use Data to load your data in excel onto the spreadsheet, Graph for all graphs, Stat for all analysis use it to answer question 2 to 4. Copy and paste all graphs and statcrunch output for full credit. 2. First explore the data using suitable statistics and graphs such as histogram, boxplot, etc. Use the results to make info In the histogram for actress mostly actresses received the Oscar in the age group between 20-40. The maximum number of actresses......

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...1.Assuming that the distribution is normal for weight relative to the ideal and 99% of the male participants scored between (-53.68,64.64), Where did 95% of the values for weight relative to the ideal lie? Round your answer to two decimal places. X =5.48, SD = 22.93 5.48 – 1.96(22.93) = AND 5.48 + 1.96(22.93) = 5.48 – 44.9428 = AND 5.48 + 44.9428 = -39.4628 AND 50.422 (-39.46, 50.42) 2.Which of the following values from Table 1 tells us about variability of the scores in a distribution? a.60.22 b.11.94 c.22.57 d.53.66 C. 22.57 3.Assuming that the distribution for General Health Perceptions is normal, 95% of the females’ scores around the mean were between what values? Round your answer to two decimal places. X = 39.71, SD = 25.46 39.71 – 1.96(25.46) = AND 39.71 + 1.96(25.46) = 39.71 – 49.9016 AND 39.71 + 49.9016 -10.4916 AND 89.6116 (-10.49, 89.61) 4.Assuming that the distribution of scores for pain is normal, 95% of the men’s scores around the mean were between what two values? Round your answer to two decimal places. X = 52.53, SD = 30.90 52.53 – 1.96(30.90) AND 52.53 + 1.96(30.90) 52.53 – 60.564 AND 52.53 + 60.564 -8.034 AND 113.094 (8.03, 113.09) 5.Were the body image scores significantly different for women versus men? Provide a rationale for your answer. Body image scores (0–100 scale) were significantly higher for women (73.1 +/- 16.93) than men (60.2 +/- 16.98), as stated in the relevant study results...

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...CCST9039 Statistics and Our Society (2014-2015, 2nd Semester) Prof. W. K. LI Department of Statistics and Actuarial Science The University of Hong Kong Chapter 1: The Genesis of Statistics Part 1/2: The Genesis of Statistics Prof. W. K. LI (SAAS) CCST9039 (14-15, 2nd) Chapter 1, part 1/2 1 / 17 (I) The Genesis of Statistics What is Statistics ? Why bother ? Origin of the word same as “Status” = State i.e. Collection of ﬁgures that describe the situation of the state Ancient records of statistical activities: Babylon, the Old Testament, · · · · · · . Prof. W. K. LI (SAAS) CCST9039 (14-15, 2nd) Chapter 1, part 1/2 2 / 17 (I) The Genesis of Statistics From the book of “Numbers” in the Old Testament (1300BC?) “And the Lord spake unto Moses in the wilderness of Sinai, in the tabernacle of the congregation, on the ﬁrst day of the second month, in the second year . . . saying, Take ye the sum of all the congregation to the children, after their families, by the house of their fathers, with the number of their names, every male by their polls; From twenty years old and upward, all that are able to go forth to war in Israel; thou and Aaron shall number them by their armies” A Census ! Prof. W. K. LI (SAAS) ( ) (Latin – “censere” means to tax) CCST9039 (14-15, 2nd) Chapter 1, part 1/2 3 / 17 From Moore & Notz (2009) Prof. W. K. LI (SAAS) CCST9039 (14-15, 2nd) Chapter 1, part 1/2 4 / 17 (I) The Genesis of Statistics The...

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...1. The median is often a better representative of the central value of a data set when the data set: Source ------------------------------------------------- Top of Form * Is bimodal. * Has a high standard deviation. * Is highly skewed. * Has no outliers. 2. The data in the Excel spreadsheet linked below provide information on the nutritional content (in grams per serving) of some leading breakfast cereals. For which nutrients is the mean nutrient content per serving greater than the median nutrient content per serving? Breakfast Cereals Source ------------------------------------------------- Top of Form * Proteins only. * Complex carbohydrates only. * Both nutrients. * Neither nutrient. 3. The histogram below plots the carbon monoxide (CO) emissions (in pounds/minute) of 40 different airplane models at take-off. The distribution is best described as is: Source ------------------------------------------------- Top of Form * Uniform. * Heteroskedastic. * Normal. * Skewed right. 4. The histogram below plots the carbon monoxide (CO) emissions (in pounds/minute) of 40 different airplane models at take-off. Which of the following statements is the best inference that can be drawn from this histogram? Source ------------------------------------------------- Top of Form * The mean amount of carbon monoxide emissions is greater than the median amount of...

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...STAT 200: Introduction to Statistics Final Examination, Fall 2015 OL1/US1 Page 1 of 6 STAT 200 OL1/US1 Sections Final Exam Fall 2015 The final exam will be posted at 12:01 am on October 9, and it is due at 11:59 pm on October 11, 2015. Eastern Time is our reference time. This is an open-book exam. You may refer to your text and other course materials as you work on the exam, and you may use a calculator. You must complete the exam individually. Neither collaboration nor consultation with others is allowed. Answer all 25 questions. Make sure your answers are as complete as possible. Show all of your work and reasoning. In particular, when there are calculations involved, you must show how you come up with your answers with critical work and/or necessary tables. Answers that come straight from programs or software packages will not be accepted. If you need to use software (for example, Excel) and /or online or hand-held calculators to aid in your calculation, please cite the sources and explain how you get the results. Record your answers and work on the separate answer sheet provided. This exam has 200 total points. You must include the Honor Pledge on the title page of your submitted final exam. Exams submitted without the Honor Pledge will not be accepted. STAT 200: Introduction to Statistics 1. Final Examination, Fall 2015 OL1/US1 True or False. Justify for full credit. Page 2 of 6 (15 pts) (a) If the variance of a data...

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... profit. To solve this problem, regression analysis method is used. In regression analysis, all three variables are taken as independent variables and thus the equation is determined. In the second regression, the food sales are removed. Here the relation between only non-food sales and size of supermarket to the profit is understood. Results: Model 1: Regression with food sales, non-food sales and size as independent variable: SUMMARY OUTPUT | | | | | | | | | | | | | | | | | | Regression Statistics | | | | | | | | Multiple R | 0.992399 | | | | | | | | R Square | 0.984855 | | | | | | | | Adjusted R Square | 0.977283 | | | | | | | | Standard Error | 1.249868 | | | | | | | | Observations | 10 | | | | | | | | | | | | | | | | | ANOVA | | | | | | | | | | df | SS | MS | F | Significance F | | | | Regression | 3 | 609.527 | 203.1757 | 130.0599 | 7.56E-06 | | | | Residual | 6 | 9.373017 | 1.562169 | | | | | | Total | 9 | 618.9 | | | | | | | | | | | | | | | | | Coefficients | Standard Error | t Stat | P-value | Lower 95% | Upper 95% | Lower 95.0% | Upper 95.0% | Intercept | -10.1702 | 3.473129 | -2.92827 | 0.026346 | -18.6687 | -1.6718 | -18.6687 | -1.6718 | Food Sales (tens of thousands of dollars) | 0.027038 | 0.012041 | 2.245505 | 0.065847 | -0.00243 | 0.056501 | -0.00243 | 0.056501 | Nonfood Sales (tens of thousands of dollars...

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...Stay at Home Moms -Breaking the Stereotypes 9/13/2015 In today’s society, no longer is it common for a woman to make the decision to stay at home and school her children. There was a time when this was far from taboo, but in the year 2015, stay at home mothers have found themselves in situations where they have to prove their worth. Women in the SAHMs (Stay at home mothers) community are labeled as lazy, are said to have it easy, are dumb, moochers, and many other stereo types. In all actuality the life of a SAHM is quite the contraire. In the 20th century the term “house wife” was more so used than the term Stay at home mom. Now, in the 21st century we changed that term because it seemed as if we knew nothing but the walls inside our homes. We are married to our husbands, not our houses. Women in the 1950’s rose to the term house wife and wore it proudly. During this time, the women who worked outside of the home were looked at as some scientific phenomenon that we didn't understand. The role of women in the 1950’s was repressive and constrictive in many ways. Society placed high importance and many expectations on behavior at home as well as in public. Women were supposed to fulfill certain roles, such as a caring mother, a diligent homemaker, and an obedient wife. The perfect mother was supposed to stay home and nurture so society would accept them. (R.C.2005) “Most of the time, when people ask, I tell them that I work from home. Technically this is true, because...

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...The Islamic University of Gaza Faculty of Commerce Department of Economics and Political Sciences An Introduction to Statistics Course (ECOE 1302) Spring Semester 2011 Chapter 7 - Sampling and Sampling Distributions Practice Exam - Solution Instructors: Dr. Samir Safi Mr. Ibrahim Abed SECTION I: MULTIPLE-CHOICE 1. Sampling distributions describe the distribution of a) parameters. b) statistics. c) both parameters and statistics. d) neither parameters nor statistics. 2. The Central Limit Theorem is important in statistics because a) for a large n, it says the population is approximately normal. b) for any population, it says the sampling distribution of the sample mean is approximately normal, regardless of the sample size. c) for a large n, it says the sampling distribution of the sample mean is approximately normal, regardless of the shape of the population. d) for any sized sample, it says the sampling distribution of the sample mean is approximately normal. 3. Which of the following statements about the sampling distribution of the sample mean is incorrect? a) The sampling distribution of the sample mean is approximately normal whenever the sample size is sufficiently large ( n ≥ 30 ). b) The sampling distribution of the sample mean is generated by repeatedly taking samples of size n and computing the sample means. c) The mean of the sampling distribution of the sample mean is equal to µ . d) The standard deviation of the sampling...

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...2.a.) > f<-read.csv("fish.csv",header=T) > attach(f) > m1<-glm(count~persons+child+factor(camper),family=poisson) > summary(m1) Call: glm(formula = count ~ persons + child + factor(camper), family = poisson) Deviance Residuals: Min 1Q Median 3Q Max -6.8096 -1.4431 -0.9060 -0.0406 16.1417 Coefficients: Estimate Std. Error z value Pr(>|z|) (Intercept) -1.98183 0.15226 -13.02 <2e-16 *** persons 1.09126 0.03926 27.80 <2e-16 *** child -1.68996 0.08099 -20.87 <2e-16 *** factor(camper)1 0.93094 0.08909 10.45 <2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 (Dispersion parameter for poisson family taken to be 1) Null deviance: 2958.4 on 249 degrees of freedom Residual deviance: 1337.1 on 246 degrees of freedom AIC: 1682.1 >plot(ftv,res1,ylim=c(-7,17),xlab="Fitted Values",ylab="Deviance Residuals",main="Deviance Residuals-Posiion Model") > qqnorm(res1) Number of Fisher Scoring iterations: 6 > round(cbind(exp(m1$coeff),exp(cbind(m1$coeff-qnorm(0.975)*sqrt(diag(vcov(m1))),m1$coeff+qnorm(0.975)*sqrt(diag(vcov(m1)))))),2) [,1] [,2] [,3] (Intercept) 0.14 0.10 0.19 persons 2.98 2.76 3.22 child 0.18 0.16 0.22 factor(camper)1 2.54 2.13 3.02 b) > res1<-residuals.glm(m1,"deviance") > ftv<-m1$fitted.values plot(ftv...

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...Answer all 20 questions. Make sure your answers are as complete as possible. Show all of your work and reasoning. In particular, when there are calculations involved, you must show how you come up with your answers with critical work and/or necessary tables. Answers that come straight from calculators, programs or software packages will not be accepted. If you need to use software (for example, Excel) and /or online or hand-held calculators to aid in your calculation, you must cite the sources and explain how you get the results. Record your answers and work on the separate answer sheet provided. This exam has 200 total points; 10 points for each question. You must include the Honor Pledge on the title page of your submitted final exam. Exams submitted without the Honor Pledge will not be accepted. STAT 200: Introduction to Statistics Final Examination, Spring 2016 OL1/US1 Page 2 of 7 1. True or False. Justify for full credit. (a) The standard deviation of a data set cannot be negative. (b) If P(A) = 0.4 , P(B) = 0.5, and A and B are disjoint, then P(A AND B) = 0.2. (c) The mean is always equal to the median for a normal distribution. (d) A 95% confidence interval is wider than a 98% confidence interval of the same parameter. (e) In a two-tailed test, the value of the test statistic is 1.5. If we know the test statistic follows a Student’s t-distribution with P(T < 1.5) = 0.98, then we fail to reject the null hypothesis at 0.05 level of significance...

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