Statistics I MTH 160
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Statistics MTH160 Chapter 7 Sample Variability 72 Sampling Distribution 73 The Sampling Distribution of Sample Means 74 Application of the Sampling Distribution of Sample Means MTH 160 Statistics Brigitte Martineau Chapter 7 72 Sampling Distribution Do you think the sample mean varies from sample to sample What is a Sampling Distribution of a Sample Statistics The distribution ofvalues for a sample statistics obtained from samples all of the same and all drawn from the same Example Consider the population of even digit on a die 2 4 6 c Find all samples of size 2 with their respective mean What is the probability of selecting any ofthese samples We say that each of these samples is Page 2 of 8 MTH 160 Statistics I Brigitte Martineau Chapter 7 0 Construct the sampling distribution for the sample mean of samples of size 2 0 Construct a histogram of the probability distribution We just created a sampling distribution of sample means also called the sampling distribution of We could also create a sampling distribution of sample range or sample minimum Let s look at the sampling distribution of sample range Sample Ranges Probabilities Page 3 of 8 MTH 160 Statistics Brigitte Martineau Chapter 7 73 The Sampling Distribution of Sample Means The Sampling Distribution of Sample Means SDSM If all possible random samples each of size n are taken from any population with mean u and a standard deviation 0 o The mean ofthe sampling distribution of g is equal to o The standard deviation ofthe sampling distribution of g is equal to o Ifthe sampled population has a normal distribution then the sampling distribution of E will The Central Limit Theorem CLT Ifthe sampled population is not normal or unknown the sampling distribution of sample means will more closely resemble Look at p 373 and 374 for illustrations of the CLT Page 4 of 8 MTH 160 Statistics Brigitte Martineau Chapter 7 Examples 0 Suppose x has a normal distribution with mean u18 and standard deviationo3 If we draw random samples of size 5 from the x distribution and represents the sample mean what can you say about the E distribution 0 Suppose x has a mean u 75 and a standard deviation 0 12 but we have no information as to whether or not the x distribution is normal lfwe draw random samples of size 30 from the x distribution and represents the sample mean what can you say about the E distribution 0 Suppose you did not know that x had a normal distribution Would you be justi ed in saying that the E distribution is approximately normal if the sample size was 71 8 Page 5 of 8 MTH 160 Statistics Brigitte Martineau Chapter 7 74 Applications of the Sampling Distribution of Sample Means When the sampling distribution of sample means is normally distributed or approximately normal by CLT we can answer probability question using the standard normal distribution But how can we transform our E distribution into a 2 distribution Examples 0 A random sample of size 36 is to be selected from a population that has a mean of 50 and a standard deviation of 10 o This sample of 36 has a mean value of E that belongs to a sampling distribution Find the shape of this sampling distribution 0 Find the mean ofthis sampling distribution 0 Find the standard error ofthis sampling distribution 0 What is the probability that this sample mean will be between 45 and 55 Page 6 of 8 MTH 160 Statistics I Brigitte Martineau Chapter 7 o What is the probability that this sample mean will have a value greater than 48 0 Consider a normal population with u 100 and 0 20 Suppose a sample of size 16 is selected at random 0 P90ltElt110 o PEgt115 o Pxgt115 Page 7 of8 MTH 160 Statistics I Brigitte Martineau Chapter 7 o The amount of fill weight of contents put into a glass jar of spaghetti sauce is normally distributed with mean u 850g and standard deviation 0 8g 0 Describe the distribution of X the amount of fill per jar Describe the distribution of g the mean weight for a sample of 24 such jars of sauce 0 Find the probability that one jar selected at random contains between 848g and 855g 0 0 Find the probability that a random sample of 24 jars has a mean weight between 848g and 855g Page 8 of 8 Statistics I 52 53 54 55 56 MTH160 Chapter 5 Probability Distributions Discrete Variables Random Variables Probability Distributions of a Discrete Random Variable Mean and Variance of a Discrete Probability Distribution The Binomial Probability Distribution Mean and Standard Deviation of the Binomial Distribution MTH 160 Brigitte Martineau Statistics Chapter 5 52 Random Variables What is a random variable Notes Examples of Random Variables Let the number of computers sold per day by a local merchant be a random variable Integer values ranging from zero to about 50 are possible values Let the time it takes an employee to get to work be a random variable Possible values are 15 minutes to over 2 hours Let the volume of water used by a household during a month be a random variable Amounts range up to several thousand gallons Let the number of defective components in a shipment of 1000 be a random variable Values range from 0 to 1000 Discrete versus Continuous Random Variables Discrete Random Variables A quantitative random variable that can assume a number ofvalues Also known as a Continuous Random Variables A quantitative random variable that can assume an number of values Also known as a Page 2 of 14 MTH 160 Statistics Brigitte Martineau Chapter 5 53 Probability Distributions of a Discrete Random Variable Let s start with an example Let s say we are looking at the random variable X Number oftails when 2 coins are tossed What are the possible values for X Let s make a table with all the possible values ofx and their respective probability PX 0 P zero tail PX 1 P one tail PX 2 P two tails The table we just created is called a What is a Probability Distribution 0 A ofthe probabilities associated with each of a random variable 0 The probability distribution is a distribution o It is used to represent Note What is a Probability Function Page 3 of 14 MTH 160 Statistics Brigitte Martineau Chapter 5 Examples 0 The probability distribution ofa modi ed die is as follow Pxx6 for x1 23 X123 ip x l i ll 0 Could you find a probability distribution that describes the probabilities obtained when rolling a regular die X123456 i P i i i i i Properties of Probability Distribution Property 1 0 S P0031 Property 2 ZPOC 1 allx Examples 0 The number of people staying in a randomly selected room at a local hotel is a discrete random variable ranging in value from O to 4 The probability distribution is known and given in the form ofa chart below IX 0 1 2 l3 I4 Px 115 215 315 415 515 a Verify that this distribution meets the 2 properties ofa probability distribution b Express the distribution as a probability function Page 4 of 14 MTH 160 Statistics I Brigitte Martineau Chapter 5 c Construct a histogram of the Hotel Room probability distribution NOTES gt The histogram ofa probability distribution uses the physical of each barto represent its assigned probability gt In the Hotel Room probability distribution the width of each bar is so the height of each bar is equal to the assigned probability which is the area of each bar gt The idea ofarea representing probability is important in the study of variables o Is Px for x1234 a probability function Why Page 5 of 14 MTH 160 Statistics Brigitte Martineau Chapter 5 54 Mean and Variance of Discrete Probability Distribution Goal To describe the and the of a population Let s review first 1 The mean ofa sample is represented by 2 s2 and s are the and ofthe 3 s2 and s are called 4 u lowercase Greek letter mu is the ofthe 5 o2 sigma squared is the of the 6 o lowercase Greek letter sigma is the of the 7 u o and o2 are called A parameter is a constant p o and o2 are typically unknown values The Mean of a Discrete Random Variable The mean u ofa discrete random variable X is found by each possible value ofx by its own probability and then adding all the together ZixPltxgti Notes 0 The mean is not necessarily a value ofthe random variable 0 The mean is a population parameterthat is usually unknown We would like to its value The Variance of a Discrete Random Variable Page 6 of 14 MTH 160 Statistics I Brigitte Martineau Chapter 5 The variance 0392 of a discrete random variable X is found by each possible value ofthe squared deviation from the mean x y2 by its own probability and then all the products together 02 Zx t2 Px th2Pltxgtl Elmo le2Px 2 The Standard Deviation of a Discrete Random Variable 2 55 Examples 0 Find the mean the variance and the standard deviation of the following probability distribution 1300 for x1 and 2 Page 7 of 14 MTH 160 Statistics I Brigitte Martineau Chapter 5 o The number of standby passengers who get seats on a daily commuter flight from Boston to New York is a random variable X with probability distribution given below Find the mean variance and standard deviation x Px 0 030 1 025 2 020 3 015 4 005 5 005 Total Page 8 of 14 MTH 160 Statistics Brigitte Martineau Chapter 5 55 The Binomial Probability Distribution Before we start this section let us review the concept of factorial Factorial means the product ofa particular set of integers and denoted by l 4 12 x 0 Suppose you are given a threequestion multiplechoice quiz You have missed the last week of class and have not read the material in your textbook You decide to take the quiz and just randomly guess all the answers Let s take the quiz 1 A B C D 2 A B C D 3 A B C D Are questions 1 2 and 3 independent or dependent events How many questions do you think you answered correctly What do you think is the probability that you got 100 on the quiz What do you think is the probability that you got 0 on the quiz What do you think would be the class average Let s look at all the possibilities using a tree diagram Page 9 of 14 MTH 160 Statistics Brigitte Martineau Chapter 5 Question For each individual question what is the probability that you got the right answer The probability that you got the wrong answer Why Let X a random variable representing the number of correct answers Px0 Px1 Px2 Px3 This last experiment is known as a Binomial Probability Experiment What is a Binomial Probability Experiment An experiment that is made of trials that possess the following properties 0 There are repeated independent trials 0 Each trial has two possible outcomes a or a o Psuccess Pfailure and 771 o The binomial random variable x is the ofthe number of successful trials that occur x may take any integer value from O to n Examples 1 Let s look at our quiz 0 Trials 0 n 0 Success Failure 0 p q 0 x Page 10 of14 MTH 160 Statistics I Brigitte Martineau Chapter 5 2 A die is rolled 20 times and the number of ves that occurred is reported as the random variable Explain why x is a binomial random variable 0 Trials 0 n 0 Success Failure 0 p I q o x The Binomial Probability Function Px q H for x 0123 71 What is J x What is p What is qH Page 11 of14 MTH 160 Statistics Brigitte Martineau Chapter 5 Examples 0 Back to our quiz let us use the formula to find Px 0 Px 2 Px3 Px1 0 Results from the 2000 census show that 42 of US grandparents are the primary caregivers fortheir grandchildren Democrat amp Chronicle Grandparents as ma and pa July 8 2002 In a group of 20 grandparents what is the probability that exactly half are primary caregivers for their grandchildren At most 19 are primary caregivers for their grandchildren First let us check if this is a Binomial experiment Trials n Success Failure I73 q x Page 12 of14 MTH 160 Brigitte Martineau Statistics Chapter 5 Using Tables p 807 809 It is also possible to use a table to compute the probabilities ofa Binomial Experiment as long as 71S 15 and that p is listed in the table 001 005 01 02 03 04 05 06 07 08 09 095 and 099 Let s use the table to answer the following questions The survival rate during a risky operation for patients with no other hope of survival is 80 What is the probability that exactly four ofthe next five patients survive this operation Success p n Failure q f boys and girls are equally likely to be born what is the probability that in a randomly selected family of six children that there will be Success p n Failure q Exactly 4 boys Exactly 2 girls At least 3 boys At most 2 boys At most 5 boys Page 13 of14 MTH 160 Statistics Brigitte Martineau Chapter 5 56 Mean and Standard Deviation of the Binomial Distribution The mean and the standard deviation ofthe binomial distribution are as follow Mean 2 u np S tan dard Deviation 0quot 2 npq Variance 0392 2 Examples 0 Find the mean and the standard deviation forthe number of sixes seen in 50 rolls ofa die 0 The probability of success on a single trial of a binomial experiment is known to be A The random variablex number of successes has a mean value of 80 Find the number of trials involved in this experiment and the standard deviation ofx 0 Consider the binomial distribution where n 4and p 03 0 Find the mean 0 Find the standard deviation 0 Using Table 2 nd the probability distribution and draw a histogram Page 14 of14 Statistics MTH160 Chapter 3 Descriptive Analysis And Presentation of Bivariate Data 32 Bivariate Data 33 Linear Correlation 34 Linear Regression MTH 160 Statistics I Brigitte Martineau Chapter 3 32 Bivariate Data What is a bivariate data 1 Both qualitative 2 One qualitative one quantitative 3 Both quantitative 1 Cross Tabulation Contingency Tables for Bivariate Data Both Qualitative A Tally the results MTB 9 Stat lt Tables lt Tally Individual Variables lt Count Gender Count f m 25 N 54 B Frequency Contingency Tables MTB 9 Stat lt Tables lt Cross Tabulation and ChiSquare lt Count Rows Gender Columns Marital Status 1 2 Missing All f 25 3 l 28 m 23 2 O 25 All 48 5 53 Page 2 of 16 MTH 160 Statistics Brigitte Martineau Chapter 3 C Relative Frequency of grand total MTB 9 Stat lt Tables lt Cross Tabulation and ChiSquare lt Total Percents Rows Gender Columns Marital Status 2 Missing All f 4717 566 5283 m 4340 377 4717 All 9057 943 10000 D Percent of Row Totals MTB 9 Stat lt Tables lt Cross Tabulation and ChiSquare lt Row Percents Rows Gender Columns Marital Status 1 2 Missing All f 8929 1071 10000 m 9200 800 10000 All 9057 943 10000 Cell Contents of Row E Percent of Column Totals MTB 9 Stat lt Tables lt Cross Tabulation and ChiSquare lt Column Percents Rows Gender Columns Marital Status Missing All f 5208 6000 5283 m 4792 4000 4717 All 10000 10000 10000 Cell Contents of Column Page 3 of 16 MTH 160 Brigitte Martineau Cross Tabulation All at the same timeNot pretty Rows 89 52 47 92 47 43 90 lOO 90 Cell Contents Gender 25 29 O8 l7 23 00 92 4O 48 57 00 57 Columns lo 60 66 00 00 77 2 Missing 3 7l 00 gtfgtfgtfO gtfgtfgtfl gtfgtfgtfgtf of Row of Column of Total lOO 52 52 lOO 47 47 lOO lOO lOO Marital Status All 28 00 83 83 25 00 l7 l7 53 00 00 00 Statistics Chapter 3 N One Qualitative and One Quantitative variables o The quantitative values are viewed as different 0 Each set identi es by levels ofthe 0 Comparison done 0 Statistics used 0 Graphs used by variable Page 4 of 16 MTH 160 Statistics Brigitte Martineau Chapter 3 Dotplot MTB 9 Graph lt Dotplot lt With Groups Dotplot of Age vs Marital Status Marital Status Boxplot MTB 9 Graph ltBoxpot lt With Groups Boxplot of Age vs Marital Status 45 4o 35 1 UI lt 30 as 2 Marital Status Page 5 of 16 MTH 160 Statistics Brigitte Martineau Chapter 3 3 Display of Two Quantitative Variables Expressed data as ordered pairs X 9 Y9 Examples 0 Height versus Weight 0 Grades versus Hours studied 0 Distance to school versus time spent driving Scatter Diagram Scatter Diagram MTB 9 Graph lt Scatterpots lt Simple Scatterplot of Arm Span vs Height 75 o u o o n n o o 70 o 39 o o o I E o o a o 3 o c o o o E 65 o o o o o lt o o o o o a o o o o 60 o o o o u 39 o o o 55 60 65 70 75 80 Height Page 6 of 16 MTH 160 Brigitte Martineau Statistics Chapter 3 33 Linear Correlation What is linear correlation o No Correlation y T 0 Positive Correlation T 0 Perfect Positive Correlation y T 0 Negative Correlation y T 0 Perfect Negative Correlation y T o Nonlinear Correlation y T Page 7 of 16 MTH 160 Statistics Brigitte Martineau Chapter 3 Coefficient of Linear Correlation r What is r Characteristics of r Positive or negative correlation 0 Weight ofa car versus mileage 0 Response time of a the fire department versus the damage done 0 Study time versus grades 0 Number of hours watching TV versus grades How to compute r Pearson s Product Moment Formula r M n 1sxsy SSxy JSSxSSy Sum of Squares Formula r We will use the Sum of Squares Formula where SSxZx2 SSyZy2 SSxyny 1 1 n Page 8 of 16 MTH 160 Brigitte Martineau Statistics Chapter 3 Let s try an example I have sampled 10 heights and 10 weights out of a survey from a previous semester Here s a table summarizing the data Sample Sample Height2 Weight2 Height Height x Weight y 2 y2 Weight xy 63 103 3969 10609 6489 67 145 4489 21025 9715 64 1 22 4096 1 4884 7808 68 1 60 4624 25600 1 0880 60 160 3600 25600 9600 75 180 5625 32400 13500 68 170 4624 28900 1 1560 67 130 4489 16900 8710 69 146 4761 21316 10074 67 154 4489 23716 10318 TOTAL 2x 668 2y 1470 2x2 44766 Zyz 220950 ny 98654 Let s construct a scatter diagram for this data Let s try to evaluate the coef cient of correlation Page 9 of 16 MTH 160 Statistics I Brigitte Martineau Chapter 3 Let s try to compute the coef cient of correlation 88 x 33 Y 33 XY SSOW r 2 NSSltxgtSSltygt Z In Minitab 9 Stat lt Basic Statistics lt Correlation Correlations Sample Height x Sample Weight y Pearson correlation of Sample Height x and Sample Weight y 0548 PValue 0101 Page10of16 MTH 160 Statistics Brigitte Martineau Chapter 3 34 Linear Regression Correlation Regression Goal of Regression Patterns of Regression I 39X 39X 39X Methods of Least Squares Page11of16 MTH 160 Statistics I Brigitte Martineau Chapter 3 o Equation of the best tting line y 2 be b1x 0 Predicted value c Criterion to find b0 and b1 The line of best fit is determined by y 2 be b1xwhere b ZxfyLM Zy mzx 1 Xxx ff SSOC and b0 n 25 071 Let s practice Remember our example about a sample of 10 weights and heights Scatterplot of Sample Weight y vs Sample Height x 190 180 I 170 O E 160 O O 393 150 1 140 g 130 O In 120 110 100 l l l l l l 50 52 54 55 58 70 72 74 75 San e Height x 88 x 33 Y 33 XY Page12of16 MTH 160 Statistics Brigitte Martineau Chapter 3 Let s compute b 1 Equation of the line of best t is Choose two values ofx within the domain and nd their predicted y in order to draw the line of best ts on your scatter plot What does the slope represent What is the yintercept Is it meaningful Page13of16 MTH 160 Statistics I Brigitte Martineau Chapter 3 Making Predictions Use the equation found earlier to predict the weight ofa person who is 66 inches tall What does it tell you Can you conclude that every person who is 66 inches will weigh What can you say Use the equation found earlier to predict the weight ofa person who is 70 inches tall What does it tell you Use the equation found earlier to predict the weight ofa person who is 80 inches tall Always make predictions within the sample domain on the input variable Page14of16 MTH 160 Statistics I Chapter 3 Brigitte Martineau In Minitab To get the equation only Stat lt Regression lt Regression To get the equation and the line of best fit on a scatterplot Stat lt Regression lt Fitted Line Plot To make some predictions Stat lt Regression lt Regression lt options Regression Analysis Sample Weight y versus Sample Height x The regression equation is Sample Weight y 66 319 Sample Height x Predictor Coef SE Coef Constant 7661 1151 7057 0582 Sample Height x 3189 1720 185 0101 S 206132 ReSg 301 ReSgadj 213 Analysis of Variance Source DF SS MS F P Regression 1 14608 14608 344 0101 Residual Error 8 33992 4249 Total 9 48600 Unusual Observations Sample Height Sample Obs x Weight y Fit SE Fit Residual St Resid 5 600 16000 12531 1339 3469 221R R denotes an observation with a large standardized residual Predicted Values for New Observations New Obs Fit SE Fit 5 Cl 95 Pl 1 14445 666 12909 15981 9449 19440 Values of Predictors for New Observations New Height Obs x 1 660 Page150f16 MTH 160 Statistics Brigitte Martineau Chapter 3 Fitted Line Plot Sample Weight y 661 3189 Sample Height x 190 39 5 206132 RSq 301 180 0 RSqadj 213 170 160 150 140 130 Sanple Weight y 120 110 100 I I I I I 60 62 64 66 68 70 72 74 76 Sa male Height x Page16of16 MTH 160 Statistics Brigitte Martineau Chapter 4 Statistics MTH160 Chapter 4 Probability Page 1 of 8 MTH 160 Statistics Brigitte Martineau Chapter 4 In which areas of your life have you heard or used probabilities lately Vocabulary 0 Experiment 0 Outcomes 0 Sample Space Notation Size ofa sample space 0 Event Any ofthe sample space Notation Therefore ifA is the event of ipping a tail Examples EX Rolling a die What is the sample space S nS Let B event ofrolling an even number B nB PB Let C event of rolling a number greater than 6 C nC PC Let D event ofrolling a number less than 7 D nD PD Page 2 of 8 MTH 160 Statistics Brigitte Martineau Chapter 4 Theoretical versus Experimental Probability If you ip a coin 10 times how many tais would you expect This is known as the probability where A is the event of flipping a tail Theoretical probability PA Does that mean you will get EXACTLY 5 tails The probability also known as the or empirical probability is the relative with which that event will occur 0f tails Number of times A occurred P A Ex erimental rob relative re uenc P P f q y 10 tosses Number of trials Usually probability is Our best estimate of Pevent is P event our experimental probability Example At the Virgin Music store in Times Square 60 people entering the store were selected at random and were asked to state their favorite type of music Ofthe 60 24 selected rock 16 selected country 8 selected classical and 12 said something other than rock country or classical Determine the empirical probability that the next person entering the store favors 1 rock music 2 country music 3 Something other than rock country or classical music 4 Does that mean that 20 ofa Americans don t like rock or country or classical music Page 3 of 8 MTH 160 Statistics I Brigitte Martineau Chapter 4 EXPERIMENT Let s take a coin and flip it 10 times How many tails do you expect Record the number of tails Trial of tails Rel Freq Cum Rel Freq 1 Cum A Prob 05 Number ofTrials Law of Large Numbers The more times you perform an experiment the closer the experimental probability will be to the theoretical probability The larger the number of trials the closerthe experimental probability statistic will be to the theoretical probability parameter Page 4 of 8 MTH 160 Statistics Brigitte Martineau Chapter 4 Example A single die is rolled Find the probability that the number on top is o A six 0 A number less than four 0 An odd number 0 A number less than four or even 0 An even number 0 Not a 2 Complements The complement of an event is the set of all sample points in the sample space that belong to that event The complement ofA is denoted by Rememberthat PAPZ1 Examples 0 The probability that you will go out tonight is 06 What is the probability that you do not go out o The probability of your instructor canceling the next test is 00001 What is the probability that the test will take place next Thursday Characteristics of probabilities o P impossible event o P event that is sure to happen A probability will always have a value between 0 If you add all probabilities ofall outcomes o P event A P not event A Page 5 of 8 MTH 160 Statistics Brigitte Martineau Chapter 4 OR Probabilities and Mutually Exclusive Events General Addition Rule For any events A and B PA or B PAPB PA and B EX A single die is tossed Find the following probabilities o P even or less than 3 o Peven or odd What does mutually exclusive mean Special Addition Rule f events A and B are mutually exclusive then PA or B PA PB Examples 0 Are the following events mutually exclusive o If PA 03 and PB 04 and ifA and B are mutually exclusive events nd the following o PZ o PE o PA email 0 PA or B Page 6 of 8 MTH 160 Statistics Brigitte Martineau Chapter 4 And Probabilities and Conditional Probability What is a conditional probability General Multiplication Rule Let A and B be two events de ned in a sample space S Then PA and B PAoPB A or PA and B PBoPA B Examples In a deck of card 2 cards are selected with replacement Find the probability that you selected 2 face cards Is the probability of selecting a face on the second card affected by the fact that you already selected a face card on the rst selection What if the previous experiment was done without replacement Is the probability of selecting a face on the second card affected by the fact that you already selected a face card on the rst selection Independent Events IfA and B are two independent events then PBPBA or PAPAB Page 7 of 8 MTH 160 Statistics I Brigitte Martineau Chapter 4 Let s practice A new grading policy has been proposed by the dean of the College of Education for all education majors All faculty and students in education were asked to give their opinion about the new grading policy The results are shown below Suppose that someone is selected at random from the College of Education either student or faculty a Compute the PFa PFa F and PFa S b Compute PF and Fa c Are the events F and Fa independent Explain d Compute PFa or 0 Are these events mutually exclusive Explain Page 8 of 8 Statistics MTH160 Chapter 1 Statistics 12 What is Statistics 14 Data Collection Understanding Statistics in the Media MTH 160 Statistics I Brigitte Martineau Chapter 1 12 Introduction of Basic Terms Statistics Descriptive statistics STATISTICS lt Inferential statistics Descriptive statistics Inferential statistics Where do you hear about statistics Let s look at some vocabulary 0 Population Finite versus infinite population 0 Sample Page 2 of 12 MTH 160 Brigitte Martineau Statistics Chapter 1 Why do we need samples Variable Data singular Data plural Experiment Parameter Statistic Page 3 of 12 MTH 160 Statistics Brigitte Martineau Chapter 1 Example The admissions of ce wants to estimate the cost oftextbooks for students at our college The plan is to randomly identify 100 students and obtain their total text book costs Describe in your own word 0 Population 0 Sample 0 Variable Data singular 0 Data plural 0 Experiment 0 Parameter o Statistic There are two types of variables Qualitative or attribute or categorical data VARIABLES lt Quantitative or numerical data Qualitative data Page 4 of 12 MTH 160 Statistics Brigitte Martineau Chapter 1 Quantitative data Examples Identify each ofthe following as being qualitative attribute or quantitative numerical data The color of pants worn by all of you Whether you will be on time or not Your satisfaction level after buying a new computer The amount oftime you spent on your math homework The number of classes you enrolled in The state you were born in The amount of lead in your tap water at home Your ight number during your last vacation Your rank after the Labor Day race The number of defective parts in a box of 50 Numerical quantitative and attribute qualitative data can be subdivided in the following categories Nominal Qualitative data Ordinal VARIABLES Discrete Quantitative data Continuous Page 5 of 12 MTH 160 Statistics Brigitte Martineau Chapter 1 Qualitative 0 Nominal variable 0 Ordinal variable Quantitative 0 Discrete variable 0 Continuous variable Examples Identify each ofthe following as being nominal ordinal discrete or continuous o The color of pants worn by all of you Whether you will be on time or not Your satisfaction level after buying a new computer The amount oftime you spent on your math homework The number of classes you enrolled in The state you were born in The amount of lead in your tap water at home Your ight number during your last vacation Your rank after the Labor Day race The number of defective parts in a box of 50 Page 6 of 12 MTH 160 Statistics Brigitte Martineau Chapter 1 14 Data Collection How to obtain the data is the rst problem statisticians have to face Statisticians are always looking for good data Any idea why Example You want to find out the result of a national election and you are looking at Rochester s results only Do you think you have good data Why Would you say your data from the Rochester sample are biased or unbiased Biased Sampling Method A sampling method that produces values that Example You are doing a survey on the number of phones per household in Monroe County You decide to call 500 family members friends and acquaintance that you know within Monroe County to nd out some results Is this a biased or unbiased sampling method Why What else is problematic in this experiment Page 7 of 12 MTH 160 Statistics I Brigitte Martineau Chapter 1 Two commonly used sampling methods that often result in biased samples are convenience and volunteer sample Convenience sample Volunteer sample There are numerous ways of selecting data to construct a sample However there are mainly ve types of unbiased sampling methods 0 Simple Random Sample Every element ofthe population has an probability of being chosen All samples of size n have an chance as well Example Here s a list of students from a previous semester Let s try to randomly select a sample of5 students out of We will need a table of random number p 805 in the textbook 1 Let s close our eyes and select the starting point at random 2 From that point go down the column and read the 2 digit number lfit is below we keep itifit is above we discard that number and keep going 3 Repeat until we have 5 data Page 8 of 12 MTH 160 Brigitte Martineau Statistics Chapter 1 Systematic Sample A sample in which every item is selected starting from a selected first element Example Let s say we want to have a X systematic sample in our case 10 Randomly select a number from the first 100X elements and then select every 100Xth item until we have the desired number of item 1 Using a table of random number randomly select a number between 1 and 10010 for a starting point 9 l First data is A Second data is A Third data is Stratified Random Sample A sample obtained by the data and then selecting a number of items from each or by means of a Example We want a sample of size 6 What would be an easy way to group the data from our class list Other suggestions Page 9 of 12 MTH 160 Statistics Brigitte Martineau Chapter 1 o Proportional Sample A sample obtained by stratifying the data and selecting a of items in to the size of each group by mean of Example Cluster Sample A sample obtained by sampling but not ofthe possible subdivisions clusters within a population Example Page 10 of12 MTH 160 Statwsucs Bngme Mamneau Chapter 1 UNDESTANDING STATISTICS IN THE MEDIA USA TODAY Snapshot 052mm Um mm a Hamburgers sleaks cmckun and may am mas summon a a s an us mmwuu lhe grill 7 7137 xi hrgus sums Ehkken 3mm cmymmmm cm usAmuAv mm mm mm ammwwm mm a Popu auon Samp e Vamab e Type of vaname Stan 5U c5 Do you have mm m ms mformauon Wm Page 11 of 12 MTH 160 Stamsucs Bngme Mamneau Chapter USA TODAY Snapshot 052020937 um 2 07 m Er M u h u r xallmxizv Checking In on vacation owners shy 2y mecnmcomplnaly m MW cmmmw sin n h 1mm am 0 D Popu auon Samp e Vamab e Type of vaname Stan 5U c5 Do you have mm m ms mformauon Wm Page12of12 Statistics I MTH160 Chapter 6 Normal Probability Distribution 62 63 64 65 66 Continuous Variables Normal Probability Distributions The Standard Normal Distribution Applications of Normal Distributions Notation Normal Approximation of the Binomial MTH 160 Statistics Brigitte Martineau Chapter 6 62 Normal Probability Distributions The normal probability distribution is a probability distribution It is the distribution in statistics Many processes have a normal or approximately normal distribution The normal probability distribution is also called It uses 2 functions 0 A function to nd the ycoordinate ofthe graph picturing the distribution i 2 e 2 0 fx 2 for all real X O39xZ 39 b o A function to determine probabilities PW S x S b IfOde this is calculus Let s try to understand this concept by looking at the curve Cl u b x Page 2 of 12 MTH 160 Statistics I Brigitte Martineau Chapter 6 Good news 0 We will NOT use these formulas but tables with probability already computed for us 0 We will learn one special normal distribution thoroughly the normal distribution 0 Then we will all other normal distributions to a standard one 0 Need to use the empirical rule do you remember 0 We will refine that empirical rule to nd the that lies between two numbers Percentage Proportion and Probability Percentage proportion and probability are basically the concepts 0 Percentage 40 is usually used when talking about a 410 of the population 0 Probability is usually used when talking about that that the next individual will possess a certain property is the graphical representation of all three when we draw a picture to illustrate the situation Examples Page 3 of 12 MTH 160 Statistics Brigitte Martineau Chapter 6 63 The Standard Normal Distribution How many normal distributions do you think exist Fortunately they are all related to the standard normal distribution also called distribution Properties of the Standard Normal Distribution 2 o The total area underthe normal curve is equal to or o The distribution is and it extends inde nitely in both directions approaching but never touching the horizontal axis 0 The distribution has a mean of and a standard deviation of o The mean divides the area in half or on each side 0 Nearly all the area is between 2 and z How does the table work See Table 3 in Appendix B p 714 The table will always give you the area between the O and a speci c value of positive 2 Examples 0 Find the area underthe standard normal curve between 2 O and z 152 Page 4 of 12 MTH 160 Brigitte Martineau Statistics Chapter 6 Find the area underthe standard normal curve to the right ofz 152 Find the area underthe standard normal curve to the left ofz 152 Find the area underthe standard normal curve between 2 152 and z 0 Find the area to the left ofz 081 Find the area to the right ofz 231 Page 5 of 12 MTH 160 Statistics Brigitte Martineau Chapter 6 c Find the area between 2 215 and z 132 c Find the area between 2 085 and z 206 Youdoit o P152ltzlt152 o Pzgt215 o P27ltzlt2 o Pzgt186 Page 6 of 12 MTH 160 Statistics I Brigitte Martineau Chapter 6 So far we have given you the zvalue also called zscore and ask you to find the probability the percent orthe area under the curve Now we will give you the probability and you will be ask to nd the zvalue associated with it o What is the zvalue associated with the 80th percentile o What zscore is associated with the 35th percentile o What zscore bound the middle 95 of the normal distribution 0 What zscore bound the middle 70 of the normal distribution Page 7 of 12 MTH 160 Statistics Brigitte Martineau Chapter 6 64 Applications of the Normal Distributions In reality how many distributions are normal with a mean of O and a standard deviation of 1 We would like to be able to apply any normal distribution using the standard 2 normal distribution In order to do so we will have to transform any variable X into 2 using a standard score Standard Score z x we saw this in chapter 2 Demonstration Let just say we are working with a normal distribution with a mean of 20 and a standard deviation of 3 Page 8 of 12 MTH 160 Statistics Brigitte Martineau Chapter 6 Examples 0 Suppose that X is normal with a mean of 100 and a standard deviation of 16 Suppose a value ofx is chosen at random Find the probability that X is between 100 and 115 o A brewery filling machine is adjusted to ll quart bottles with a mean of 320 oz of ale and a variance of0003 Periodically a bottle is checked and the amount ofale is noted 0 Assuming the amount of ll is normally distributed what is the probability that the next randomly checked bottle contains more than 3202 oz 0 The results ofa test are normally distributed with a mean of 72 and a standard deviation of 13 Your instructor has decided to give A to the top 5 ofthe class What is the minimum grade you need to achieve an A Page 9 of 12 MTH 160 Statistics I Brigitte Martineau Chapter 6 c Find the value of the 33rd percentile if we have a normal distribution with a mean of 100 and a standard deviation of 15 o A radar unit is used to measure the speed ofautomobiles on an express way during rush hour traf c The speeds of individual automobiles are normally distributed with a mean of 62 mph 0 Find the standard deviation ofall speeds if 3 ofthe automobiles travel faster than 72 mph Using the standard deviation found in the previous step nd the percentage of these cars that are traveling slower than 55 mph 0 Using the standard deviation found in step 1 nd the 95th percentile for the variable speed 0 Page 10 of12 MTH 160 Statistics I Brigitte Martineau Chapter 6 65 Notation le is a normal random variable with mean u and standard deviationo this is often denoted x N MI Examples Suppose X is a normal random variable with u 35 and o 6 A convenient notation to identify this random variable is Suppose X is a normal random variable with u 100 and 02 9 A convenient notation to identify this random variable is We need a convenient notation to indicate the area under the standard normal distribution since zscores are used throughout statistics in a variety of ways Zn algebraic name of zscore such that a Let s practice 0 2015 represents Page 11 of12 Statistics MTH 160 Chapter 6 Brigitte Martineau 0 Z represents 070 c Find the zscore that bound the middle 90 ofthe normal distribution Note Table 4 Appendix B on p 811 of your textbook will help you nd some common onetailed and twotailed situations Examples Find 2005 c Find the zscore that bound the middle 80 ofthe normal distribution Page 12 of12 Statistics I MTH160 Chapter 2 Descriptive Analysis And Presentation of SingleVariable Data 22 23 24 26 27 28 Graphs Pareto Diagrams and StemandLeaf Displays Frequency Distribution and Histograms Measures of Central Tendency Measures of Dispersion Measures of Position Interpreting and Understanding Standard Deviation The Art of Statistical Deception to read MTH 160 Statistics Brigitte Martineau Chapter 2 22 Graphs Pareto Diagrams and StemandLeaf Displays Why graphing Graphs for qualitative data Circle graph and Bar graph 1 Circle Graph also known as Pie Chart of Smokers Category I n I y Y 14 25 9 n 40 741 In Minitab Graph gt Pie Chart gt select a variable you can also select various options within the pie chart menu Page 2 of 22 MTH 160 Statistics Brigitte Martineau Chapter 2 2 Bar Graph 0 O O O O O 0 Chart of Living Arrangements 40 30 E g 20 U 10 1 3 4 Living Arrangements 1 Living wth Parents 3 Liv39ng wth Spouse 2 Living wth Others 4 Liv39ng Abne In Minitab Graph gt Bar Chart gt select the type gt OK gt select a variable you can also select various options within the bar chart menu Page 3 of 22 MTH 160 Brigitte Martineau Pareto Diagram a special type of bar graph Example Statistics Chapter 2 The nal daily inspection defect report for a cabinet manufacturer is given in the table below Management has given the cabinet production line the goal for reducing their defects by 50 What defects should they give special attention to in working toward this goal Daily Defect Inspection Report 140 120 10039 Count 4O 20 l l i 0 I I I I I I Blemi Scrat Chi Stai Othe De 43 40 25 12 10 5 319 296 185 89 74 37 319 615 800 889 963 1000 Percent n Minitab Stat gt Quality tools gt Pareto Chart gt select a variable Page 4 of 22 MTH 160 Statistics Brigitte Martineau Chapter 2 Graphs for quantitative data dotplots and stemandleaf Graphs for quantitative data are useful to display the distribution But what is a distribution 1 Dotplot O O O O Dotplot of Height Dotplot of Height vs Gender H iii lllllliuiu 8 i 50 53 55 5399 72 75 78 39 I l i 39 39 Height 50 53 55 59 72 75 78 Height In Minitab Graph gt Dotplot gt select a variable Page 5 of 22 MTH 160 Statistics I Brigitte Martineau Chapter 2 23 Frequency Distribution and Histograms Frequency distributions and histograms are used to summarize large data sets What is a frequency distribution Grouped versus Ungrouped Example of an ungrouped frequency distribution Variableofkids Data 2 2 O O O 1 2 1 1 1 O 0 Examples of a grouped frequency distribution Grouping Rules Procedure for Grouping Page 6 of 22 MTH 160 Statistics Brigitte Martineau Chapter 2 Example A video store has computed the number of movies rented for every day ofthe last month 74 142 179 127 198 105 98 87 189 154 189 207 76 95 108 163 205 96 149 174 123 147 108 101 185 125 87 119 138 162 Classes Frequency Midpoint What is a histogram Page 7 of 22 MTH 160 Brigitte Martineau Histograms can have different shapes Symmetrical Skewed to right JShaped Statistics Chapter 2 Uniform rectangular Skewed to left Bimodal Page 8 of 22 MTH 160 Statistics Brigitte Martineau Chapter 2 24 Measures of Central Tendency Some examples of measures of Central tendency are What do they measure What is an average THE MEAN Symbol for the mean of the population Symbol for the mean of the sample x Formula XZ 71 What is the mean of 10 2030 What is the mean of 10 20 300 What happened Example You scored 70 80 65 and 90 on your 4 rst tests What score do you need on your fth test in orderto have a mean of 75 Page 9 of 2 MTH 160 Statistics Brigitte Martineau Chapter 2 0 THE MEDIAN What is it Symbol How to nd the median Examples Find the median of 4 8 3 8 2 9 2 11 3 Find the median of4 8 3 8 2 9 2 11 3 15 THE MODE What is it Symbol Bimodal or no mode Example Page 10 of 22 MTH 160 Brigitte Martineau 0 THE MIDRANGE What is it Symbol lowest value highest value midran e g 2 Formula Examples Consider the following 2 sets of data Calculate the Averages for each Statistics Chapter 2 Data Set Mean Median Mode Midrange Best Measure 1 2 3 4 90 51015 20 50 10 20 30 4010 The results of3 tests of MTH 160 are shown in the following table Explain what is going on Mean Median Exglanation Test 1 74 73 Test 2 73 80 Test 3 65 6O Page 11 of 22 MTH 160 Statistics I Brigitte Martineau Chapter 2 25 Measures of Dispersion Everyday you buy a can of Sprite Does it mean that you drink the EXACT same quantity of Sprite everyday WHY Can you think about any experiment or action in life where there is absolutely no variability Variability can be found Measures of Dispersion are used to measure the What are the common measures ofdispersion Why do we need measures ofdispersion Small variation Large variation 0 RANGE What does the range measure Formula Range Highest Value Lowest Value 0 VARIANCE Variance of a population Variance of a sample XXX if Formula s2 n 1 STANDARD DEVIATION What is the standard deviation Standard deviation ofa population Standard deviation ofa sample Formula s Page 12 of 22 MTH 160 Statistics Brigitte Martineau Chapter 2 Example Find the range the variance and the standard deviation of the following set of numbers Range Variance Standard Deviation Step 1 Find the mean Y Step 2 Fill in the table below Data SGt x X SHORTCUT FORMULA n n l 2 2 S Sum XXX ff Page 13 of 22 MTH 160 Statistics I Brigitte Martineau Chapter 2 Find the range the variance and the standard deviation of the following data 8 8 12 14 6 6 Range Variance Standard Deviation Data Set X Z x x SHORTCUT FORMULA 2 2x2 x 2 S n n l 2 Sum Sum XXX X The results of2 tests of MTH 160 are shown in the following table Explain Mean Sta39ld Explanation DeVIatIon Test 1 74 4 Test 2 73 22 Page 14 of 22 MTH 160 Statistics I Brigitte Martineau Chapter 2 26 Measures of Position Measures of position are used to Most popular measures of position are QUARTILES PERCENTILES Relationship between quartiles and percentiles 1St quartile is equivalent to 2nd quartile is equivalent to 3rd quartile is equivalent to Page 15 of 22 MTH 160 Statistics Brigitte Martineau Chapter 2 How to find percentiles and quartiles Rank the data in ascending order Compute A i 100 3 lfA is an integer The position of the percentile is at A 05 A5 The percentile is halfway between the value ofthe data in the Ath position and the value ofthe next data I lfA is a fraction or a decimal The position of the percentile is at the next larger integer after A The percentile is the value ofthe data at that position mentioned above Examples The following data represents the pH levels ofa random sample of swimming pools in a California town 56 56 58 59 60 60 61 62 63 64 67 68 68 68 69 70 73 74 74 75 Find the 34th and the 60th percentile as well as the 1St and 3rd quartile Page 16 of 22 MTH 160 Statistics I Brigitte Martineau Chapter 2 THE MIDQUARTILE The midquartile is another measure of Formula The mean median mode midrange and midquartile are all measures of central tendency Are they all equal in value Can you find an example where they would be 5NUMBER SUMMARY The 5number summary indicates how much the data are spread in each 3 4 5 The BoxandWhisker display also called boxplot displays the 5number summary 0 Vertical or horizontal o The box is used to depict o The whiskers are line segments used to depict The line through the box represents One line segment represents 0 The other line segment represents Boxplot of Shoe Size o The outlier Shoe Size 8 o Page 17 of 22 MTH 160 Statistics I Brigitte Martineau Chapter 2 EXAMPLE A random sample of students in a sixth grade class was selected Their weights are given in the table below Find the 5number summary for this data and construct a boxplot 63 64 76 76 81 83 85 86 88 89 90 91 92 93 93 93 94 97 99 99 99 101 108 109 112 ZSCORE also called standard score What is a zscore Formula Be careful 1 The calculated value ofz is rounded to the nearest 2 The zscore measures the number of above or below the 3 zscores range from to 4 zscores may be used to make comparisons of Page 18 of 22 MTH 160 Statistics I Brigitte Martineau Chapter 2 EXAMPLES A certain data set has mean 76 and standard deviation 10 Find the zscores for 90 and 60 Bill and Joe both got 79 on their statistics test Bill is in section 1 where the mean was 75 and the standard deviation was 10 Joe is in section 5 where the mean was 77 and standard deviation was 21 Who has the best relative score Page 19 of 22 MTH an Statishcsi EngmeMamneau ChaptevZ 7 Interpreting and Understanding Standard Deviation Standan deviaimn is a measuve u Them ave 2 miesiu descnbe daiaihai my mum s1anuam uswanun 1 EMPIRICAL RLILE Havanabie is nuvmaiiy msmbumu mum data he mmquot 7 51aman deviaimn mum mean mum data he mmquot isiandavd deviaimns mum mean 3 Appmmaieiyi a mum data he mmquot isiandavd deviaimns mum mean Nate 4 99 7 lt 95 lt 55 gt x72 H m x 7 m m Page 2n uvzz MTH 160 Statistics I Brigitte Martineau Chapter 2 EXAMPLE A random sample of plum tomatoes was selected from a local grocery store and their weights recorded The mean weight was 65 ounces with a standard deviation of04 ounces lfthe weights are normally distributed a What percentage ofweights falls between 57 and 73 b What percentage ofweights falls above 77 The Empirical rule can be used to nd out whether or not a distribution is approximately normal 1 Find the and the 2 Compute the actual proportion of data within 1 2 and 3 standard deviations ofthe mean A Compare with the empirical rule A lfthe proportions found are reasonably close to those ofthe empirical rule then the data are approximately normally distributed Page 21 of 22 MTH 160 Statistics Brigitte Martineau Chapter 2 CHEBYSHEV S THEOREM The proportion of any distribution that lies within k standard deviations of the mean is at least lik Z where k is any positive number larger than 1 This theorem applies to ALL distributions of data Notes 1 2 3 Illustration at least 1 irk X 7k EXAMPLE At the close of trading a random sample of 35 technology stocks was selected The mean selling price was 6775 and the standard deviation was 123 Use Chebyshev s theorem with k 2 3 to describe the distribution Page 22 of 22
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