INTRO TO STATISTICS 1
INTRO TO STATISTICS 1 STA 2023
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Date Created: 09/18/15
DISTRIBUTION OF SOME SAMPLE STATISTICS Rule 1 Sampling Distribution of f9 A X l 1 p N p p 19 IF I I XBnp Minxpzm Minx1 p210 MN210gtltn Rule 2 Distribution of X Normal T heory Oquot X N XT IF X N NHX 0X Rule 3 Distribution ofXCLT X 39 N XOT IFn is large Make sure to check that ALL conditions are satis ed before you use any of these rules Example on page 156 of notes Machines ll bottles of cola The bottle label says each one contains 295 ml but there will be variability in the contents it is a random variable Suppose we know that the distribution of the contents of these bottles X is approximately Normal with mean 298 ml standard deviation 3 m1 ie X N N298 3 a Find the probability that one randomly selected bottle contains less than 295 ml Since we are given thatXNN298 3 X 295 298 PX lt 295P Xlt PZlt 10001515 Thus 0X 3 there is a 1515 chance that a randomly selected bottle will contain less than 295 ml of coke b Find the probability that the average content of a bottle in a randomly selected six pack is less than 295 m1 3 Since X N N298 3 X N N298 E Le X N N2981225 by rule 2 X u 295 298 039 1225 is 0 71 of all samples of size six will have average contents below 295 ml Hence PX lt 295 P j PZ lt 245 00071 That c Between what two values would be the central 95 of the means of samples of size 6 We are asked tofind two numbers x1 and x2 such thatPx1 lt 2 lt x2 09500 In part b we have established that N N2981225 Also from the tables of the standard normal distribution we find that PZ lt196 09750 PZ lt 196 00250 and hence P 196 lt Z lt196 09500 Using all ofthese we get 095 Px1lt Xlt x2 Px1 298 X p lt x2 298 1225 039 1225 P x1 298ltZlt x2 298 1225 1225 P 196lt Zlt196 x 298 So 1961127 which gives x1298 7196X1225 2955992956 Similarly x2 298 196 W yields x2 300 4 That is 95 ofall samples ofsize 6from this population will have sample means between 295 6 and 3 00 4 d What would be the average contents of the heaviest 10 of sixpacks Again using the fact that X N N298l225 and from the standard normal distribution tables 010 PZ gt 128 since PZ lt 128 090 we have X 010 PZgt 128 P UXgt128 72 Pgt pg 128x039X Pgt 298128gtlt1225 Pgt 29957 Thus heaviest 10 of all samples of size 6 from this population will have an average in excess of 299 5 7 e Why could we use the normal table to nd the probabilities in this problem Although the sample size n 6 is small and hence we cannot use the CLT rule 3 we do not need to actually do not prefer to use it because we are given that the population has a normal distribution and hence the mean of a random sample from this population has a normal distribution by rule 2 STA 2023 BPresnell amp DWackerly Lecture 13 156 Thought Vital papers will demonstrate their vitality by moving from where you left them to where you can t nd them Assignments Today P 236 240 For tomorrow Exercises 565 569 560 563 564 568 For Wednesday P 254 264 COMPUTER DEMO For Thursday Exercises 61 63 64 68 Last Time 0 Working with the Normal distribution 0 Tables to find probabilities 0 Key DRAW PICTURES STA 2023 BPresnell amp DWackery Lecture 13 157 Normal Approximation to the Binomial Distribution p 236 240 Suppose a N binnp o a has a binomial distribution 0 Sample size n 0 Probability ofa success PS 2 p o p 2 up 0 2 Mpg p 185 Chpt 4 o If n is large the probabilities involving a can be approximated with probabilities based on a Normal distribution with mean and standard deviation H 1 771970 1 Van STA 2023 BPresnel amp DWackery Lecture 13 158 See Figure 518 p 237 Normal Density A u 5 cs 1581 Binomial Histogram n10p5 P3 Xlt5 STA 2023 BPresnell amp DWackerly Lecture 13 159 5 5 P 5Pz up Van39 5 5 PCB Z 5 Pz 2 up Vnpq P3 g a S 5 P 323 Van39 o The 5 s are called corrections for continuity o The largest value of interest gets a little larger by 5 to get to edge of box in the binomial histogram o Smallest values of interest gets a little smaller 0 Probabilities involving z s obtained from z table STA 2023 BPresnell amp DWackerly Lecture 13 160 How large should n be so that the normal approximation is good enough quot to use Figure 519 p 238 p 240 o u 30 2 up 3 Mpg completely contained in the interval from O to n O 1 2 3 n1 n u 3o 30 0 This turns out to be the same as n29XIargerofpandq smaller ofp and q Ex 1025 n 1027 712 1023 712 8 p 2 n293 36 9 p 1 71293 81 STA 2023 BPresnell amp DWackerly Lecture 13 161 Ex Not in Book Motel records 10 of those with reservations DO NOT show up 0 Capacity 200 rooms 0 Reservations 215 o What is the probability that there will be a room for all who show up 0 What is the probability that more than 190 show up 0 a showup RH S o a is binomial n 2 gtIltp gtllt u 2 2 0 71m STA 2023 BPresnel amp DWackery Lecture 13 162 o 71 big enough to use normal approximation gtIlt n 2 gtllt o What is the probability that there will be a room for all who show up gtIlt Want 1333 gtIlt 1393 g w P z 3 gtIlt Pz g 159 TI W 200 201 PX 200 gtIlt Pz g 159 gtIlt Thus 1333 3 200 w STA 2023 BPresnel amp DWackery Lecture 13 163 o What is the probability that more than 190 show up gtIlt Want 1393 1393 gt 190 2 1393 1333 2 191 w 96 6 P gt z 44 P gt z 44 gt PQZ 6amp gtIlt Pz Z 68 gtIlt Thus 1393 gt 190 w STA 2023 BPresnell amp DWackery Lecture 13 164 Systematic approach to approximating Binomial Probabilities 1 Compute p 2 up and 0 2 n q 2 Set up needed probability involving a binomial random variable rewriting if necessary so that have only 3 andor 2 with no lt or gt Then make appropriate continuity corrections 3 Subtract off the mean divide by the std dev and use the normal tables to approximate the probability STA 2023 BPresne amp DWackery Lecture 13 165 Ex Coin tossing In 10 tosses approximate the prob of getting 45 or 6 heads vofheadsn 10p5q5 p 2 up 2 105 2 5 a z W 1055 1581 P lt z lt 2 P 949 lt z lt 949 Note The exact prob in this example is 065625 binomial table gives 656 Approx is very good here even for n 2 10 STA 2023 BPresnell amp DWackerly Lecture 14 166 Thought A truly Wise person never plays leapfrog With a unicorn Assignments Today P 254 264 For Thursday Exercises 61 63 64 68 For Monday SPRING BREAK Last Time 0 Normal Approximation to the Binomial Distribution 0 71 large enough n2 9 X lt largerofpandqgt smaller ofp and q 0 Write probabilities for the binomial variable with the uu sign P8ltmlt15P9 m 14 0 Use Continuity Correction STA 2023 BPresnell amp DWackery Lecture 14 167 Sampling Distributions 0 Recall p 4 6 objective of statistics is to make an inference about a population based on information contained in a sample from the population 0 Desired inference often phrased on terms of one or more parametersp 254 A Earameter is a meaningful number associated with a Eopulation Mean pr Variance 02 Standard Deviation or Popn Range Popn Median etc o Meaningful numbers computed from the observations in a Sample are called tatistics p 254 5192 etc used to make inferences about parameters STA 2023 BPresnell amp DWackerly Lecture 14 168 Results of Repeated Computation of the Statistic T Different samples yield different values for f E is a RANDOM VARIABLE The values of i tend to pile up in certain regions There is a probability distribution associated with the values of f This probability distribution is called the SAMPLING DISTRIBUTION of the statistic f p 255 STA 2023 BPresnell amp DWackerly Lecture 14 169 Ex Consider a spinnerthat can land on 1 2 or 3 each with probability o For the spinner J 02 E C2 p12 67 0 Spin the spinner twice record f average of 2 numbers Sample E Prob Sample f Prob 11 1 31 2 5 12 32 25 5 13 33 3 g 1 21 g 22 2 5 23 25 g STA 2023 BPresnell amp DWackery Lecture 14 170 The sampling distribution of E 5 20 25 30 COIN Mi What is the mean of E 195 2 EM 5 STA 2023 BPresnell amp DWackerly Lecture 14 171 o The mean of the sampling distribution of E 15 is equal to the true population mean u ME U ID 266 o The standard deviation of the sampling distribution popn std dev sample size of E 05 is equal to a gtllt 05 2 p 266 often called the standard error of the mean gtllt Note Bigger 71 smaller 05 Ex Population with mean 50 standard deviation 12 Take 16 observations DIME 0 Standard error 05 2 W STA 2023 BPresnell amp DWackerly Lecture 14 172 A point estimator for a parameter Defn 64 p 261 o a rule or formula telling how to use the use the data in a sample to compute a single number that we intend to be close to the value of the population parameter 0 E sample mean is a point estimator for pr popn mean 0 52 sample variance is a point estimator for 02 popnvanance STA 2023 BPresnel amp DWackery Lecture 14 173 An unbiased estimator for a population parameter if Defn 65 p 261 the mean of the sampling distribution of the estimator equals the parameter ME H o E is an estimator for u Unbiased Estimator I undere stlmates H OVCI C sumates I Tends to over and underestimate the same proportion of the time o Divided by n 1 when we computed the sample variance 52 to get an unbiased estimator for the population variance 02 STA 2023 BPresnel amp DWackery Lecture 14 174 Biased Estimator 3952 underestimates u overestimates Tends to overestimate too often If we have two unbiased estimators prefer the one with the SMALLER standard error P l close to pr u P close to pr STA 2023 BPresnell amp DWackerly Lecture 1 STA 2023 Spring 2001 Thought for the day Never wrestle with a pig You both get dirty and the pig likes it Lecturer Dennis Wackerly 222 GriffinFloyd Hall 3921941 Ext 227 Office Hours Tuesday 900 AM 1200 noon Text Statistics 8th Ed by McClave and Sincich Bundle includes Minitab statistics software and A Minitab Guide to Statistics by Meyer and Krueger Notes in this class are not to be used for commercial purposes Notetakers for A Notes are not welcome STA 2023 BPresnell amp DWackerly Lecture 1 Course Web page 0 httpwebstatufledudwack 0 FULL version of syllabus download immediately 0 Formula Sheet 0 Projects 0 Answers to projects 2 days before quizexam o How to get started with Minitab o How to convert McClave data set to Minitab format 0 Course notes Friday for following week 0 Sample quizzes exams STA 2023 BPresnel amp DWackery Lecture 1 HOW TO SUCCEED o REGULARLY attend both lectures and discussions 0 Read and use the text 0 ACTIVELY do the assignments 0 DON T FALL BEHIND For Tomorrow Pages 2 10 39 42 Exercises 113 119 123 232 mean only 235 236 means only For Wednesday Pages 43 46 50 54 STA 2023 BPresnell amp DWackerly Lecture 1 4 Chapter 1 Populations Samples Statistical Inference STATISTICS a branch of science that is concerned with collecting and interpreting data 1 How can I COMMUNICATE data to others 0 DESCRIPTIVE STATISTICS p 2 Summarize Data Histograms Section 22 Pie Charts Stem and Leaf Diagrams Numerical Summaries 2 How can data be used to REACH CONCLUSIONS 0 INFERENTIAL STATISTICS p 3 STA 2023 BPresnell amp DWackerly Lecture 1 Population a large body of units representing ALL of the items of interest to the data collector Defn 14 94 Variable a characteristic of an individual population unit that is of interest Defn 15 p4 Sample A subset of items selected from the population Defn 16 p5 0 Contains the information that we actually analyse EXAMPLE Population of interest the group of UF students who graduated December 16 2000 0 Characteristics Variables of interest Income of parents GPA Have job or not Number of semesters to graduate STA 2023 BPresnell amp DWackerly Lecture 1 EXAMPLE According to a survey of 1002 adults published in USA Today August 18 1997 11 of the TV viewers who had seen the Lubriderm See you later alligator skin lotion commercials liked them a lot 0 Population of interest ALL TV viewers who have seen the commercials in the U S 0 Characteristic Variable of interest Like commericals or not 0 Sample TV viewers actually interviewed in the survey 0 lnferences of possible interest Estimate the proportion of ALL viewers in US who like the commercials Advertiser caresll The survey also indicated that 8 of the male and 13 of the female viewers liked the commericals Is the campaign more popular among female viewers STA 2023 BPresnell amp DWackerly Lecture 1 EXAMPLE We wish to assess the effectiveness of a new medication in lowering blood pressure in hypertensive adults 0 HOW 0 One way administer the drug to 30 hypertensive adults that are representative of all hypertensive adults Population ALL hypertensive Adults Sample The 30 individuals used in the study Possible Inference estimate the average decrease in BP for hypertensive adults who take the medication STA 2023 BPresnell amp DWackerly Lecture 1 8 Statistical inference usually takes one of the following forms p6 0 estimation of some characteristic of the population 0 prediction of a future observation 0 decision making including testing hypotheses about the population Each type of inference Population involves using the sample data to make inferences quotquotquot about the population 4 Summary The objective of statistics is to make an inference about an entire population based on information contained in a sample taken from that population STA 2023 BPresnell amp DWackerly Lecture 1 EXAMPLE According to the Gainesville Sun August 19 1993 the average total SAT scores in 1993 for students from Florida and Iowa were as follows 0 Florida 882 0 Iowa 1103 What are the populations of interest 0 Presumably the groups of all college bound students in each of the two states 0 Actually In Florida 52 of all high school seniors took the test In Iowa only 5 of HS seniors took the test most bound for top Northeastern schools o Is the comparison of the average SAT scores given in the Sun reasonable Probably not the populations are actually very different STA 2023 BPresnell amp DWackerly Lecture 1 10 5 Elements of a Statistical nferencep 8 1 2 Clear specification of the population of interest Identification of one or more Variables to be investigated A sample of population units The inference about the population Measure of the goodness or reliability of the inference Example 11 is a reasonable estimate of the proportion of all TV viewers who like the See you later alligator commericals but how good is the estimate Is it close to the real percentage How close Within 1 or 5 Example Suppose we decide that there is a difference in the percentages of male and female TV viewers who like the commericals Is it likely that we made a mistake How likely STA 2023 BPresnell amp DWackerly Lecture 1 11 Types of Data 0 Quantitative Data p 8 measured on a numeric scale height weight temperature admission rate 0 Qualitative Data p 8 classified into groups no numerical interpretation therapy worksdoes not religious affiliation species of fish Graphical Methods summarize quantitative data through graphical methods histograms Thursday Section 22 Numerical Methods for data summary and inference typically make use of the observed values in the population or sample to compute meaningful numbers Some Notation Section 23 a a variable to be measured 131 132 the first second etc measurements 2 13 add up Sum all values of a Z add up the squares of the v values Z 232 square the sum of the v values STA 2023 BPresne amp DWackery Lecture 1 12 EXAMPLE Each individual in a random sample of 5 freshmen was asked how many movies she saw last month 6 8 0 3 5 vvvvv x1 x2 x3 x4 5 2222 68035 22 223 6282023252 36640925 134 2602 2 222 484 STA 2023 BPresne amp DWackery Lecture 1 13 Measures of Central Tendency Where is the middle The Mean of a set of v values is the sum of the measurements divided by the number of terms in the setp41 p41 2 37239 n where n is the number of LIZ values The mean is denoted 5 if the 33 s represent a SAMPLE p42 u if the 33 s represent a Population p42 EXAMPLE For our sample of 5 students 2 22 222 44 n 5
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