Stat 110, Introduction to Statistical Reasoning
Stat 110, Introduction to Statistical Reasoning STAT 110 003
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STAT 110 003
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This 13 page Class Notes was uploaded by Rachel Stein on Thursday January 21, 2016. The Class Notes belongs to STAT 110 003 at University of South Carolina taught by Leslie Hendrix in Fall 2015. Since its upload, it has received 65 views. For similar materials see Introduction to Statistical Reasoning in Statistics at University of South Carolina.
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Date Created: 01/21/16
Where Do Data Come From? The field of statistics is applicable in every discipline. To give an idea of how widespread its use is, consider the following examples illustrating the scope of applications: From ratemyprofessors.com, STAT 110 instructors often get comments about the practice test looking like the real exam…..should you believe this claim? Here are some interesting stats from Exam 2 in a recent semester Summary statistics: Column n Mean Variance Std. Dev. Std. Err. Median Range Min Max Q1 Q3 Exam 2 379 78.51715 318.37735 17.843132 0.9165401 82 102 0 102 70 90 • • Claim found on MedPageToday, Jan. 14, 2013, “Fast Foods Tied to Allergies, Excema in Kids” http://www.medpagetoday.com/Pediatrics/GeneralPediatrics/36848 A claim found on MedPageToday, June 29, 2014, “Run for your Life? Right!” http://www.medpagetoday.com/Cardiology/Prevention/46982 Another claim found on MedPageToday, July 17, 2012, “IVIG Stops Alzheimer’s in Its Tracks” http://www.medpagetoday.com/MeetingCoverage/AAIC/33780 51% of adults age 18 and older are married today – 72% in 1960 http://www.pewsocialtrends.org/2011/12/14/barely-half-of-u-s-adults-are-married-a-record-low/?src=sdt-carousel Statistics is the science (or art) of data. ____Individuals________are the objects described by a set of data. Individuals may be people, but they may also be animals or things. A __Variable_______is any characteristic of an individual. A variable can take different values for different individuals. The actual measurements recorded for individuals are called _Data_____. Example 1 What are the individuals? Name Major Points Grade Students Advani, Sura Comm 397 B What are the variables? Barton, David Hist 323 C Brown, Annette Lit 446 A Major, points, grade Chiu, Sun Psyc 405 B Cortez, Maria Psyc 461 A Example 2 Mak Vehicle Transmis Cylind Cit Highw BM Subcom Automati 6 19 27 BM Subcom Manual 6 20 29 Buic Midsize Automati 6 20 30 Che SUV Automati 6 16 21 Individuals? Vehicles Chapter 1 Page 2 Variables? Make, model, vehicle type, ect Example 3 In an agricultural study in Kansas, researchers want to know which of three fertilizer compounds produces the highest wheat yield (in kg/plot). An experimenter uses 15 plots of land. Each fertilizer is applied to 5 plots of land. After harvest, the resulting yield is measured. Individuals? A. Farmers B. Fertilizer types 60.5 53.8 73.2 63.4 59.4 59.5 C. Wheat 48.2 61.1 66.3 yield D.Plots of 55.5 58.8 70.2 land Variables? Fertilizer type, wheat yield Ways to Gather Data 1. Observational Study An observational study observes individuals and measures variables of interest but does not attempt to influence the responses. A response variable is a variable that measures an outcome or result of a study. The purpose of an observational study is to describe some group or situation. Chapter 1 Page 3 The _Population_______ for a statistical study is the entire group of individuals about which we want information. A ____Sample____ is the part of the population from which we actually collect information and is used to draw conclusions about the whole. 2. Sample Survey A Sample survey is a type of observational study that surveys a group of individuals by studying only some of its members (selected because they represent the larger group of individuals) It is a survey because the individuals provide their own responses It is a sample survey because the individuals participating in the survey are a sample of the population A census is a sample survey that attempts to include the entire population as the sample. The US Census is required by the constitution every 10 years You can see 2010 Census Data and info here http://www.census.gov/2010census/ Lots of really cool data can be found here. Let’s take a look at some in class. We’ll always miss some people in the census count… http://www.cnsnews.com/news/article/dozens-us-cities-line-contest- 2010-censu Example 4 The University of Pennsylvania’s National Annenberg Election Survey conducted a poll from July 30 to August 5, 2004. They asked: Do you favor or oppose Federal funding of research on diseases like Alzheimer’s using stem cells taken from human embryos? The survey reported that the poll consisted of 1345 randomly selected adults in the United States. Population? All US Adults Sample? 1345 randomly selected US adults Chapter 1 Page 4 Example 5 The American Community Survey (ACS) contacts 3 million households, including some in every county in the US. This new Census Bureau survey asks each household questions about their housing, economic, and social status. Population? A. All US Adults B. All US Counties C.All US Households D.All 3 Million Households Sample? Example 6 Video adapter cables have pins that plug into slots in a computer monitor. The cable will not work if pins are bent or broken. A store chooses 5 cables from each lot and inspects the pins. If any of the cables have bent or broken pins, the entire lot is sent back. Population? A.Lot of adapter cables B. Cable pins C. 5 Cables Sample? A. Lot of adapter cables B. Cable pins C.5 Cables Example 7 A sociologist wants to know the opinions of employed adult women about government funding for day care. She obtains a list of the 580 members of a women’s club and mails a questionnaire to 100 of these women selected at random. Only 41 questionnaires are returned. Chapter 1 Page 5 Population? Employed adult women Sample? 41 women who returned the survey What percentage of the women contacted responded? 41% 3. Experiment An experiment deliberately imposes some treatment on individuals in order to observe their responses. The purpose of an experiment is to study whether the treatment causes a change in the response. Virtually all scientific research involves conducting well-designed experiments. Researchers hope the results from these experiments support a research hypothesis. Example 8 Salmonella bacteria are widespread in human and animal populations, and there are over 2,000 known serotypes. The reported incidence of salmonella illnesses in humans is about 17 cases per 100,000 people. A food scientist wants to see how withholding feed from pigs prior to slaughter can reduce the size of gastrointestinal tract lacerations during the actual slaughtering process. This is an important issue since pigs infected with salmonella may contaminate the food supply through these lacerations (among other routes, including fecal matter and meat juices). He chose 45 pigs from 3 farms. Individuals = pigs prior to slaughter Population = all pigs prior to slaughter Sample = 45 pigs from 3 farms Three treatments (we’ll give a formal definition of “treatment” in later chapter): Treatment 1: no food withheld prior to transport Treatment 2: food withheld 12 hours prior to transport Treatment 3: food withheld 24 hours prior to transport Chapter 1 Page 6 Data were measured on many variables: body temperature prior to slaughter, weight prior to slaughter, treatment assignment, the farm from which each pig originated, number of lacerations recorded, size of laceration (cm) How should we assign pigs to one of the three treatments? Randomly Why would one want to use animals from three farms? Infection, dirty farm, different practices Why might body temperature or prior weight be of interest? Temp and weight could indicate existing illness Example 9 Classify the Data Collection Type for the following questions of interest: – Is your school’s football team called for fewer penalties in home games than away games? A. Sample survey B.Observational study C. Experiment – Do college students perform better on exams when Mozart is playing softly in the background than when no music is playing? A. Sample survey B. Observational Study C.Experiment – Are college students satisfied with the quality of education they are receiving? A.Sample survey B. Observational Study C. Experiment Chapter 1 Page 7 Word of Caution: Statistical conclusions hold “on average” for groups of individuals. They don’t tell us much about one individual. Chapter 1 Page 8 Samples, Good and Bad Goal of Sampling We want to make a statement about a large group of individuals (the population), but oftentimes it is not practical or even possible to measure each individual in the population. In this case, we choose a sample of individuals that is (hopefully) representative of the population. What happens when our sample is not representative of the population? How to Sample Badly The design of a statistical study is __biased___ if it systematically favors certain outcomes. A ___response sample______________ chooses itself by responding to a general appeal. -individuals volunteer themselves to be in the sample -also called a _________self-selection sample_______________ Selection of whichever individuals are easiest to reach is called __convenience sample______. -researcher chooses who to ask to participate -individuals can still choose not to participate Convenience samples and voluntary response samples are often biased. Example 1 Ann Landers once asked the readers of her nationally syndicated newspaper advice column, “If you had it to do over again, would you have children?” She received nearly 10,000 responses, almost 70% saying “no.” Is it true that 70% of parents regret having children? No, the people who wanted to respond could Problems, (1) voluntary response survey, (2) strong feelings Example 2 A student at the university is conducting a survey to find the opinion of her fellow students on the availability of student parking on campus. She stands outside of a dorm and polls fellow students as they leave the dorm. Which bad sampling method is this? ----Convenience sample Example 3 The popular radio Ace&TJ Show recently asked fans to vote on their website to the following question A nurse at KATE MIDDLETON'S hospital who was pranked by two Australian DJs last week was found DEAD in her home on Friday. Police suspect SUICIDE. The DJs are off the air until further notice...a decision they made along with their radio station. Should the radio DJ's be fired? This is an example of which type of sampling? A. A convenience sample B.A voluntary response sample The most basic, good sampling method is known as the Simple Random Sample. The simple random sample is at the heart of all good sampling schemes. A simple random sample (SRS) of size n individuals from the population is chosen in such a way that: – Every set of n individuals has an equal chance to be the sample actually selected – Every individual has an equal chance of being chosen for the sample The easiest way to do this is to place names in a hat (the population) and draw out a handful (the sample). Step 1: Label. Assign a numerical label to every individual in the population. Be sure that all labels have the same number of digits if you plan to use a table of random digits. Step 2: Software or Table. Use random digits to select labels at random. Use software whenever possible – tables are old fashioned! http://bcs.whfreeman.com/scc8e/ - Choose “Statistical Applets”, then “Simple Random Sample”. There are lots of other computer generators available: www.randomizer.org, TI-83, 84, and 89 calculators, Chapter 2 Page 2 www.dougshaw.com/sampling, Statistical packages like R, SAS, Minitab, etc…. If you are using a table of random digits…. Population labels must each contain the same number of digits Spaces in the random digits table have no meaning (they are just place holders) You can start anywhere you like in the table (across rows, up a column, down a column,…) Some people start their population labels at 0 and some start them at 1 (be aware) Skip repeated codes and those outside the range of labels Example 4 Take a Simple Random Sample (SRS) of 3 people. Step 1: Label your “population” elements. Step 2: (Using random sampling generator) Obtain the sample. Step 2: (Using random digits table) Obtain the sample. Since we are using the pesky table of random digits, be sure each label (code) has the same number of digits! Use the following line from a random digits table. Note: In practice you would choose any line you want, but in class we will use the same line so we learn how to use the table. Chapter 2 Page 3 05497 12005 13659 81273 Example 5 Take a Simple Random Sample (SRS) You are reporting on apartments in Columbia. You decide to select 5 complexes at random for in-depth interviews with residents. 01-Abbott Arms 08-Claire Tower 15-Keswick 02-Asbury Arms 09-Colony East 16-Landmark 03-Ashland 10-Cornell Arms 17-Paces Run 04-Bent Tree 11-Fairways 18-Ravenwood 05-Briargate 12-Fox Run 19-Riverview 06-Brook Pines 13-Green Oaks 20-Stone Ridge 07-Cedarwood 14-Hunter’s Green 21-Whaley’s Mill Use the following portion of Table A at line 140 (read across the row) to sample 5 complexes. 12975 13218 13048 45144 72321 21940 00360 02428 96767 35964 23822 96012 The fifth complex chosen is A. Ravenwood B. Green Oaks C. Riverview D. Fox Run Chapter 2 Page 4 Some final thoughts - Can you Trust a Sample? We can’t trust results from convenience and voluntary response samples, because they are chosen in ways that invite bias. We have more confidence in results from a SRS, because it avoids bias. The first question to ask of any sample is whether it was chosen at random. Clearly, the SRS is a handy tool for getting a random sample, but it is not sophisticated enough to deliver the kind of information we want in many cases. We need more sampling options…coming up soon Chapter 2 Page 5
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