Statistics 1

Suppose we want to find out how many residents of Imperial Valley believe in the statewide legalization of marijuana provided the revenue from the legalization resulted in free college tuition for all students in California. Which sampling technique do you believe will generate the most accurate results?

Student A: Asked all 5000 local high school students and Imperial Valley College students and all 9652 Imperial Valley College students.

Student B: Randomly selected 500 most likely voters how they would vote on this measure.

Chapter Problem Why was the Literary Digest poll so wrong?Founded in 1890, the Literary Digest magazine was famous for its success in conducting polls to predict winners in presidential elections. The magazine correctly predicted the winners in the presidential elections of 1916, 1920, 1924, 1928 and 1932. In the 1936 presidential contest between Alf Landon and Franklin D. Roosevelt, the magazine sent out ten million ballots and received 1,293,669 ballots for Landon and 972,897 ballots for Roosevelt, so it appeared that Landon would capture 57% of the vote. The size of this poll is extremely large when compared to the sizes of other typical polls, so it appeared that the poll would correctly predict the winner once again. James A. Farley, Chairman of the Democratic National Committee at the time, praised the poll by saying this: “Any sane person cannot escape the implication of such a gigantic sampling of popular opinion as is embraced in The Literary Digest straw vote. I consider this conclusive evidence as to the

desire of the people of this country for a change in the National Government. The Literary Digest poll is an achievement of no little magnitude. It is a poll fairly and correctly conducted.” Well, Landon received 16,679,583 votes to the 27,751,597 votes cast for Roosevelt. Instead of getting 57% of the vote as suggested by The Literary Digest poll, Landon received only 37% of the vote. The Literary Digest magazine suffered a humiliating defeat and soon went out of business. In that same 1936 presidential election, George Gallup used a much smaller poll of 50,000 subjects, and he correctly predicted that Roosevelt would win. How could it happen that the larger Literary Digest poll could be wrong by such a large margin? What went wrong? As you learn about the basics of statistics in this chapter, we will return to the Literary Digest poll and explain why it was so wrong in predicting the winner of the 1936 presidential contest.

Statistics 2

Polling Data New Hampshire Primary:November 10, 2011 – January 9, 2012 Predictions 2012 Presidential Elections 2008 Presidential Election Predictions and Final Results

www.realclearpolitics.com

Statistics 3

2012 Predictions: Rasmussen reports

www.rasmussenreports.com

www.realclearpolitics.com

Statistics 4

1.1 What can we learn from the Literary Digest and Gallup polls?

What conjectures can we make as to why the predicted Literary Digest poll so far off from the actual results?

o Receiving 2.3 million respondents vs. a sample of 50,000? Respondents constitute a sample, whereas the population consists of the entire

collection of all adults eligible to vote. DEFINITIONS: Data, Statistics, population, census, sample

Data are a collection of observations (such as measurement, genders, survey responses. Statistics is the science of

o planning studies and experiments, o obtaining data, o organizing the data, o summarizing, presenting, analyzing, and interpreting the data o and finally, drawing conclusions based on the data so that people can make

informed decisions. A population is the complete collection of all individuals to be studies

o All individuals including, but are not limited to scores, people, measurements o The collection is complete in the sense that the set includes all individuals that are

to be studied. A census is the collection of data from every member of the population. A sample is a subcollection of members selected from the population

Example 1: The Literary Digest poll was a ______________________ of 2.3 million respondents. What would the population consist of? Remember—garbage in, garbage out! Sample data must be collected through a process of ________________________ selection. If sample data are not collected in an appropriate way, the data may be completely ________________!

Statistics 5

1.2 STATISTICAL THINKING Key Concept…

When conducting a statistical analysis of data we have collected or analyzing a statistical analysis done by someone else, we should not rely on blind acceptance of mathematical calculations. It is imperative that we consider these factors: 휎Context of the data 휎Source of the data 휎Sampling method 휎Conclusions 휎Practical implications

In learning to think statistically, common sense and practical considerations are usually

more important than the implementation of cookbook formulas and calculations. Example 2: Here is some data! Your grade in this dependent upon doing the following:

In the next 2 minutes, please summarize the data. By class Thursday, present a power point to our class summarizing, analyzing and

interpreting the data.

650 24249 0 1050 20666 0 967 19413 0 500 21992 0 1700 21399 0 2000 22022 0 1100 25859 0 1300 20390 0 1400 23738 0 2250 23294 0 800 19063 0 3500 30131 0 1200 18698 0 1250 25348 0

2250 25642 1 3000 23074 1 1750 28349 1 1525 24644 1 1500 23245 1 1500 24378 1 1250 23246 1 1200 23695 1 1600 23258 1 425 19325 1 1450 20397 1 900 17256 1 675 19545 1 1450 20780 1

At this point, you should be saying to yourself: _________________________

Why?

Statistics 6

Example 2:

In the next 2 minutes, please summarize the data. By class Thursday, present a power point to our class summarizing, analyzing and

interpreting the data. Description: These data for the 1991 season of the National Football League were reported by the Associated Press. QB is the salary (in thousands) of quarterback, TOTAL: total team salaries (in thousands of dollars) and NFC 1 means National football conference, NFC 0 means American Football Conferrence.

Number of cases: _______

Variable Names:

Statistics 7

Example 3: Refer to the data in the table below.

The x-values are weights (in pounds) of cars; The y-values are the corresponding highway fuel consumption amounts (in mi/gal).

Car Weights and Highway Fuel Consumption Amounts WEIGHT 4035 3315 4115 3650 3565 FUEL CONSUMPTION

26 31 29 29 30

a. Context of the data.

i. Are the x-values matched with the corresponding y-values? That is, is each x-value somehow associated with the corresponding y-value in some meaningful way?

ii. If the x and y values are matched, does it make sense to use the difference between each x-value and the y-value that is in the same column? Why or why not?

b. Conclusion. Given the context of the car measurement data, what issue can be addressed by conducting a statistical analysis of the values?

c. Source of the data. Comment on the source of the data if you are told the car manufacturers supplied the values. Is there an incentive for car manufacturers to report values that are not accurate?

d. Conclusion. If we use statistical methods to conclude that there is a correlation between the weights of cars and the amounts of fuel consumption, can we conclude that adding weight to a car causes it to consume more fuel?

Statistics 8

Example 4: Form a conclusion about statistical significance. Do not make any formal calculations. Either use results provided or make subjective judgements about the results.

One of Gregor Mendel’s famous hybridization experiments with peas yielded 580 offspring with 152 of those peas (or 26%) having yellow pods. According to Mendel’s theory, 25% of the offspring should have yellow pods. Do the results of the experiment differ from Mendel’s claimed rate of 25% by an amount that is statistically significant?

1.3 TYPES OF DATA Key Concept… A goal of statistics is to make inferences, or generalizations, about a population. We discussed the meaning of population and sample, now we need to learn to learn how distinguish between cases when we have data of the entire population or data from a sample only and from different types of numbers. 휎Parameter 휎Statistic 휎Quantitative Data 휎Categorical Data

DEFINITIONS: Parameter and a statistic

A parameter is a numerical measurement describing some characteristic of a ________________________.

A statistic is a numerical measurement describing some characteristic of a

_____________________.

Statistics 9

Example 5: Determine whether the given value is a statistic or a parameter.

a. 45% of the students in a calculus class failed the first exam.

b. 25 calculus students were randomly selected from all the sections of calculus I. 38% of

these student failed the first exam.

DEFINITIONS: Quantitative and Categorical data

Quantitative (aka numerical) data consist numbers representing counts or measurements.

o Examples: grades on exam 1 of statistics students, length of hair, age (in years) of respondents

Categorical (aka qualitative or attribute) data consist names or labels that are not

numbers representing counts or measurements. o Examples: Student ID numbers, number on the Jersey of a soccer player, social

security numbers, phone numbers Example 6: Please provide two examples each of:

a. Quantitative data

b. Categorical data

Statistics 10

,

DEFINITIONS: Discrete and continuous data

Discrete data result when the number of possible values is either a finite number or a countable number.

o Examples: The number of students in this room.

Continuous (aka numerical) data result from infinitely many possible values that correspond to some continuous scale that cover a range of values without gaps, interruptions or jumps.

o Example: Age, volume of water in a jug, length of your hair Example 7: Please provide two examples each of:

a. Discrete data

b. Continuous data Definitions: Levels of Measurement

Nominal level of measurement --characterized by data that consists of names, labels, or categories only. The data cannot be arranged in an ordering scheme (such as low to high).

o Yes/No/undecided; Political Party Ordinal level of measurement – Data that can be arranged in some order, but

differences (obtained by subtraction) between data values either cannot be determined or are meaningless.

o Course grades, ranks of colleges Ratio level of measurement – similar to interval level, with the additional property that

there is also a natural zero starting point (where zero indicates that none of the quantity is present). For values at this level, differences and ratios are both meaningful.

o Distances, Prices Interval level of measurement – similar to ordinal level, with the additional property

that the difference between any two data values is meaningful. However, data at this level do not have a natural zero starting point.

o Temperatures, years

Statistics 11

Example 8: Please provide an example of each level of measurement:

(a) Nominal (b) Ordinal (c) Ratio (d) Interval

1.4 & 1.5: CRITICAL THINKING & COLLECTING SAMPLE DATA

Key Concepts…

We focus on meaning of information obtained by studying data as well as the methods used in collecting sample data. Goals in this section are to: 휎Learn how to interpret information based on data. 휎Thinking carefully about the context of the data, the source of the data, the method used in data collection, the conclusions reached and practical implications. 휎Misuse of statistics 휇 Evil intent on the part of a dishonest person 휇 Unintentional errors on the part of people who do not know better. σ If sample data are not collected in the appropriate way, the data may be so completely useless that no

amount of statistical torturing can salvage them.

"Lies, damned lies, and statistics" is a phrase describing the persuasive power of numbers, particularly the use of statistics to bolster weak arguments, and the tendency of people to disparage statistics that do not support their positions. It is also sometimes colloquially used to doubt statistics used to prove an opponent's point.

Statistics 12

Sampling Techniques

Voluntary Response Sample (self-selected sample) – respondents themselves decide whether to be included

o Literary Digest study Simple Random Sample – Every subject in the population has an equally likely

chance of being selected (each subject has the same probability of being selected). Any pair (or triples, quads, etc) has an equally likely chance of being selected.

o Drawing names from a hat o First draw: Population size “N” – each subject has probability 1/N o Second draw: Population size “N – 1” – each subject has prob 1/(N-1)

Random Sample – members from a population are selected in such a way that each individual member in the population has either an equal or unequal chance of being selected; that is the probability of selected different subjects need not be equal.

o NBA draft Systematic Sample – select a starting point and then select every kth element in

the population o Put 30 numbers in a hat (there are 30 students on my roster). o Choose every 6th student to survey.

Convenience Sampling – use results that are very easy to obtain. o It is convenient for me to take my sample at the grocery store at 10 a.m. close to

the 55+ living. BTW – I am asking people whether they believe that abortion should be legal/illegal.

Stratified Sampling – subdivide the population into at least two different subgroups (strata) so that subjects within the same subgroup share the same characteristics (such as gender or age bracket), then we draw a sample from each subgroup (or stratum)

o Suppose a group want to learn the chances of a bond issue passing. A community is composed of two distinct income groups – low and high. Population of community is stratified into low and high income and a simple random sample from the low and high income groups.

Cluster sampling – first divide the population into sections (or clusters), then randomly select some of those clusters, and then survey every member in the cluster.

o Randomly select airline flights – then survey every passenger on the plane. Multistage sampling – use a different sampling method in different stages

Statistics 13

Example 9: Determine what sampling technique: Voluntary, Random, simple random, systematic, cluster, convenience, or stratified sample would be useful in each situation.

(a) An experimenter wants to estimate the average water consumption per family in a city. He chooses a random starting point on every block and uses the water bill of every 4th house. ____________________

(b) Six different health plans were randomly selected and all of their members were surveyed about customer satisfaction.

____________________

(c) I surveyed all of my students to obtain sample data consisting of the number of credit cards students possess. _____________________

(d) In a study of college programs, 820 students are randomly selected from those

majoring in communications, 1463 were randomly selected from those majoring in mathematics and 760 were randomly selected from those majoring in history. _____________________

(e) Wes set up a booth outside the cafeteria and waited for students to approach him. He asked students whether professor’s at IVC should be allowed to assign homework. _____________________

(f) On your desk, you have five flash cards. Each flash card either has A, B, C, D

or F on it. Draw your flash card and that is your grade for the course. _____________________

Statistics 14

Misuse of Data

Reported Results – take the data yourself rather than allowing someone to report their data.

o I will measure the height of each of you rather than you report to me how tall you are.

Small Samples – conclusions shall not be reached on samples that are “not large enough”. The question arises – how large of sample is large enough – and this brings us to a discussion on the normal distribution.

o I will ask five students on their feeling about euthanasia. Misused Percentages – misleading or unclear percentages. (% review at end of section)

o Continental Airlines claimed that they have improved 100% on lost luggage in the last six months. The New York Times interpreted (correctly) to mean that no baggage is now being lost – an accomplishment not yet enjoyed by Continental Airlines.

Loaded Questions – if survey questions are not worded carefully, the results of a study can be misleading. Survery questions can be “loaded” or intentionally worded to elicit a desired response.

Order of questions -- results from a poll conducted in Germany o Would you say that traffic contributes more or less to air pollution than industry?

Traffic presented first, 45% blamed traffic, 27% blamed industry o Would you say that industry contributes more or less to air pollution than traffic?

Industy presented first, 24% blamed traffic and 57% blamed industry. Missing Data

o Sometimes missing data is out of control of experimenter (subjects dropping out of a study for reasons unrelated to study)

o Low income people less likely to respond to a questions such as income

o U.S. Census – missing data from homeless people – o people that do not respond to phone surveys (or don’t have land lines)

Self-Interest study

o Aware of monetary gains from results. Our survey 250 people from hiring professionals demonstrated that you

are more likely to get hired if you purchase Tistoni shoes for men (price tag: $38,000). (By the way, I work for Tistoni and as part of my marketing, I took every one of these executives to dinner).

Deliberate Distortions – see graphs next page

Small samples – make sure your sample size is large enough!

Statistics 15

Example 10: Do you want to work for company A or company B. The x-axis represents time and the y-axis represents salary. All aspects of both Company A and Company B are equivalent. Company A Company B

Bias – everything sampling technique will generate some bias. Goal of the researcher should be to minimize bias. These are two biases that can be minimized. The non-response bias – at least report the non-response, or take another random sample from the differing population.

Correlation vs. Causality Example 11: It is a known fact that as ice cream sales increase, shark attacks increase.

Therefore, we can conclude that sharks love to eat people that eat ice cream!! Obviously, ‘we’ ice cream eaters taste better!

Is this a valid conclusion? Why/ why not?

Sala

ry

Salary

Sala

ry

Salary

Non-response bias – answers of those that respond potentially differ from respondents that did not respond.

Response Bias – respondents answering questions in the way that they believe the surveyor wants them to respond.

Statistics 16

Correlation vs. Causality and Confounding

Example 12: David’s study demonstrated that tall people read better.

(a) Does being tall CAUSE people to read better?

(b) Is it possible that there is a correlation between being tall and reading better?

(c) What are some possible confounding variables?

Finding a statistical association between two variables and to conclude that one of the variables causes (or directly affects) the other variable.

Two variables may seem to be linked (smoking and pulse rate), but the increase in pulse rate may/may not be caused by smoking.

The relationship of the shark attacks vs. ice cream sales as well as smoking vs. pulse rate are correlations.

Even though we may find a number of cases to be true – we cannot conclude that one variable caused the other

o Need to consider confounding variables.

Confounding Variables – not able to distinguish among the affects of different factors.

Moral of the story – CORRELATION DOES NOT IMPLY CAUSALITY!!!

Statistics 17

Studies

Example 13: Identify whether each study is cross-sectional, retrospective or prospective.

(a) Qualcomm funded a project that studied the affects of 4th – 6th gradestudents who were taught by a math specialist ability to communicate mathematics over a period of 4 years.

(b) Physicians at the Mount Sinai Medical Center plan to study emergency personnel who worked at the site of the terrorist attacks in New York City on September 11, 2001. They plan to study these workers from now until several years into the future.

(c) University of Toronto researchers studied 669 traffic crashes involving drivers with cell phones. They found that cell phone use quadruples the risk of a collision.

Observational Study – we observe and measure specific characteristics, but we do not attempt to modify the subjects being studied.

o Cross-Sectional Study – Data are observed, collected and measured at one

particular point in time. Aims to provide data on the entire population. o Retrospective study (Observational Study)

Case-Control study – Data are observed, collected and measured from a particular subset of the population.

Used frequently in epidemiology – compare subjects with a particular condition “the cases” with those that do not have the condition “the controls” but who are otherwise similar.

Data are collected from the past by going back in time (through examination of records, interviews, and so on.

A type of case-control study (look back historically)

o Prospective (longitudinal) study Data are collected in the future from groups sharing common

factors (called cohorts).

Experiment – we apply some treatment and then proceed to observe its effect on the subjects (Subjects in experiments are called experimental units)

**Note: In an observational study, no treatment is given.

Statistics 18

Experimental Design

Designing an experiment is important. A faulty design can result in ‘GIGA’ (as your author suggests) – garbage in, garbage out!

Famous example: The Salk Vaccine Experiment o 1954 – test the effectiveness of the Salk vaccine in preventing polio. o 200,745 children given a treatment

201,229 injected with placebo (essentially, a sugar pill that has no affect) 200,745 injected with Salk vaccine injection

o Result: 115 injected with placebo developed paralytic polio 33 developed paralytic polio of those injected with vaccine.

Types of Experiments

Randomization – subjects assigned to different groups through a process of random selection. Equivalent to flipping a coin!

o The children in the Salk Vaccine experiment were assigned treatment or control group based on random selection.

o Use chance as a way to create similar groups. Replication – replication of an experiment on more than one subject.

o Sample sizes should be large enough so that the erratic behavior characteristic of small samples do not disguise the true effects of differing treatments.

o Larger sample sizes increase the change of recognizing different treatment effects; but large sample sizes do not necessarily indicate a good sample.

Completely Randomized Experimental Design – assign subjects to different treatment groups through a process of random selection (Salk Vaccine)

Randomized Block Design – Group subjects that are similar but blocks differ in ways

that might affect the outcome of an experiment (i.e. gender and assigning treatment for heart medication ). A block is a group of subjects that are similar.

Rigorously Controlled Design – subjects are assigned to different treatment groups

in ways that are important to experiment

Matched Pairs Design – Compare exactly two treatment groups by using subjects matched in pairs that somehow are related and/or share similar characteristics. Can be subject (twin studies)…Coke/Pepsi anyone??

Statistics 19

Blinding & Placebo Affect

Sampling Errors

There will always be sampling error, no matter how well the experimental design is planned.

If we randomly sample 1000 IVC students and asked if they obtained a high school diploma or a GED the result will differ slightly from another 1000 students asked the same question. (we often hear the term “margin of error” in reporting statistics. This will be discussed later).

Sampling Error – the difference between a sample result and the true population result; such an error results from change sample fluctuations.

Non-sampling error – occurs when the sample data are incorrectly collected, recorded, or analyzed (such as selected a biased sample, using a defective measurement instrument, or copying the data incorrectly).

o The student that ends up with 1005% in the class had an incorrect test score input in the grade book!

Blinding is a technique used in an experiment in which the subject doesn’t know whether he or she is receiving the treatment or the placebo.

In a double-blind experiment, both the subject and the investigator do not know whether the subject received the treatment or the placebo.

Placebo affect occurs when an untreated subject reports an improvement in symptoms.

** Note: Salk Vaccine was a double-blind experiment.

Statistics 20

Example 14: Choose something you want to study and design an experiment.

Statistics 21

Homework Chapter 1

PERCENTAGE REVIEW

“of” often means multiply

Percent means per hundred so

Percentage of: Change the % to then multiply.

Fraction to percentage: Divide by denominator and multiply by 100.

Decimal to percentage: Multiply the decimal by 100 and put in the percent symbol.

Percentage to decimal: Remove the percent symbol and divide by 100.

Perform the indicated operation.

a. 12% of 1200

b. Write 5/8 as a percentage.

c. 12% of 1200

d. Write 5/8 as a percentage.

%100

nn

1100

1.1 NA 1.2 1-18, 23, 26, 28 1.3 1-32, 34 1.4 1,3, 4, 5, 6, 8, 9, 10, 12, 13, 15-19, 21, 24, 25, 28, 30 1.5 1-4,6, 9, 11, 12, 13, 15, 16, 18, 19, 21-26, 27, 29, 31