Lecture 1: Mon, Jan 13

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Lecture 1 Mon, Jan 13

• Introduction/Syllabus (web page)
• Some useful guidelines
• Course Outline
• Review (Ch 9,10)
• Key Statistical Concepts
• Sampling Distributions
• Confidence Intervals Inference
• JMP-IN

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Guidelines

• Lectures
• Daily Reading Even Numbered Problems.
• Always try to relate new concepts to existing
  examples
• Solutions to examples will be provided in class.
• All lectures can be downloaded off the course
  website. (Click on the lecture schedule and click
  on the date to obtain the lecture)
• Print outs of lectures will not be provided by
  the instructor.
• All necessary JMP instructions for the
  assignments will be provided in class, or
  available through the JMP help link in the course
  website.

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Guidelines (Contd.)

• Assignments and Exams
• A typical exam and/or assignment will have 60-70
  straightforward material, along with 30-40 of
  harder material.
• Each assignment will be worth 10 pts and all
  problems will be graded.
• Total of 7-8 assignments
• All assignments will typically be due a week from
  the date they are assigned.

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Guidelines (Contd.)

• All assignments and exams are cumulative in some
  sense (For example You could be asked to
  construct confidence intervals (Ch10,12,13) for
  the estimate of the slope (Ch 18,19,20)).
• Assignments and exams not collected will be kept
  outside the instructors office.
• The instructors office hours is primarily meant
  for addressing conceptual issues. For homework
  related questions students are encouraged to use
  the TAs office hours and Statlab hours.

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Guidelines (Contd.)

• JMP-IN
• Used extensively for assignments
• Familiarity with outputs for exams
• Recommended JMP-IN text is good reference
• Other general guidelines
• Feedback on lecture style, assignments, office
  hours are encouraged.
• Constant interaction encouraged to better
  understand material.

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Guidelines (Contd.)

• All re-grade requests (neatly written) should be
  handed to the instructor on or before the due
  date.
• Communications via email should be used only in
  case of emergencies, for appointments, and short
  questions.
• Only a tentative guideline of the exam format
  will be provided before the exams.

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Guidelines (Contd.)

• The final grade is determined based on the
  assignments, midterms, and the final. No other
  special work/projects can be used as
  supplements.
• These guidelines, rules for the course, can be
  changed at any time by the instructor.

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Guidelines (Contd.)

• Preparation for exams
• Work on lectures
• The book (remember you are required to have one
  the red thing)
• Work on assignments
• Lastly, work on the practice exams (without
  looking at the solutions)

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Course Outline

• Inference (Use Sample to infer population
  characteristics)
• Confidence Intervals
• Tests of Hypotheses (Test a variety of claims)
• Analysis of Variance (compare more than two
  groups)
• Regression (Relationships among variables)
• Simple Linear (Single independent variable)
• Multiple Linear (More than one independent
  variable)
• Polynomial (Models that allow powers of Indep.
  Vars.)
• Assorted Topics
• Chi-Squared Tests (Tests for Qualitative Data)
• Times Series Models (Detect patterns over time)
• Forecasting (Predict outcome for future time
  periods)

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Key Statistical Concepts

• Statistics the art of data analysis. Involves
  classifying, summarizing, organizing, and
  interpreting numerical information.
• Population the set of all items of interest in a
  statistical problem.
• Sample a subset of items in the population.
• Descriptive Statistics a body of methods used to
  summarize and organize the characteristics of
  sample data.
• Inferential Statistics a body of methods used to
  draw inferences about characteristics of
  populations based on sample data.

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• Variable characteristic or property of an
  individual item of a population or sample.
• Observation the value assigned to a variable.
• Parameter descriptive measure of a population.
• Statistic descriptive measure of a sample.
• Statistical Inference process of making an
  estimate, prediction or decision about a
  population based on information contained in a
  sample.
• Measure of Reliability a statement about the
  degree of uncertainty.

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Example Cola Wars

• Cola wars is the popular term for the intense
  competition between Coca-Cola and Pepsi displayed
  in their marketing campaigns. Their campaigns
  have featured movie and television stars, rock
  videos, athletic endorsements, and claims of
  consumer preference based on taste tests.
  Suppose, as part of a Pepsi marketing campaign,
  1,000 cola consumers are given a blind taste test
  (ie, a taste test in which the two brand names
  are disguised). Each consumer is asked to state
  their gender, age and a preference for brand A or
  brand B.

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Answers to Key Questions

• a. Population of interest the collection or set
  of all cola consumers.
• b. Variables of interest gender, age and cola
  preference.
• c. Sample 1,000 cola consumers selected from the
  population of all cola consumers.
• d. Inference of interest generalization of the
  cola preferences of the 1,000 sampled consumers
  to the population of all cola consumers. In
  particular, the preferences of the consumers in
  the sample can be used to estimate the percentage
  of all cola consumers who prefer each brand.

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• e. When the preferences of 1,000 consumers who
  are used to estimate the preference of all
  consumers in the region, the estimate will not
  exactly mirror the preferences of the population.
  For example, if the taste test shows that 56 of
  the 1,000 consumers chose Pepsi, it does not
  follow (nor is it likely) that exactly 56 of all
  cola drinkers in the region prefer Pepsi.
• Nevertheless, we can use sound statistical
  reasoning to ensure that our sampling procedure
  will generate estimates that are almost certainly
  within a specified limit of the true percentage
  of all consumers who prefer Pepsi.
• For example, such reasoning might assure us
  that the estimate of the preference for Pepsi
  from the sample is almost certainly within 5 of
  the actual population preference. The implication
  is that the actual preference for Pepsi is
  between 51 ie, (56-5) and 61 ie, (565)-
  that is, (56 5) This interval represents a
  measure of reliability for the inference.

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Sampling Distributions

• Two widely used formulas from Stat101 are

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Central Limit Theorem for the Sample Mean

• If a random sample is drawn from any
  population
• 1) The sampling distribution of the sample
  mean is approximately normal for
  sufficiently large sample size.
• 2) The larger the sample size, the more the
  sampling distribution of will resemble a
  normal distribution.

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The Sampling Distribution of the Mean of Random
Variables
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The Sampling Distribution of the Sum of Random
Variables
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Sampling Distribution of a Proportion

• The mean, variance and SD of p-hat are
• So, the variable
• is approximately a standard normal RV.

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How large should n be?

• In general, the sample size required to apply
  the Central Limit Theorem depends on the
  population distribution.
• But, as a rule of thumb, many people
  (including the book) use the minimum sample size
  of n30.

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Sampling Distribution to Statistical Inference

• Classical inference
• 1) Assume a value of the parameter
• 2) Find a test statistic and its distribution
• 3) Calculate the value of the test statistic from
  the data.
• 4) If the probability of seeing a value at least
  as extreme as what you observed is small, then
  you can reject the hypothesized parameter value.

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Inference for the Mean

• In order to optimize service, a bus company is
  trying to estimate the true rate of customers
  that arrive at a particular stop within a 10
  minute period during rush hour.
• The company collected data from 5pm to 6pm every
  day for a month (n180), and found a sample mean
  of arrivals per 10 min. Does this
  support the managers hypothesis that the true
  arrival rate is

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• Compute the probability of seeing an sample
  average that is more extreme than 8.1, assuming
  the population mean is 9.
• Conclusion it is very unlikely we would have
  seen data this extreme if the true mean were 9.

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Confidence Intervals

• Confidence interval estimates provide a range of
  plausible values for the unknown parameter.
• Before the experiment, the probability that the
  confidence interval will cover the true parameter
  value is
• After the experiment, we say that, with
  confidence,
  the interval covers the true parameter value.
  Equivalently, if we repeat our experiment over
  and over, and construct 95 confidence intervals
  each time, we would expect about 95 of the
  intervals to cover the true value of the mean.

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• The general form for a
• confidence interval for a parameter
• where is a point estimate for .
• For example a C.I. for the
  mean is given by

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Information and Confidence Intervals

• Small interval ? more information.
• Larger interval ? less information.

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Inference using Confidence Intervals

• 1) Assume a particular value for mu.
• 2) Collect data construct confidence interval
• 3) If the hypothesized value of mu is not
  contained in the interval ? evidence that the
  value is incorrect.

C.I.
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Exercise 10.24

• A statistics professor is investigating how many
  classes university students miss each semester.
  To answer this question, she took a random sample
  of 100 students and asked them how many classes
  they had missed in the previous semester.
• Estimate the mean number of classes missed by all
  students at the university. Use a 99 confidence
  level and assume that the population SD is known
  to be 2.2 classes.

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• Given
• The 99 confidence interval is

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Components of a Confidence Interval
UCL
LCL
width
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Take Away

• Be comfortable with topics from Stat101
• Sampling Distributions
• Confidence Intervals (sigma known)
• Inference using sampling distributions and CI
• Use of z-tables, t, F, Chi Squared.
• Basic JMP-IN (opening data files, descriptive
  statistics)
• Reading Ch 11, 12.