Elementary Statistics

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Elementary Statistics

• Sampling

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Where are we?

• Scientific Method
• Formulate a theory -- hypothesis
• Collect data to test the theory--??
• Analyze the results
• Interpret the results and make a decision
  --p-value approach

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Fair selection method?

• A sampling method that gives each unit in the
  population a known, non-zero chance of being
  selected is called a probability sampling method.
  
• Simple Random Sampling
• Stratified Random Sampling
• Systematic Sampling
• Cluster Sampling
• Multistage Sampling.

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• Let's Do It! 2.8 Faculty Salaries
• A study is being conducted of the faculty
  salaries for a public
• university. Of the 1200 tenure-track faculty, 480
  are full
• professors, 336 are associate professors and the
  remaining are
• assistant professors. A simple random sample of
  100 full
• professors, 50 associate professors and 50
  assistant professors will
• be taken and information about salary will be
  obtained.
• (a) What type of sampling method is used to
  obtain the
• 200 selected faculty members?
• (b) Using your calculator (with a seed value of
  18) or the
• random number table (Row 14, Column 1) to give
  the
• labels for the first 5 full professors to be
  selected.
• (c) The table summarizes the average salary for
  the sampled
• faculty members by rank.
• Give the overall estimate of the average salary
  for all faculty

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Stratified random sampling

• Note with stratified random sampling, not all
  samples of size n have the same chance.
• Note ideally, the variability of the units
  within each stratum should be small compared to
  the variability between the strata.

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Systematic Sampling

• For a 1-in-k systematic sample, you order the
  units of the population in some way, and randomly
  select one of the first k units in the ordered
  list. This selected unit is the first unit to be
  included in the sample. You continue through the
  list selecting every kth unit from then on.
• Example 2.18 -- A 1-in-4 Systematic Sample
• 19 units in population, A through S, take a
  1-in-4 systematic sample.
• A B C D E F G H I
• 1 2 3 4
• J K L M N O P Q R S

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Cluster Sampling

• In cluster sampling, the units of the population
  are grouped into clusters. One or more clusters
  are selected at random. If a cluster is
  selected, all of the units that form that cluster
  are included in the sample.
• Notice the difference between stratified sampling
  method and cluster sampling method.

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Summary
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Study studies

• Should you believe it?
• How the data is collected?
• How the data is measured?
• How the data is summarized?

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Type of studies

• Baseball and the Divorce Rate
• Do couples who go to baseball games have better
  relationships than those who dont? The Center
  for Marital and Family Studies at the University
  of Denver observed that in cities with major
  league baseball teams, the divorce rate is 23
  lower than in cities without baseball teams. A
  spokesperson for the center stated, I dont want
  to say having a team lowers the divorce rate, but
  it might help.
• observational study
• explanatory variable -gt response variable
• Baseball Team Status -gt divorce rate
• Treatment - Two level

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Type of studies

• Greedy behavior-intentional or instinctive.
• The November 1997 issue of the Journal of
  Personality and Social Psychology reports
  research into greedy behavior. The subjects were
  randomly divided into two equal-sized groups.
  Group1 was told that they were sharing a box of
  sand and were instructed to take out what they
  considered a fair portion. They were also told
  they would receive 2 for every pound they
  scooped out. Group2 was told that they would
  receive 10 if they accurately scooped a
  specified amount of sand.
• experiment

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Type of studies

• In an experiment, the researcher actively imposes
  some treatment on the units or subjects in order
  to observe the responses.
• In an observational study the researcher simply
  observes the subjects or units and records
  variables of interest. The researcher does not
  attempt to manipulate or influence the responses.

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Type of variables

• A response variable measures an outcome of the
  study. It is the variable that is thought to
  depend in some way on the explanatory variable.
• An explanatory variable or factor is a variable
  that is thought to explain or cause the observed
  outcomes. It is the variable that explains the
  changes in the response variable.
• The possible values of the explanatory variable
  are called the levels of that explanatory
  variable. A treatment is a specific combination
  of the levels of the explanatory variables.

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Lets do it! 3.1 -- Explanatory Variable versus
Response Variable

• For each set of variables listed below, identify
  which variable should be the response variable
  and which should be the explanatory variable.
• (a)V1 The amount of money earned for a part-time
  job_________
• V2 The number of hours worked________________
  ______
• (b) V1 The weight of a package
  __________________________
• V2 The first-class postage rate at the post
  office__________
• (c)V1 The salary of a high school
  teacher _____________
• V2 The number of years of teaching
  experience______________

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Confounding problem

• A confounding variable is a variable whose effect
  on the response variable cannot be separated from
  the effect of the explanatory variable on the
  response variable.
• Note any study can be wrongly interpreted
  because of confounding variable.

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Two kinds of Observation studies

• A retrospective study is a study of past events.
  Researchers identify subjects that have
  experienced certain responses and look back to
  see if the subjects also had various factors or
  explanatory variables. Subjects may be asked to
  recall past events. A retrospective study works
  from the responses to the potential explanatory
  variables.
• A prospective study is a study of ongoing or
  future events. Researchers identify subjects
  which have various explanatory variables or
  factors and follow them into the future and
  record the responses. A prospective study works
  from the potential explanatory variables to the
  responses.

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Difficulty in Observation study

• Can not control the explanatory variable
• Can not control for the confound variable
• Hard to adjust for unknown confound variable

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Studies and Experiments

• Studies
• Observation studies
• Experiments
• Variables
• Response variables
• Explanatory variables
• Level value of Explanatory variables
• Treatment different combination of variables
• Confounding variables

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Understanding Experiments

• Object
• Experimental units or subjects(person)
• Explanatory variables
• Factors that can be controlled by experimenter.
• Treatment
• Particular combination of the levels
• Displayed by design-layout table

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Review example

• P200 3.51 A New Zealand study that followed
  1000 children through age 18 has found that those
  who were breast-fed as children were far better
  in school. The author s of the study conjecture
  that fatty acids found in breast milk, but not in
  formula, may boost brain development.
• A) A criticism of the study is that breast-fed
  children tend to have mothers who were older,
  better educated, and wealthier. A trait that by
  itself could account for the differences in
  academic achievement is called a
• Response variable
• Control variable
• Confounding variable
• Is this an observational study or an experiment?
• Give the response variables for this study.

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Producing Hard Gears

• A manufacturer of automatic car transmissions
  uses steel gears. An important characteristic of
  these gears is their hardness. An experiment is
  conducted to compare gears made from three
  different steel alloys, that is, three different
  compositions of steel. Gears made from each of
  these steel alloys were heat-treated or hardened
  at two different temperature levels, 1600 ?F and
  1800 ?F. Twelve gears of each of the three
  different steel alloys were available -- six were
  randomly assigned to each of the two temperature
  levels.
• Experimental units is ?
• Response variable is ?
• Explanatory variables are ?
• Levels are ?
• Treatments are?
• The number of experiment
• unit required ?

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LDI 3.5 The look of salsa

• An experiment has been conducted to improve
  the overall look (appeal) of salsa made by a
  college campus Mexican restaurant. The manager
  wants to investigate the effect of chunkiness of
  the tomatoes and amount of peppers on the
  appearance rating. Four sizes of chunkiness of
  the tomatoes (small, medium, large, and extra
  chunky) and three pepper amounts (few, medium, a
  lot) will be used in the experiment. Five bowls
  of salsa will be made for each treatment
  combination and volunteer customers will be asked
  to score the jars of salsa on overall appearance.
• (a) Explain why this is an experiment and not an
  observational study.
• (b) For this experimental situation ---
• (i) The experimental units are
• (ii) The response variable is
• (iii) The factors or explanatory variables are
• How many levels are there for each factor?
• (iv) How many treatments are there?
• (v) How many experimental units are required for
  the experiment?

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Plan an Experiment--Design

• Experimental Design Does Taking Vitamin C Lead
  to Fewer Colds?Consider the following
  experiment a researcher believes that taking a
  daily dose of vitamin C will help reduce the
  incidence of colds in elementary-school children.
  To assess this claim, 100 elementary school
  children, who volunteered and for whom parental
  consent was received, were given a daily dose of
  vitamin C for a three-month winter period. The
  number of colds during this season was recorded
  for each subject. For this experiment we have
  100 subjects, one response variable, the number
  of colds, and one explanatory variable, vitamin
  C, at one level, a daily dose.
• Based on the data, does it appear that taking a
  daily dose of vitamin C
• helps reduce the incidence of colds?
• Main Problem No comparison

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