Hypothesis Testing

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Hypothesis Testing

• GTECH 201
• Lecture 16

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Overview of Todays Topic

• Formulation
• Evaluation
• Refining and Restating
• Statistical Tests

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What is a Hypothesis?

• Unproven or unsubstantiated statement
• You need to know the literature before you can
  formulate a hypothesis statement
• Data collection should support hypothesis testing
  and evaluation
• If hypothesis is tested and found to be correct,
  then results can be refined (different scenarios
  can be tested)
• If partially correct, then hypothesis statement
  needs to be refined (reworded)

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Hypothesis Testing

• Multi-step procedure that leads the researcher
  from the hypothesis statement to the decision
  regarding the hypothesis
• 6- step process
• State null and alternate hypotheses
• Select appropriate statistical test
• Select level of significance
• Delineate regions of rejection and nonrejection
  of hypotheses
• Calculate test statistic
• Make regarding null hypothesis

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Step 1

• State null and alternate hypotheses
• Null hypothesis
• A hypothesis to be tested
• Usually represented as
• Alternative hypothesis
• A hypothesis considered as an alternate to the
  null hypothesis
• Usually represented as

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Guidelines for Setting up H0, HA

• Hypothesis tests concerning one parameter
• Population mean, m
• A null hypothesis for a hypothesis test
  concerning a population mean should always
  specify a single value for that parameter
• ( ) sign must appear in the null hypothesis
• Therefore

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Guidelines, part 2

• Alternative hypothesis
• The choice of the alternative hypothesis depends
  on and should reveal the purpose of the
  hypothesis test
• Null hypothesis and alternative hypothesis are
  mutually exclusive
• Three choices are possible

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Guidelines, part 3

• An alternate hypothesis with a sign is called
  a two-tailed test
• The population mean, is different from a
  specified value,
• When a lt sign appears in the alternate
  hypothesis, the test is called a left-tailed test
• When a gt sign appears in the alternate
  hypothesis, the test is called a right-tailed
  test

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Setting up Hypotheses

• A snack food company produces 454 gms bags of
  pretzels. Although the actual weights deviate
  slightly from the 454 gms, and vary from one bag
  to another, the quality control team insists that
  the mean net weight of bags be maintained at 454
  gms. If the mean net weight of the bags is lower
  or higher, it is likely to cause problems.
• If you work for the quality control team and you
  want to decide whether the packaging machine is
  working properly, how would you set up a
  hypothesis test?

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Stating Hypotheses

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Select Appropriate Test

• One sample difference of means t test
• Objective
• Compare a random sample mean to a population mean
  for difference
• Requirements and assumptions
• Random sample
• Normally distributed population
• Variable is measured at interval or ratio scale
• Hypotheses
• Test Statistic

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Test Statistic

• sample mean
• population mean
• standard error of the mean
• population standard deviation

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Level of Significance

• ? 0.10 (90) 0.05 (95) 0.01 (99.7)
• Errors
• Type I error Rejecting the null hypothesis
  when it is in fact true
• Type II error Not rejecting the null hypothesis
  when it is in fact false

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Identify Regions of Rejection

• Of null hypothesis
• Two-tailed
• Left tailed (directional)
• Right tailed (directional)
• Calculate test statistic
• Make decision regarding null or alternate
  hypothesis

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To Work in Class

• We want to investigate demographic change in an
  area
• 3500 households (HH)
• You take a sample of 250 HH
• Sample mean 2.68 sample variance 4.3
• ? 0.10 (90)
• Now, we want to find out if the mean HH size in
  this one area is typical or representative of the
  national mean household size (2.61)
• Use the six step process to compare how closely
  the samples that you have taken compare with the
  national average HH size of 2.61

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Limits of Hypothesis Testing

• Pre-selecting level of significance
• Lacks a theoretical basis
• Used for convenience
• Binary nature of null and alternative hypothesis
• P-value or Probability value
• Accepted approach
• The exact significance level associated with the
  calculated test statistic is determined

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More About P-Value

• We can define P-value as
• The exact probability of getting a test statistic
  value of a given magnitude, IF the null
  hypothesis is true
• What is the probability of making a Type I error
• Type I error occurs when the null hypothesis is
  rejected using the hypothesis testing procedure,
  even though in reality the null hypothesis is true

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Comparing Classical and P Value Approaches

• Classical
• State hypotheses
• Decide on significance level
• Select test
• Delineate regions of rejection/nonrejection
• Calculate the test statistic
• State your conclusion in words

• P- Value
• State hypotheses
• Decide on significance level
• Compute the value of the test statistic
• Determine P-value
• P reject null hypothesis otherwise do not
  reject
• State your conclusion in words

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Guidelines for Using P-Value
Evidence against H0

• Weak or none
• Moderate
• Strong
• Very strong

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Example

• A random sample of 18 people with income below
  the poverty level reveals their daily intake of
  calcium
• mean 747.4 mg
• standard deviation 188 mg
• Use the P-value approach to determine whether the
  data provides sufficient evidence at the 5
  significance level to conclude that the mean
  calcium intake of all Americans with income below
  the poverty level is less than the required daily
  allowance of 800 mg

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Parametric and Nonparametric Tests

• Parametric tests
• Require knowledge about population parameters
• Assumptions made about population distribution
• E.g., population is normally distributed
• Sample data measured on Interval/Ratio scale
• Non-parametric tests
• Requires no knowledge about population parameters
• Distribution-free
• Some non-parametric tests are designed to be
  applied for nominal, ordinal data ( ) we
  will talk about these in the next lecture

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Choices/Options

• Run only a parametric test
• Run only a non-parametric test
• Run both tests
• Goal
• State the problem
• Decide what inferential technique will be useful
• Identify formulae associated with the technique
• Interpret the results