Statistics

1
Statistics Data Analysis

• Course Number B01.1305
• Course Section 31
• Meeting Time Wednesday 6-850 pm

Hypothesis Testing
2
Class Outline

• Review of midterm exam
• Hypothesis Testing
• One-sample tests
• Two-sample tests
• P-values
• Relationship with Confidence Intervals

3
Review of Last Class

• Statistical Inference
• Point Estimation
• Confidence Intervals

4
Reminder Statistical Inference

• Problem of Inferential Statistics
• Make inferences about one or more population
  parameters based on observable sample data
• Forms of Inference
• Point estimation single best guess regarding a
  population parameter
• Interval estimation Specifies a reasonable
  range for the value of the parameter
• Hypothesis testing Isolating a particular
  possible value for the parameter and testing if
  this value is plausible given the available data

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Point Estimators

• Computing a single statistic from the sample data
  to estimate a population parameter
• Choosing a point estimator
• What is the shape of the distribution?
• Do you suspect outliers exist?
• Plausible choices
• Mean
• Median
• Mode
• Trimmed Mean

6
Confidence Intervals

• Specification of a probably range for a
  parameter
• Used to understand how statistics may vary from
  sample to sample
• States explicit allowance for random sampling
  error (not selection biases)
• We have 95 confidence that the population
  parameter falls within the bounds of the interval
• Orthe interval is the result of a process that
  in the long run has a 95 probability of being
  correct

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Hypothesis Testing

• Chapter 8

8
Overview

• A research hypothesis typically states that there
  is a real change, a real difference, or real
  effect in the underlying population or process.
  The the opposite, null hypothesis, then states
  that there is no real change, difference, or
  effect
• The basic strategy of hypothesis testing is to
  try to support a research hypothesis by showing
  that the sample results are highly unlikely,
  assuming the null hypothesis, and more likely,
  assuming the research hypothesis
• The strategy can be implemented in equivalent to
  raise by creating a formal rejection region, by
  obtaining a plea value, were like seeking whether
  the null hypothesis value falls within a
  confidence interval
• There are risks of false positive and a false
  negative errors
• Tests of a mean usually are based on the
  t-distribution
• Tests of the proportion may be done by using a
  normal approximation

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Overview

• Very often sample data will suggest that
  something relevant is happening in the underlying
  population
• A sample of potential customers may show that a
  higher proportion prefer a new brand to the
  existing one
• A sampling of telephone response time by
  reservation clerks may show an increase in
  average customer waiting time
• A sample of the service times may indicate
  customers are receiving poorer service fan in the
  company thinks it is providing
• The question of whether the apparent defects in
  the sample is an indication of something
  happening in the underlying population and more
  if he apparent effect is merely a fluke

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What is Hypothesis Testing

• Method for checking whether an apparent result
  from a sample could possibly be due to
  randomness
• Checks on how strong the evidence is
• Are sample data reflecting a real effect or
  random fluke?
• Results of a hypothesis test indicate how good
  the evidence is, not how important the result is

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Motivating Case Study 1

• FCC has been receiving complaints from customers
  ordering new telephone service
• Big telecommunications company tells the FCC that
  the average time a new customer has to wait for
  new service installation is 72 hours (excluding
  weekends) with a standard deviation of 24 hours
• The FCC randomly samples 100 new customers from
  the telecom company and asks how long each had to
  wait for new service installation

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Testing Hypotheses

• Research Hypothesis, or Alternative Hypothesis is
  what the is trying to prove
• Denoted Ha
• Null Hypothesis is the denial of the research
  hypothesis. It is what is trying to be disproved
• Denoted H0

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Hypothesis Testing Components

• Define research hypothesis direction
• One-sided (lt or gt)
• Two-sided (?)
• Strategy is to attempt to support the research
  hypothesis by contradicting the null hypothesis
• The null hypothesis is contradicted if when
  assuming it is true, the sample data are highly
  unlikely and more likely given the research
  hypothesis
• Test Statistic Summary of the sample data

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Basic Logic

• Assume that H0 m72 is true
• Calculate the value of the test statistic
• Sample mean, proportion, etc.
• If this value is highly unlikely, reject H0 and
  support Ha
• We can use the sampling distribution to determine
  what values of the test statistic are
  sufficiently unlikely given the null hypothesis

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Rejection Region

• Specification of the rejection region must
  recognize the possibility of error
• Type I Error Rejecting the null hypothesis when
  in fact it is true
• In establishing a rejection region, we must
  specify the maximum tolerable probability of this
  type of error (denoted a)
• Type II Error Failing to reject the null
  hypothesis when in fact it is false (beyond
  scope)
• Rejection region can be based on sampling
  distribution of the sample statistic
• Remember, we want to reject the null hypothesis
  if the value of the test statistic is highly
  unlikely assuming H0 is true
• Can uses the tails of a normal distribution

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Rejection Region
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Rejection Region (cont)

• To determine whether or not to reject the null
  hypothesis, we can compute the number of standard
  errors the sample statistic lies above the
  assumed population mean
• This is done by computing a z-statistic for the
  sample mean

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Rejection Region (cont)
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Example

• The FCC sample of 100 randomly selection new
  service customers resulted in a mean of 80 hours.
• Setup the hypothesis test
• Calculate the test statistic
• Interpret the hypothesis

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Example

• A researcher claims that the amount of time urban
  preschool children age 3-5 watch television has a
  mean of 22.6 hours and a standard deviation of
  6.1 hours.
• A market research firm believes this is too low
• The television habits of a random sample of 60
  urban preschool children are measured and
  resulted in the following
• Sample mean 25.2
• Should the researchers claim be rejected at an a
  value of 0.01?

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Summary for Z Test with s Known
22
Example

• A researcher claims that the amount of time urban
  preschool children age 3-5 watch television has a
  mean of 22.6 hours and a standard deviation of
  6.1 hours.
• A market research firm believes this is
  incorrect, but does not know in which direction
• The television habits of a random sample of 60
  urban preschool children are measured and
  resulted in the following
• Sample mean 25.2
• Should the researchers claim be rejected at an a
  value of 0.01?

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Z-values Worth Remembering
z0.05 1.645 z0.025 1.96 z0.01
2.326 z0.005 2.576
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P-Value

• Probability of a test statistic value equal to or
  more extreme than the actual observed value
• Recall basic strategy
• Hope to support the research hypothesis and
  reject the null hypothesis by showing that the
  data are highly unlikely assuming that the null
  hypothesis is true
• As the test statistic gets farther into the
  rejection region, the data become more unlikely,
  hence the weight of evidence against the null
  hypothesis becomes more conclusive and p-value
  become smaller

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P-Value (cont)

• Small p-values indicate strong, conclusive
  evidence for rejecting the null hypothesis
• Computation is straightforward in our z-test
  example
• Compute the p-value for our telecom example

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P-Value (cont)

• P-value is also referred to as attained level of
  significance
• Results of a test are said to be statistically
  significant at the specified p-value
• Statistically significant says the difference
  between what is observed and what is assumed
  correct is most likely not due to random
  variation
• It DOES NOT MEAN the difference is important!
• It DOES NOT tell you that the difference is
  meaningful from business perspective (practical
  significance)
• With large enough sample size, any difference can
  become meaningful

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P-Value for a z Test
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Hypothesis Testing with the t Distribution

• Population standard deviation is rarely known
• Basic ideas of hypothesis testing are not
  changed, we simply switch sampling distributions

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T Test for Hypotheses about m
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Example

• Airline institutes a snake system waiting line
  at its counters to try to reduce the average
  waiting time
• Mean waiting time under specific conditions with
  the previous system was 6.1.
• A sample of 14 waiting times is taken
• Sample mean 5.043
• Standard deviation 2.266
• Test the null hypothesis of no change against an
  appropriate research hypothesis using a0.10.
• Calculate the rejection region
• Calculate the t-statistic
• Perform and interpret the hypothesis test
• Calculate the associated p-value

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Example

• Performance based benefits are a way of giving
  employees more of a stake in their work
• A study was conducted to find out how managers of
  343 firms view the effectiveness of various kinds
  of employee relations programs
• Each rated the effect of employee stock ownership
  on product quality using a scale from 2 (large
  negative effect) to 2 (large positive effect).
• Sample Mean 0.35
• Standard Error 0.14
• Do managers view employee stock ownership as a
  worthwhile technique?
• Create a 95 confidence interval for the
  population parameter
• Perform a hypothesis test that the population
  mean isnt equal to zero

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Example

• To help your restaurant marketing campaign target
  the right age levels, you want to find out if
  there is a statistically significant difference,
  on the average, between the age of your customers
  and the age of the general population in town,
  which is 43.1 years.
• A random sample of 50 customers shows an average
  of 33.6 years with a standard deviation of 16.2
  years
• Perform a two-sided test at the 1 significance
  level
• What is the p-value?

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t-Test Assumptions

• Hypothesis tests allow for random variation, but
  not for bias
• Measurements are statistically independent
• Underlying population distribution should be
  symmetric
• Skewness affects p-value

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Hypothesis Testing a Proportion

• We can also perform hypothesis tests for
  proportions / percentages by using a normal
  approximation to the binomial distribution

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Testing a Population Proportion
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Example

• A company figures out that the launch of their
  new product will only be successful if more than
  23 of consumers try the product
• Based on a pilot study based on 205 consumers,
  you expect 44.1 of consumers to try it
• How sure are you that the percentage of people
  who will try the new product is above the
  break-even point of 23?

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Using A Confidence Interval

• Construct a confidence interval (say at 95
  confidence) in the usual way
• If m0 is outside the interval, it is not a
  reasonable value for the population parameter and
  you fail to reject the research hypothesis
• Why does this work?
• Confidence interval says that the probability
  that the population parameter is in the random
  confidence interval is 0.95.
• If the null hypothesis was true, then the
  probability that m0 is in the interval is also
  95
• When the null is true, you will make the correct
  decision in 95 of all cases

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R Tutorial on Hypothesis Testing
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Testing Two Samples

• Can test whether two samples are significantly
  different or not, on the average
• Unpaired test test whether two independent
  columns of numbers are different
• Paired test test whether two columns of numbers
  are different when there is a natural pairing
  between them

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R Tutorial on Two Sample Hypothesis Testing
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Next Time

• Regression Analysis