Section 9.1 ~ Fundamentals of Hypothesis Testing

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Section 9.1 Fundamentals of Hypothesis Testing

• Introduction to Probability and Statistics
• Ms. Young

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Objective
Sec. 9.1

• After this section you will understand the goal
  of hypothesis testing and the basic structure of
  a hypothesis test, including how to set up the
  null and alternative hypotheses, how to determine
  the possible outcomes of a hypothesis test, and
  how to decide between these possible outcomes.

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Statistical Claims

• Of our 350 million users, more than 50 log on
  to Facebook everyday
• Using Gender Choice could increase a womans
  chance of giving birth to a baby girl up to 80
• According to the U.S. Census Bureau, Current
  Population Surveys, March 1998, 1999, and 2000,
  the average salary of someone with a high school
  diploma is 30,400 while the average salary of
  someone with a Bachelor's Degree is 52,200.
• How could we determine whether these claims are
  true or not?
• Hypothesis Testing

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Formulating the Hypothesis
Sec. 9.1

• A hypothesis is a claim about a population
  parameter
• Could either be a claim about a population mean,
  µ, or a population proportion, p
• All of the claims on the previous slide would be
  considered hypotheses
• A hypothesis test is a standard procedure for
  testing a claim about a population parameter
• There are always at least two hypotheses in any
  hypothesis test
• The null hypothesis the claim does not hold
  true
• The alternative hypothesis the claim does hold
  true

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Null Hypothesis
Sec. 9.1

• The null hypothesis, represented as (read as
  H-naught), is the starting assumption for a
  hypothesis test
• The null hypothesis always claims a specific
  value for a population parameter and therefore
  takes the form of an equality
• Take the claim, using Gender Choice could
  increase a womans chance of giving birth to a
  baby girl up to 80 for example. If the product
  did not work, it would be expected that there
  would be an approximately equally likely chance
  of having either a boy or a girl. Therefore, the
  null hypothesis (the claim not working) would be

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Alternative Hypothesis
Sec. 9.1

• The alternative hypothesis, represented as ,
  is a claim that the population parameter has a
  value that differs from the value claimed in the
  null hypothesis, or in other words, the claim
  does hold true
• The alternative hypothesis can take one of the
  following forms
• left tailed
• Ex. A manufacturing company claims that their
  new hybrid model gets 62 mpg. A consumer group
  claims that the mean fuel consumption of this
  vehicle is less than 62 mpg.
• This alternative hypothesis would be considered
  left-tailed since the claimed value is smaller
  (or to the left) of the null value
• right tailed
• Ex. The claim that Gender Choice increases a
  womans chance of having a baby girl up to 80
  would be testing values above the null value of
  .5, and would therefore be right-tailed

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Alternative Hypothesis Contd
Sec. 9.1

• two tailed
• Ex. A wildlife biologist working in the African
  savanna claims that the actual proportion of
  female zebras in the region is different from the
  accepted proportion of 50.
• Since the claim does not specify whether the
  alternative hypothesis is above 50 or below 50,
  it would be considered two-tailed in which case
  the values above and below would be tested
  

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Possible Outcomes of a Hypothesis Test
Sec. 9.1

• There are two possible outcomes to a hypothesis
  test
• Reject the null hypothesis in which case we have
  evidence in support of the alternative hypothesis
• Not reject the null hypothesis in which case we
  do not have enough evidence to support the
  alternative hypothesis
• NOTE Accepting the null hypothesis is not a
  possible outcome since it is the starting
  assumption.
• The test may provide evidence to NOT REJECT the
  null hypothesis, but that does not mean that the
  null hypothesis is true
• Be sure to formulate the null and alternative
  hypotheses prior to choosing a sample to avoid
  bias

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Example 1
Sec. 9.1

• For the following case, describe the possible
  outcomes of a hypothesis test and how we would
  interpret these outcomes
• The manufacturer of a new model of hybrid car
  advertises that the mean fuel consumption is
  equal to 62 mpg on the highway (µ 62 mpg). A
  consumer group claims that the mean is less than
  62 mpg (µ lt 62 mpg).
• Possible outcomes
• Reject the null hypothesis of µ 62 mpg in which
  case we have evidence in support of the consumer
  groups claim that the mean mpg of the new hybrid
  is less than 62
• Do not reject the null hypothesis, in which case
  we lack evidence to support the consumer groups
  claim
• Note this does not necessarily imply that the
  manufacturers claim is true though

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Drawing a Conclusion from a Hypothesis Test
Sec. 9.1

• Using the claim that Gender Choice could increase
  a womans chance of giving birth to a baby girl
  up to 80, suppose that a sample produces a
  sample proportion of, .
• Although this supports the alternative hypothesis
  of , is it enough evidence to reject
  the null hypothesis?
• This is where statistical significance comes into
  play (introduced in section 6.1)
• Recall that something is considered to be
  statistically significant if it most likely DID
  NOT occur by chance
• There are two levels of statistical significance
• The 0.05 level which means that if the
  probability of a particular result occurring is
  less than 0.05, or 5, then it is considered to
  be statistically significant at the 0.05 level
• The 0.01 level which means that if the
  probability of a particular result occurring is
  less than 0.01, or 1, then it is considered to
  be statistically significant at the 0.01 level
• The 0.01 level would represent a stronger
  significance than the 0.05 level

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Hypothesis Test Decisions Based on Levels of
Statistical Significance
Sec. 9.1

• We decide the outcome of a hypothesis test by
  comparing the actual sample result (mean or
  proportion) to the null hypothesis. We must
  choose a significance level for the decision.
• If the chance of a sample result at least as
  extreme as the observed result is less than 0.01,
  then the test is statistically significant at the
  0.01 level and offers STRONG evidence for
  rejecting the null hypothesis
• If the chance of a sample result at least as
  extreme as the observed result is less than 0.05,
  then the test offers MODERATE evidence for
  rejecting the null hypothesis
• If the chance of a sample result at least as
  extreme as the observed result is greater than
  the chosen level of significance (0.01 or 0.05),
  then we DO NOT reject the null hypothesis

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P-Values
Sec. 9.1

• A P-Value, or probability value, is the value
  that represents the probability of selecting a
  sample at least as extreme as the observed sample
• In other words, it is the value that allows us to
  determine if something is statistically
  significant or not
• NOTE notice that the P-Value is represented
  using a capitol P, whereas the population
  proportion is represented using a lowercase p.
• We will learn how to actually calculate the
  P-Value in the following sections
• A small P-value indicates that the observed
  result is unlikely (therefore statistically
  significant) and provides evidence to reject the
  null hypothesis
• A large P-value indicates that the sample result
  is not unusual, therefore not statistically
  significant - or that it could easily occur by
  chance, which tells us to NOT reject the null
  hypothesis

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Example 2
Sec. 9.1

• You suspect that a coin may have a bias toward
  landing tails more often than heads, and decide
  to test this suspicion by tossing the coin 100
  times. The result is that you get 40 heads (and
  60 tails). A calculation (not shown here)
  indicates that the probability of getting 40 or
  fewer heads in 100 tosses with a fair coin is
  0.0228. Find the P-value and level of statistical
  significance for your result. Should you conclude
  that the coin is biased against heads?
• The P-Value is 0.0228
• This value is smaller than 5 (.05), but not
  smaller than 1 (.01), so it is statistically
  significant at the 0.05 level which gives us
  moderate reason to reject the null hypothesis and
  conclude that the coin is biased against heads

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Putting It All Together
Sec. 9.1
Step 1. Formulate the null and alternative
hypotheses, each of which must make a claim about
a population parameter, such as a population mean
(µ) or a population proportion (p) be sure this
is done before drawing a sample or collecting
data. Based on the form of the alternative
hypothesis, decide whether you will need a left-,
right-, or two-tailed hypothesis test. Step 2.
Draw a sample from the population and measure the
sample statistics, including the sample size (n)
and the relevant sample statistic, such as the
sample mean (x) or sample proportion (p). Step 3.
Determine the likelihood of observing a sample
statistic (mean or proportion) at least as
extreme as the one you found under the assumption
that the null hypothesis is true. The precise
probability of such an observation is the P-value
(probability value) for your sample result. Step
4. Decide whether to reject or not reject the
null hypothesis, based on your chosen level of
significance (usually 0.05 or 0.01, but other
significance levels are sometimes used).