BCOR 1020 Business Statistics

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BCOR 1020Business Statistics

• Lecture 19 April 1, 2008

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Overview

• Chapter 9 Hypothesis Testing
• Logic of Hypothesis Testing
• Testing a Proportion (p)

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Chapter 9 Logic of Hypothesis Testing

• What is a statistical test of a hypothesis?
• Hypotheses are a pair of mutually exclusive,
  collectively exhaustive statements about the
  world.
• One statement or the other must be true, but they
  cannot both be true.
• We make a statement (hypothesis) about some
  parameter of interest.
• This statement may be true or false.
• We use an appropriate statistic to test our
  hypothesis.
• Based on the sampling distribution of our
  statistic, we can determine the error associated
  with our conclusion.

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Chapter 9 Logic of Hypothesis Testing

• 5 Components of a Hypothesis Test
• Level of Significance, a maximum probability of
  a Type I Error
• Usually 5
• Null Hypothesis, H0 Statement about the value
  of the parameter being tested
• Always in a form that includes an equality
• Alternative Hypothesis, H1 Statement about the
  possible range of values of the parameter if H0
  is false
• Usually the conclusion we are trying to reach (we
  will discuss.)
• Always in the form of a strict inequality

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Chapter 9 Logic of Hypothesis Testing

• 5 Components of a Hypothesis Test
• Test Statistic and the Sampling Distribution of
  the Test statistic under the assumption that H0
  is true (the equality in H0)
• Z and T statistics for now
• Decision Criteria do we reject H0 or do we
  accept H0?
• P-value of the test
• or
• Comparing the Test statistic to critical regions
  of its distribution under H0.

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Chapter 9 Logic of Hypothesis Testing

• Error in a Hypothesis Test

• Type I error Rejecting the null hypothesis when
  it is true.
• a P(Type I Error) P(Reject H0 H0 is True)
• Type II error Failure to reject the null
  hypothesis when it is false.
• b P(Type II Error) P(Fail to Reject H0 H0
  is False)

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Chapter 9 Logic of Hypothesis Testing

• General Approach to Hypothesis Testing
• Select acceptable error levels
• a (always chosen before conducting the test)
• b (often computed after the test has been
  conducted)
• Select Null and Alternative Hypotheses
• Based on the problem objectives
• A statistical hypothesis is a statement about the
  value of a population parameter q.
• A hypothesis test is a decision between two
  competing mutually exclusive and collectively
  exhaustive hypotheses about the value of q.

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Chapter 9 Logic of Hypothesis Testing

• The Direction of the Hypothesis Test
• indicated by H1

gt indicates a right-tailed test lt indicates a
left-tailed test ? indicates a two-tailed test
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Chapter 9 Logic of Hypothesis Testing
When to use a One- or Two-Sided Test

• A two-sided hypothesis test (i.e., q ? q0) is
  used when direction (lt or gt) is of no interest to
  the decision maker
• A one-sided hypothesis test is used when - the
  consequences of rejecting H0 are asymmetric,
  or - where one tail of the distribution is of
  special importance to the researcher.
• Rejection in a two-sided test guarantees
  rejection in a one-sided test, other things being
  equal.

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Chapter 9 Logic of Hypothesis Testing

• General Approach to Hypothesis Testing
• Define an appropriate test statistic and its
  distribution under the null hypothesis
• This will vary depending on what parameter we are
  testing a hypothesis about and the assumptions we
  use.
• Define your decision criteria
• How do you decide whether to reject or accept
  H0?

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Chapter 9 Logic of Hypothesis Testing
Decision Rule

• A test statistic shows how far the sample
  estimate is from its expected value, in terms of
  its own standard error.
• The decision rule uses the known sampling
  distribution of the test statistic to establish
  the critical value that divides the sampling
  distribution into two regions.
• Reject H0 if the test statistic lies in the
  rejection region.

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Chapter 9 Logic of Hypothesis Testing
Decision Rule for Two-Tailed Test

• Reject H0 if the test statistic lt left-tail
  critical value or if the test statistic gt
  right-tail critical value.

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Chapter 9 Logic of Hypothesis Testing
Decision Rule for Left-Tailed Test

• Reject H0 if the test statistic lt left-tail
  critical value.

Decision Rule for Right-Tailed Test

• Reject H0 if the test statistic gt right-tail
  critical value.

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Chapter 9 Logic of Hypothesis Testing
Decision Rule

• a, the probability of a Type I error, is the
  level of significance (i.e., the probability that
  the test statistic falls in the rejection region
  even though H0 is true).

a P(reject H0 H0 is true)

• A Type I error is sometimes referred to as a
  false positive.
• For example, if we choose a .05, we expect to
  commit a Type I error about 5 times in 100.

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Chapter 9 Logic of Hypothesis Testing
Decision Rule

• A small a is desirable, other things being equal.
• Chosen in advance, common choices for a are
  .10, .05, .025, .01 and .005 (i.e., 10,
  5, 2.5, 1 and .5).
• The a risk is the area under the tail(s) of the
  sampling distribution.
• In a two-sided test, the a risk is split with a/2
  in each tail since there are two ways to reject
  H0.

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Chapter 9 Logic of Hypothesis Testing
Decision Rule

• b, the probability of a type II error, is the
  probability that the test statistic falls in the
  acceptance region even though H0 is false.

b P(fail to reject H0 H0 is false)

• b cannot be chosen in advance because it depends
  on a and the sample size.
• A small b is desirable, other things being equal.

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Chapter 9 Logic of Hypothesis Testing
Power of a Test

• The power of a test is the probability that a
  false hypothesis will be rejected.
• Power 1 b
• A low b risk means high power.
• Larger samples lead to increased power.

Power P(reject H0 H0 is false) 1 b
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Chapter 9 Logic of Hypothesis Testing
Relationship Between a and b

• Both a small a and a small b are desirable.
• For a given type of test and fixed sample size,
  there is a trade-off between a and b.
• The larger critical value needed to reduce a risk
  makes it harder to reject H0, thereby increasing
  b risk.
• Both a and b can be reduced simultaneously only
  by increasing the sample size.

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Chapter 9 Logic of Hypothesis Testing
Choice of a

• The choice of a should precede the calculation of
  the test statistic.

Significance versus Importance

• The standard error of most sample estimators
  approaches 0 as sample size increases.
• In this case, no matter how small, q q0 will be
  significant if the sample size is large enough.
• Therefore, expect significant effects even when
  an effect is too slight to have any practical
  importance.

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Chapter 9 Testing a Proportion

• Recall our motivating example for the
• discussion of confidence intervals
• Suppose your business is planning on bringing a
  new product to market.
• There is a business case to proceed only if
• the cost of production is less than 10 per unit
• and
• At least 20 of your target market is willing to
  pay 25 per unit to purchase this product.
• How do you determine whether or not to proceed?
• You will likely conduct experiments/surveys to
  estimate these variables and make appropriate
  inferences

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Chapter 9 Testing a Proportion

• Motivating Example (continued)
• The percentage of your target market that is
  willing to pay 25 per unit to purchase this
  product can be modeled as the probability of a
  success for a binomial variable.
• You can conduct survey research on your target
  market.
• If 200 people are surveyed and 44 say they would
  pay 25 to purchase this product, what can you
  conclude?
• We will test an appropriate hypothesis to
  determine whether the proportion of our target
  market that is willing to pay 25 per unit
  exceeds 20 (as required by the business case).

(Overhead)
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Chapter 9 Testing a Proportion

• To Test a Hypothesis on a Proportion, p
• Select your level of significance, a.
• Usually we use a 0.05 (or 5).
• Select your null and alternative hypotheses based
  on the problem statement.
• Choose from

• H0 p gt p0
• H1 p lt p0

(ii) H0 p lt p0 H1 p gt p0
(iii) H0 p p0 H1 p p0
where p0 is the null hypothesized value of p
(based on the problem statement).
Since we are controlling a, the conclusion we
want to test is in H1.
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Chapter 9 Testing a Proportion

• To Test a Hypothesis on a Proportion, p
• Define the test statistic and its distribution
  under the null hypothesis
• Start with the point estimate of p, p X/n
• Recall that for a large enough n np gt 10 and n(1
  p) gt 10, p is approximately normal with

and
If the null hypothesis is true, particularly if
p p0, p is approximately normal with
and
Based on this, we define the following test
statistic which has an approximate standard
normal distribution under H0
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Chapter 9 Testing a Proportion

• To Test a Hypothesis on a Proportion, p
• Define the decision criteria
• We will compare our test statistic to critical
  values (based on a) of its distribution under H0.
• If the test statistic is more extreme than the
  critical value of the null distribution, we
  reject H0 in favor of H1.
• Our comparison and depends on which of the three
  alternative hypotheses we are considering.

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Chapter 9 Testing a Proportion

• To Test a Hypothesis on a Proportion, p
• Define the decision criteria (continued)

• For the hypothesis test H0 p gt p0 vs. H1 p lt
  p0, we will reject H0 in favor of H1 if Z lt Za.

(ii) For the hypothesis test H0 p lt p0 vs. H1 p
gt p0, we will reject H0 in favor of H1 if
Z gt Za.
(iii) For the hypothesis test H0 p p0 vs. H1
p p0, we will reject H0 in favor of H1
if Z gt Za/2.
If we decide to reject H0 in favor of H1, then we
say that there is statistically significant
evidence that H0 is false and H1 is true. The
probability that this conclusion is wrong is no
greater than a.
If we decide not to reject H0 in favor of H1,
then we say that H0 is plausible (but we do not
know the probability that this statement is wrong
yet).
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Chapter 9 Testing a Proportion

• To Test a Hypothesis on a Proportion, p
• Form Hypotheses
• Collect data
• Conduct test
• Report results
• Repeat and refine as necessary
• Back to our motivating example

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Chapter 9 Testing a Proportion

• Example
• We will test an appropriate hypothesis to
  determine whether the proportion of our target
  market that is willing to pay 25 per unit
  exceeds 20 (as required by the business case).
  If 200 people are surveyed and 44 say they would
  pay 25 to purchase this product, what can you
  conclude?
• Our point estimate, p X/n 44/200 0.22
  exceeds 20. Does this mean we can conclude that
  the population proportion exceeds 20?
• No! We must conduct the appropriate hypothesis
  test.

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Chapter 9 Testing a Proportion

• Example (continued)
• We will choose the level of significance, a
  0.05. If we reject H0, the probability that we
  have made an error (i.e. that H0 is true) will be
  no greater than 5.
• We must choose the appropriate null and
  alternative hypotheses. Recall that the
  conclusion we are hoping to test should be in the
  alternative.
• Since we want to determine whether p exceeds 20,
  
• we want to test

(ii) H0 p lt p0 H1 p gt p0
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Chapter 9 Testing a Proportion

• Example (continued)
• We calculate our test statistic for the test of
  the proportion

N(0,1) if H0 is true.

• We define our rejection criteria and make a
  decision based on the data.

• For the hypothesis test H0 p lt p0 vs. H1 p gt
  p0, we will reject H0 in favor of H1 if Z gt Za.

Since we chose a 0.05, Za Z.05 1.645.
Since Z 0.71 is not greater than Z.05 1.645,
we will fail to reject H0 in favor of H1.
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Chapter 9 Testing a Proportion

• Example (continued)
• We state the conclusion of our hypothesis test in
  clear language.
• Since Z 0.71 is not greater than Z.05 1.645,
  we will fail to reject H0 in favor of H1. (This
  is probably not clear to someone who is not a
  statistician.)
• Based on the data in our study, there is not
  statistically significant evidence that the
  proportion of our target market that is willing
  to pay 25 per unit exceeds 20.

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Chapter 9 Testing a Proportion

• Calculating the p-value of the test
• The p-value of the test is the exact probability
  of a type I error based on the data collected for
  the test. It is a measure of the plausibility of
  H0.
• P-value P(Reject H0 H0 is True) based on our
  data.
• Formula depends on which pair of hypotheses we
  are testing

• For the hypothesis test H0 p gt p0 vs. H1 p lt
  p0,

(ii) For the hypothesis test H0 p lt p0 vs. H1 p
gt p0,
(iii) For the hypothesis test H0 p p0 vs. H1
p p0,
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Chapter 9 Testing a Proportion

• Example Lets calculate the p-value of the
• test in our example
• We found Z 0.71
• Since we were testing H0 p lt p0 vs. H1 p gt p0,

Interpretation If we were to reject H0 based on
the observed data, there is a 28 chance we would
be making a type I error. Since this is larger
than a 5, we will not reject H0.
Area 0.2839
-0.71