Statistics with Economics and Business Applications

1
Statistics with Economics and Business
Applications
Chapter 8 Test of Hypotheses for Means and
Proportions Null and alternative hypotheses,
test statistic, type I and II errors,
significance level, p-value
2
Review

• I. Whats in last lecture?
• Small-Sample Estimation of a Population Mean
  
  Chapter 7
• II. What's in the next two lectures?
• Hypotheses tests for means and proportions
  Read Chapter 8

3
Introduction

• Setting up and testing hypotheses is an essential
  part of statistical inference. In order to
  formulate such a test, usually some theory has
  been put forward, either because it is believed
  to be true or because it is to be used as a basis
  for argument, but has not been proved.
• Hypothesis testing refers to the process of using
  statistical analysis to determine if the
  differences between observed and hypothesized
  values are due to random chance or to true
  differences in the samples.
• Statistical tests separate significant effects
  from mere luck or random chance.
• All hypothesis tests have unavoidable, but
  quantifiable, risks of making the wrong
  conclusion.

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Introduction

• Suppose that a pharmaceutical company
• is concerned that the mean potency m of an
  antibiotic meet the minimum government potency
  standards. They need to decide between two
  possibilities

• The mean potency m does not exceed the required
  minimum potency.
• The mean potency m exceeds the required minimum
  potency.
• This is an example of a test of hypothesis.

5
Introduction

• Similar to a courtroom trial. In trying a person
  for a crime, the jury needs to decide between one
  of two possibilities
• The person is guilty.
• The person is innocent.
• To begin with, the person is assumed innocent.
• The prosecutor presents evidence, trying to
  convince the jury to reject the original
  assumption of innocence, and conclude that the
  person is guilty.

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Five Steps of a Statistical Test

• A statistical test of hypothesis consist of
  five steps
• Specify statistical hypothesis which include a
  null hypothesis H0 and a alternative hypothesis
  Ha
• Identify and calculate test statistic
• Identify distribution and find p-value
• Make a decision to reject or not to reject the
  null hypothesis
• State conclusion

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Null and Alternative Hypothesis

• The null hypothesis, H0
• The hypothesis we wish to falsify
• Assumed to be true until we can prove otherwise.
• The alternative hypothesis, Ha
• The hypothesis we wish to prove to be true

Court trial Pharmaceuticals H0 innocent
H0 m does not exceeds required potency Ha
guilty Ha m exceeds required potency
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Examples of Hypotheses

• You would like to determine if the diameters of
  the ball bearings you produce have a mean of 6.5
  cm.

• H0 ???6.5
• Ha ????6.5
• (Two-sided or two tailed alternative)

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Examples of Hypotheses

• Do the 16 ounce cans of peaches meet the claim
  on the label (on the average)?
• Notice, the real concern would be selling the
  consumer less than 16 ounces of peaches.

• H0 ? ? 16
• Ha ??lt 16
• One-sided or one-tailed alternative

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Comments on Setting up Hypothesis

• The null hypothesis must contain the equal sign.
  
• This is absolutely necessary because the
  distribution of test statistic requires the null
  hypothesis to be assumed to be true and the value
  attached to the equal sign is then the value
  assumed to be true.
• The alternate hypothesis should be what you are
  really attempting to show to be true.
• This is not always possible.

There are two possible decisions reject or fail
to reject the null hypothesis. Note we say fail
to reject or not to reject rather than
accept the null hypothesis.
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Two Types of Errors

• There are two types of errors which can
• occur in a statistical test
• Type I error reject the null hypothesis when it
  is true
• Type II error fail to reject the null
  hypothesis when it is false

Actual Fact Your Decision H0 true H0 false
Fail to reject H0 Correct Type II Error
Reject H0 Type I Error Correct
Actual Fact Jurys Decision Guilty Innocent
Guilty Correct Error
Innocent Error Correct
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Error Analogy

• Consider a medical test where the hypotheses are
  equivalent to
• H0 the patient has a specific disease
• Ha the patient doesnt have the disease
• Then,
• Type I error is equivalent to a false negative
• (I.e., Saying the patient does not have the
  disease when in fact, he does.)
• Type II error is equivalent to a false positive
• (I.e., Saying the patient has the disease when,
  in fact, he does not.)

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Two Types of Errors

• Define
• a P(Type I error) P(reject H0 when H0 is
  true)
• b P(Type II error) P(fail to reject H0 when H0
  is false)

We want to keep the both a and ß as small as
possible. The value of a is controlled by the
experimenter and is called the significance
level. Generally, with everything else held
constant, decreasing one type of error causes the
other to increase.
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Balance Between ??and ?

• The only way to decrease both types of error
  simultaneously is to increase the sample size.
• No matter what decision is reached, there is
  always the risk of one of these errors.
• Balance identify the largest significance level
  a as the maximum tolerable risk you want to have
  of making a type I error. Employ a test procedure
  that makes type II error b as small as possible
  while maintaining type I error smaller than the
  given significance level a.

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Test Statistic

• A test statistic is a quantity calculated from
  sample of data. Its value is used to decide
  whether or not the null hypothesis should be
  rejected.
• The choice of a test statistic will depend on
  the assumed probability model and the hypotheses
  under question. We will learn specific test
  statistics later.
• We then find sampling distribution of the test
  statistic and calculate the probability of
  rejecting the null hypothesis (type I error) if
  it is in fact true. This probability is called
  the p-value

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P-value

• The p-value is a measure of inconsistency
  between the hypothesized value under the null
  hypothesis and the observed sample.
• The p-value is the probability, assuming that H0
  is true, of obtaining a test statistic value at
  least as inconsistent with H0 as what actually
  resulted.
• It measures whether the test statistic is likely
  or unlikely, assuming H0 is true. Small p-values
  suggest that the null hypothesis is unlikely to
  be true. The smaller it is, the more convincing
  is the rejection of the null hypothesis. It
  indicates the strength of evidence for rejecting
  the null hypothesis H0

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Decision

• A decision as to whether H0 should be
  rejected results from comparing the p-value
  to the chosen significance level a
• H0 should be rejected if p-value ? a.
• H0 should not be rejected if p-value gt a.

When p-valuegta, state fail to reject H0 or not
to reject rather than accepting H0. Write
there is insufficient evidence to reject H0.
Another way to make decision is to use critical
value and rejection region, which will not be
covered in this class.
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Five Steps of a Statistical Test

• A statistical test of hypothesis consist of
  five steps
• Specify the null hypothesis H0 and alternative
  hypothesis Ha in terms of population parameters
• Identify and calculate test statistic
• Identify distribution and find p-value
• Compare p-value with the given significance level
  and decide if to reject the null hypothesis
• State conclusion

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Large Sample Test for Population Mean

• Step 1 Specify the null and alternative
  hypothesis
• H0 m m0 versus Ha m ? m0 (two-sided test)
• H0 m m0 versus Ha m gt m0 (one-sided test)
• H0 m m0 versus Ha m lt m0 (one-sided test)
• Step 2 Test statistic for large sample (n30)

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Intuition of the Test Statistic

• If H0 is true, the value of should be
  close to m0, and z will be close to 0. If H0 is
  false, will be much larger or smaller than m0,
  and z will be much larger or smaller than 0,
  indicating that we should reject H0. Thus

Ha m ? m0

• z is much larger or smaller than 0 provides
  evidence against H0
• z is much larger than 0 provides evidence
  against H0
• z is much smaller than 0 provides evidence
  against H0

Ha m gt m0
Ha m lt m0
How much larger (or smaller) is large (small)
enough?
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Large Sample Test for Population Mean

• Step 3 When n is large, the sampling
  distribution of z will be approximately standard
  normal under H0. Compute sample statistic

z is defined for any possible sample. Thus it is
a random variable which can take many different
values and the sampling distribution tells us the
chance of each value. z is computed from the
given data, thus a fixed number.
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Large Sample Test for Population Mean

• Ha m ? m0 (two-sided test)

• Ha m gt m0 (one-sided test)

• Ha m lt m0 (one-sided test)

P(zgtz), P(zgtz) and P(zltz) can be found from
the normal table
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Example

• The daily yield for a chemical plant has
  averaged 880 tons for several years. The quality
  control manager wants to know if this average has
  changed. She randomly selects 50 days and
  records an average yield of 871 tons with a
  standard deviation of 21 tons. Conduct the test
  using a.05.

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Example
Decision since p-valuelta, we reject the
hypothesis that µ880. Conclusion the average
yield has changed and the change is statistically
significant at level .05.
In fact, the p-value tells us more the null
hypothesis is very unlikely to be true. If the
significance level is set to be any value greater
or equal to .0024, we would still reject the null
hypothesis. Thus, another interpretation of the
p-value is the smallest level of significance at
which H0 would be rejected, and p-value is also
called the observed significance level.
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Example
A homeowner randomly samples 64 homes similar
to her own and finds that the average selling
price is 252,000 with a standard deviation of
15,000. Is this sufficient evidence to conclude
that the average selling price is greater than
250,000? Use a .01.
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Example
Decision since the p-value is greater than a
.01, H0 is not rejected. Conclusion there is
insufficient evidence to indicate that the
average selling price is greater than 250,000.
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Small Sample Test for Population Mean

• Step 1 Specify the null and alternative
  hypothesis
• H0 m m0 versus Ha m ? m0 (two-sided test)
• H0 m m0 versus Ha m gt m0 (one-sided test)
• H0 m m0 versus Ha m lt m0 (one-sided test)
• Step 2 Test statistic for small sample

Step 3 When samples are from a normal
population, under H0 , the sampling distribution
of t has a Students t distribution with n-1
degrees of freedom
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Small Sample Test for Population Mean

• Step 3 Find p-value. Compute sample statistic
  

• Ha m ? m0 (two-sided test)
• Ha m gt m0 (one-sided test)
• Ha m lt m0 (one-sided test)

P(tgtt), P(tgtt) and P(tltt) can be found from
the t table
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Example

• A sprinkler system is designed so that the
  average time for the sprinklers to activate after
  being turned on is no more than 15 seconds. A
  test of 5 systems gave the following times
• 17, 31, 12, 17, 13, 25
• Is the system working as specified? Test using
• a .05.

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Example

• Data 17, 31, 12, 17, 13, 25
• First, calculate the sample mean and standard
  deviation.

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Approximating the p-value

• Since the sample size is small, we need to
  assume a normal population and use t
  distribution. We can only approximate the p-value
  for the test using Table 4.

Since the observed value of t 1.38 is smaller
than t.10 1.476, p-value gt .10.
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Example
Decision since the p-value is greater than .1,
than it is greater than a .05, H0 is not
rejected. Conclusion there is insufficient
evidence to indicate that the average activation
time is greater than 15 seconds.
Exact p-values can be calculated by computers.
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Large Sample Test for Difference Between Two
Population Means
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Large Sample Test for Difference Between Two
Population Means

• Step 1 Specify the null and alternative
  hypothesis
• H0 m1-m2D0 versus Ha m1- m2? D0
• (two-sided test)
• H0 m1-m2D0 versus Ha m1- m2gt D0
• (one-sided test)
• H0 m1-m2D0 versus Ha m1- m2lt D0
• (one-sided test)
• where D0 is some specified difference that
  you wish to test. D00 when testing no difference.

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Large Sample Test for Difference Between Two
Population Means

• Step 2 Test statistic for large sample sizes
  when
• n130 and n230

Step 3 Under H0, the sampling distribution of z
is approximately standard normal
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Large Sample Test for Difference Between Two
Population Means

• Step 3 Find p-value. Compute

• Ha m1- m2? D0 (two-sided test)
• Ha m1- m2gt D0 (one-sided test)
• Ha m1- m2lt D0 (one-sided test)

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Example
Avg Daily Intakes Men Women
Sample size 50 50
Sample mean 756 762
Sample Std Dev 35 30

• Is there a difference in the average daily
  intakes of dairy products for men versus women?
  Use a .05.

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Example
Decision since the p-value is greater than a
.05, H0 is not rejected. Conclusion there is
insufficient evidence to indicate that men and
women have different average daily intakes.
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Small Sample Testing the Difference between Two
Population Means
Note that both population are normally
distributed with the same variances
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Small Sample Testing the Difference between Two
Population Means

• Step 1 Specify the null and alternative
  hypothesis
• H0 m1-m2D0 versus Ha m1- m2? D0
• (two-sided test)
• H0 m1-m2D0 versus Ha m1- m2gt D0
• (one-sided test)
• H0 m1-m2D0 versus Ha m1- m2lt D0
• (one-sided test)
• where D0 is some specified difference that
  you wish to test. D00 when testing no difference.

41
Small Sample Testing the Difference between Two
Population Means

• Step 2 Test statistic for small sample sizes

Step 3 Under H0, the sampling distribution of t
has a Students t distribution with n1n2-2
degrees of freedom
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Small Sample Testing the Difference between Two
Population Means

• Step 3 Find p-value. Compute

• Ha m1- m2? D0 (two-sided test)
• Ha m1- m2gt D0 (one-sided test)
• Ha m1- m2lt D0 (one-sided test)

43
Example
Two training procedures are compared by
measuring the time that it takes trainees to
assemble a device. A different group of trainees
are taught using each method. Is there a
difference in the two methods? Use a .01.
Time to Assemble Method 1 Method 2
Sample size 10 12
Sample mean 35 31
Sample Std Dev 4.9 4.5
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Example
Time to Assemble Method 1 Method 2
Sample size 10 12
Sample mean 35 31
Sample Std Dev 4.9 4.5
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Example
Decision since the p-value is greater than a
.01, H0 is not rejected. Conclusion there is
insufficient evidence to indicate a difference in
the population means.
df n1 n2 2 10 12 2 20
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The Paired-Difference Test

• We have assumed that samples from two
  populations are independent. Sometimes the
  assumption of independent samples is
  intentionally violated, resulting in a
  matched-pairs or paired-difference test.
• By designing the experiment in this way, we can
  eliminate unwanted variability in the experiment
• Denote data as

Pair 1 2 n
Population 1 x11 x12 x1n
Population 2 x21 x22 x2n
Difference d1 x11- x21 d2x12- x22 dnx1n- x2n
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The Paired-Difference Test

• Step 1 Specify the null and alternative
  hypothesis
• H0 m1-m2D0 versus Ha m1- m2? D0
• (two-sided test)
• H0 m1-m2D0 versus Ha m1- m2gt D0
• (one-sided test)
• H0 m1-m2D0 versus Ha m1- m2lt D0
• (one-sided test)
• where D0 is some specified difference that
  you wish to test. D00 when testing no difference.

48
The Paired-Difference Test

• Step 2 Test statistic for small sample sizes

Step 3 Under H0, the sampling distribution of t
has a Students t distribution with n-1 degrees
of freedom
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The Paired-Difference Test

• Step 3 Find p-value. Compute

• Ha m1- m2? D0 (two-sided test)
• Ha m1- m2gt D0 (one-sided test)
• Ha m1- m2lt D0 (one-sided test)

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Paired t Test Example
A weight reduction center advertises that
participants in its program lose an average of at
least 5 pounds during the first week of the
participation. Because of numerous complaints,
the states consumer protection agency doubts
this claim. To test the claim at the 0.05 level
of significance, 12 participants were randomly
selected. Their initial weights and their
weights after 1 week in the program appear on the
next slide. Set up and perform an appropriate
hypothesis test.
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Paired Sample Example continued
52
Paired Sample Example continued
Each member serves as his/her own pair. weight
changesinitial weightweight after one week
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Paired Sample Example continued
Decision since the p-value is smaller than a
.05, H0 is rejected. Conclusion there is strong
evidence that the mean weight loss is less than 5
pounds for those who took the program for one
week.
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Key Concepts

• I. A statistical test of hypothesis consist of
  five steps
• Specify the null hypothesis H0 and alternative
  hypothesis Ha in terms of population parameters
• Identify and calculate test statistic
• Identify distribution and find p-value
• Compare p-value with the given significance level
  and decide if to reject the null hypothesis
• State conclusion

55
Key Concepts

• II. Errors and Statistical Significance
• Type I error reject the null hypothesis when it
  is true
• Type II error fail to reject the null
  hypothesis when it is false
• The significance level a P(type I error)
• b P(type II error)
• The p-value is the probability of observing a
  test statistic as extreme as or more than the one
  observed also, the smallest value of a for which
  H 0 can be rejected
• When the p-value is less than the significance
  level a , the null hypothesis is rejected

56
Key Concepts

1. Test for a population mean H0 µµ0

large
small
57
Key Concepts

• IV. Test for Difference Between Two Population
  Mean H0 µ1-µ2D0

large
small
58
Key Concepts

• V. The Paired-Difference Test

• Ha m1- m2? D0 (two-sided test)
• Ha m1- m2gt D0 (one-sided test)
• Ha m1- m2lt D0 (one-sided test)