Hypothesis Testing

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Chapter 8

• Hypothesis Testing
• Null and Alternative Hypotheses and Errors in
  Testing
• Large Sample Tests about a Mean Rejection
  Points
• Small Sample Tests about a Population Mean
• Hypothesis Tests about a Population Proportion

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Introduction

• A Hypothesis Test is a statistical procedure
• that involves formulating a hypothesis and
• using sample data to decide on the validity of
• the hypothesis.
• In this chapter we will focus on hypothesis tests
  about population means and proportions (since we
  are knowledgeable about how sample means and
  proportions are distributed).

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• In order to test any hypothesis (even non
  statistical hypotheses) you need three elements
• 1. The Hypotheses (Both the hypothesis that you
  are testing and the alternative hypothesis, which
  is the opposite of the hypothesis)
• 2. An unbiased test statistic- A measure that
  you will use to evaluate the hypotheses.
• 3. A rejection rule- the rule that you will use
  to ultimately decide if a hypothesis should be
  rejected.

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Non Statistical Examples

• Example 1
• Hypothesis I should be admitted to Harvard Law
  School
• Alternative Hypothesis I should not be admitted
  to Harvard Law School
• Test Statistic LSAT
• Rejection Rule Harvard may have a cut off LSAT
  score for admitting students.
• Example 2
• Hypothesis OJ is innocent
• Alternative Hypothesis OJ is guilty
• Test Statistic A jury of 12 of his peers.
• Rejection Rule If 12 of 12 jurors rule that he
  is guilty beyond a reasonable doubt.

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Hypothesis Testing Rules

• In order to test a hypothesis, you must first
  find the test statistic and rejection rule that
  is appropriate for evaluating your hypotheses
  (i.e., we could not evaluate OJs innocence based
  upon his LSAT score).
• All hypothesis tests will always end in 1 of 2
  ways
• 1. You conclude that you must reject the
  hypothesis (This is the same as concluding that
  you have proven the alternative true) or
• 2. You conclude that you do not have enough
  evidence to reject the hypothesis (This is the
  same as concluding that you do not have enough
  evidence to prove that the alternative is true).

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Developing Null and Alternative Hypotheses

• Hypothesis testing can be used to determine
  whether a statement about the value of a
  population parameter should or should not be
  rejected.
• The null hypothesis, denoted H0, is a statement
  of the basic proposition being tested. The
  statement is not rejected unless there is
  convincing sample evidence that it is false.
• The alternative or research hypothesis, denoted
  Ha, is an alternative (to the null hypothesis)
  statement that will be accepted only if there is
  convincing sample evidence that it is true.

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Developing Null and Alternative Hypotheses

• Testing the Validity of a Claim
• If you wish to find evidence to contradict a
  claim, the claim should be stated as the null
  hypothesis.
• If you wish to prove a claim to be true, then you
  should state the claim as the alternative
  hypothesis.
• Claims that test whether the mean is equal to a
  specific value must be stated as the null
  hypothesis.

Anderson, Sweeney, and Williams
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A Summary of Forms for Null and Alternative
Hypotheses about a Population Mean

• The equality part of the hypotheses always
  appears in the null hypothesis.
• In general, a hypothesis test about the value of
  a population mean ?? must take one of the
  following three forms
• Where ?0 is a specific value
  

H0 ?gt ?0 Ha ? lt ?0
H0 ?lt ?0 Ha ? gt ?0
H0 ? ?0 Ha ? ? ?0
One-tailed
One-tailed
Two-tailed
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Example 8.1 Metro EMS

• A major west coast city provides one of the most
  comprehensive emergency medical services in the
  world. Operating in a multiple hospital system
  with approximately 20 mobile medical units, the
  service goal is to respond to medical emergencies
  with a mean time of 12 minutes or less.
• The director of medical services wants to
  formulate a hypothesis test that could use a
  sample of emergency response times to determine
  whether there is sufficient evidence to prove
  that they are not meeting their goal.

Anderson, Sweeney, and Williams
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• Null and Alternative Hypotheses
• Hypotheses Conclusion and Action
• H0 ?lt??? The emergency service is
  meeting the response goal no
  appropriate follow-up action is
  necessary.
• Ha???gt??? The emergency
  service is not meeting
• the response goal appropriate
  follow-up action is necessary.
• Where ? mean response time for the
  population
• of medical emergency
  requests.
• By defining the claim as the null hypothesis we
  can set out to try to find sufficient evidence to
  reject the claim.

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Example 8.2 The Potato Chip Manufacturer

• Many people eat chips with their soda. Suppose a
  potato chip
• manufacturer is concerned that the bagging
  equipment may not be
• functioning properly when filling 10-oz bags.
  You have been asked to
• set up a hypothesis test that will help determine
  if there is a problem with
• the bagging equipment. What null and alternative
  hypothesis would you
• use?

Pelosi and Sandifer
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• Hypotheses Conclusion and
  Action
• H0 ???0 The machine is working
  properly no appropriate follow-up
  action is necessary.
• Ha??????0 The machine is
  not working properly
• 
  appropriate follow-up action is necessary.
• Where ? mean filling weight for the
  machine.

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Type I and Type II Errors

• Since hypothesis tests are based on sample data,
  we must allow for the possibility of errors.
• A Type I error is rejecting H0 when it is true.
• The person conducting the hypothesis test
  specifies the maximum allowable probability of
  making a Type I error, denoted by ? and called
  the level of significance.

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• A Type II error is accepting H0 when it is false.
• Generally, we cannot control for the probability
  of making a Type II error, denoted by ?.
• Statistician avoids the risk of making a Type II
  error by using the phrase do not reject H0
  instead of accept H0.

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Large Sample Tests about Mean (n?30)

• If the sampled population is normal or if n is
  large,

H0 ?gt ?0 Ha ? lt ?0 Test Statistic Reject
ion Rule Reject H0 if z lt- z?????
H0 ?lt ?0 Ha ? gt ?0 Test Statistic Reject
ion Rule Reject H0 if z gt z ????????

• H0 ? ?0
• Ha ? ? ?0
• Test Statistic
• Rejection Rule
• Reject H0 if z gt z???

If ? is unknown, use s to estimate ?
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Steps for Computing z? (The z value with an upper
tail area of ?.

1. In order to determine the z value with an upper
   tail area of ?, we need the area beneath the
   normal curve between the mean and the z value of
   interest.

area .5- ?
?
2. Go to the area section of the standard
normal table and find the area
closest to the area computed in 2. The
corresponding z value is z?.
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• One-Tailed Test about a Population Mean Large n
• ? P(Type I Error)

Sampling distribution of (assuming H0 is
true)
Reject H0
Do Not Reject H0
??
Anderson, Sweeney, and Williams
?0
z?
(Critical value)
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• Two-Tailed Test about a Population Mean Large n
• ? P(Type I Error)

Sampling distribution of (assuming H0 is
true)
Reject H0
Reject H0
???/2
???/2
Anderson, Sweeney, and Williams
z
z?/2
-z?/2
?0
(Critical values)
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Steps of Hypothesis Testing

• Determine the null and alternative hypotheses.
• Specify the level of significance ?.
• Select the test statistic that will be used to
  test the hypothesis.
• Using the Test Statistic
• State the rejection rule for H0 and use ??to
  determine the critical value for the test
  statistic.
• Collect the sample data and compute the value of
  the test statistic.
• Use the value of the test statistic and the
  rejection rule to determine whether to reject H0.

Anderson, Sweeney, and Williams
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Example 8.1 (Revisited)

• Recall example 8.1. Suppose we collected a
  sample of n 40 EMS calls and computed
  13.25 minutes and s 3.2 minutes. Using
  ?.05 conduct a hypothesis test to see if you can
  find the evidence to refute their claim that the
  average response time is less than 12 minutes.
• (The sample standard deviation s can be used
  to
• estimate the population standard deviation
  ?.)
• Step 1 H0 ??lt12 ?
• Ha? ?gt?12
• Step 2 ?????? ?.05
• Step 3

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• Step 4 Reject H0 if z gt za 1.645
  (a0.05)
• Step 5
• Step 6. Is zgt 1.645?
• Since 2.47 gt 1.645, we reject H0.
• Conclusion We are 95 confident that Metro
  EMS
• is not meeting the response goal of 12
  minutes
• appropriate action should be taken to improve
• service.

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Example 8.3

• Consider a company that is trying a new and
  cheaper package design
• for its product. The average sales for this
  product are currently
• 1500/month. Suppose they wish to prove that
  sales will decrease as a
• result of the new method. In order to test this
  claim they used n36
• test stores and computed an average sale,
  1450 with s 250. (Use ?.10)
• Step 1 H0 ??gt1500 ?
• Ha? ?lt?1500
• Step 2 ?????? ?.1
• Step 3

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• Step 4 Reject H0 if z lt- za 1.28
  (a0.10)
• Step 5
• Step 6. Is zlt-1.28?
• Since -1.2 is not less than -1.28, we cannot
  reject H0.
• Thus we cannot find sufficient evidence to prove
  that the new advertising
• method will decrease sales

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Example 8.4 The Chapperel Steel Company

• Another recent management approach is to have
  employees become actual
• partners of the business. Chapperel Steel
  Company has done exactly this
• and the company feels that one of the benefits of
  this concept is that the
• average number of sick days will decrease. Prior
  to implementing this
• program, Chapperel had an average of 7.2 sick
  days per employee. Set up
• the null and alternative hypothesis to test if
  the average number of sick
• days per employee is different from 7.2.
• After implementing this program, a sample of 40
  employees provides a
• sample mean of 6.5 day and a standard deviation
  of 2.5 days. Test this
• hypothesis test using ?.01.

Pelosi and Sandifer
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• Step 1 H0 ????7.2 ?
• ????Ha? ? ? ?7.2
• Step 2 ?.01
• Step 3
• ??????
• Step 4 Reject H0 if z gt za/2 2.575
  (?.01)

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• Step 5
• Step 6 Is z gt 2.575 ?
• Since -1.77 1.77 does not exceed 2.575,
  we cannot reject this claim. Thus we cant
  refute the claim that this program does alter the
  number of employee sick days.

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Example 8.5 Glow Toothpaste

• The production line for Glow toothpaste is
  designed to fill tubes of toothpaste with a mean
  weight of 6 ounces.
• Periodically, a sample of 30 tubes will be
  selected in order to check the filling process.
  Quality assurance procedures call for the
  continuation of the filling process if the sample
  results are consistent with the assumption that
  the mean filling weight for the population of
  toothpaste tubes is 6 ounces otherwise the
  filling process will be stopped and adjusted.

Anderson, Sweeney, and Williams
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• Step 1 H0 ????? ?
• Ha? ??????
• Step 2 Assume a .05 level of significance.
• Step 3
• Step 4 ?????Assuming a .05 level of
  significance,
• Reject H0 if z gt za/2 1.96 (?.05)

Anderson, Sweeney, and Williams
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• Step 5
• Assume that a sample of 30 toothpaste tubes
• provides a sample mean of 6.1 ounces and standard
• deviation of 0.2 ounces.
• Let n 30, 6.1 ounces, s .2
  ounces
• Step 6 Is z gt 1.96 ?
• Since 2.74 gt 1.96, we reject H0. Thus, the
  mean filling weight for the population of
  toothpaste tubes is not 6 ounces.

Anderson, Sweeney, and Williams
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Confidence Interval Approach to aTwo-Tailed Test
about a Population Mean

• Select a simple random sample from the population
  and use the value of the sample mean to
  develop the confidence interval for the
  population mean ?.
• If the confidence interval contains the
  hypothesized value ?0, do not reject H0.
  Otherwise, reject H0.

Anderson, Sweeney, and Williams
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• Confidence Interval Approach to a Two-Tailed
  Hypothesis Test
• The 95 confidence interval for ? is
• or 6.0284 to 6.1716
• Since the hypothesized value for the population
  mean, ?0 6, is not in this interval, the
  hypothesis-testing conclusion is that the null
  hypothesis,
• H0 ? 6, can be rejected. (As shown in the
  previous slide, following traditional hypothesis
  testing steps.)

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Small Sample Tests about Mean

• If the sampled population is normal, and ? is
  known, use the Large sample
• formulas on slide 12.

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Small Sample Tests about Mean (nlt30)

• If the sampled population is normal, nlt30, and ?
  is unknown

H0 ?gt ?0 Ha ? lt ?0 Test Statistic Reject
ion Rule Reject H0 if t lt- t?????
H0 ?lt ?0 Ha ? gt ?0 Test Statistic Reject
ion Rule Reject H0 if tgt t ????????

• H0 ? ?0
• Ha ? ? ?0
• Test Statistic
• Rejection Rule
• Reject H0 if t gt t???

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Summary of Hypothesis Testing Procedures for a
Population Mean
Yes
No
n gt 30 ?
No
Popul. approx. normal ?
? known ?
Yes
Yes
Use s to estimate ?
No
? known ?
No
Use s to estimate ?
Yes
Slide 15
Slide 15 use s for ?
Increase n togt 30
Slide 33
Slide 15
Anderson, Sweeney, and Williams
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Example 8.6 Highway Patrol

• A State Highway Patrol periodically samples
  vehicle speeds at various locations on a
  particular roadway. The sample of vehicle speeds
  is used to test the hypothesis
• H0 m lt 65.
• The locations where H0 is rejected are deemed
  the best locations for radar traps.
• At Location F, a sample of 16 vehicles shows a
  mean speed of 68.2 mph with a standard deviation
  of 3.8 mph. Use an a .05 to test the
  hypothesis.

Anderson, Sweeney, and Williams
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• Let n 16, 68.2 mph, s 3.8 mph
• Step 1 H0 m lt 65
• Ha m gt 65
• Step 2 a .05
• Step 3

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A Summary of Forms for Null and Alternative
Hypotheses about a Population Proportion

• The equality part of the hypotheses always
  appears in the null hypothesis.
• In general, a hypothesis test about the value of
  a population proportion p must take one of the
  following three forms (where p0 is the
  hypothesized value of the population proportion).

H0 p gt p0 H0 p lt p0 Ha p lt p0
Ha p gt p0
H0 p p0 Ha p ? p0
One-tailed
One-tailed
Two-tailed
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Tests about a Population ProportionLarge-Sample
Case (np gt 5 and n(1 - p) gt 5)

• H0 pgt p0
• Ha pltp0
• Test Statistic
• Rejection Rule
• Reject H0 if z lt- z?????

H0 pltp0 Ha pgt p0 Test Statistic Rejecti
on Rule Reject H0 if z gt z ????????

• H0 p p0
• Ha p ? p0
• Test Statistic
• Rejection Rule
• Reject H0 if z gt z???

where
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Example 8.7 NSC

• For a Christmas and New Years week, the
  National Safety Council estimated that 500 people
  would be killed and 25,000 injured on the
  nations roads. The NSC claimed that 50 of the
  accidents would be caused by drunk driving.
• A sample of 120 accidents showed that 67 were
  caused by drunk driving. Use these data to test
  the NSCs claim with a 0.05.

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Example NSC

• Step 1 H0 p .5
• Ha p .5
• Step 2 a 0.05
• Step 3

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Step 4 Reject H0 if z gt za/2 1.96
(a.05) Step 5 Step 6 Is z gt 1.96
? No, so do not reject H0. Thus, we do not
have enough evidence to reject the claim that 50
of the accidents would be caused by drunk
driving.
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The End