Chapter 7 Hypothesis Testing

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Section 7-1 7-2 Overview and Basics of
Hypothesis Testing
Created by Erin Hodgess, Houston, Texas
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Chapter 7Hypothesis Testing

• 7-1 Overview
• 7-2 Basics of Hypothesis Testing
• 7-4 Testing a Claim About a Mean ? Known
• 7-5 Testing a Claim About a Mean ? Not Known
• 7-3 Testing a Claim About a Proportion

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7.2 Fundamentals of Hypothesis Testing

• Recall Inferential Statistics draw conclusions
  about a population based on sample data
• Confidence Intervals Estimate the value of a
  population parameter (using sample statistics).
• Hypothesis TestsTests a claim (hypothesis) about
  a population parameter (using sample statistics).

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Basic Idea for Hypothesis Tests

• State the hypothesis.
• Gather sample data (evidence).
• Make a decision about your hypothesis, based on
  sample data.

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Rare Event Rule for Inferential Statistics

• If, under a given assumption, the probability of
  a particular observed event is exceptionally
  small, we conclude that the assumption is
  probably not correct.

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Components of a Hypothesis Test

• The Null and Alternative Hypotheses
• The test statistic and making decisions
• Types of errors in hypothesis tests
• Writing conclusions

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A. The Hypotheses

• Null Hypothesis, H0
• statement about the parameter (µ, p, s)
• assumed true until proven otherwise
• must contain equality sign
• Alternative Hypothesis, H1 or Ha
• statement about the parameter that must be true
  if H0 is false.
• must contain

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The Hypotheses

• Symbolic Form of the Hypotheses
• Note The original claim may be in the null or
  the alternative hypothesis.

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B. Making Decisions

• State Hypotheses. Assume H0 is true.
• Collect sample data (test statistic)
• Make a decision about H0
• Reject H0 , or
• Fail to Reject the H0
• Note a) Reject Ha ? Accept Ha
• b) Fail to Reject the H0 ? Cant
  Accept Ha

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C. Errors in Decision Making

• Type I error is the mistake of rejecting the
  null when it is true Reject Ho
  Ho is true
• A Type II error is the mistake of failing to
  reject the null when it is false FTR Ho
  Ho is false
• The symbol ???(alpha) is used to represent the
  probability of a type I error. P(Type I error)
  ?
• The symbol ? (beta) is used to represent the
  probability of a type II error. P(Type II error)
  ?

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D. Writing Conclusions

• All conclusions are written in terms of the
  original claim.
• How the conclusion is worded depends on
• Whether the original claim was in the null or the
  alternative, and
• Whether the decision was Reject Ho (which implies
  you accept the Ha) or FTR Ho (which implies you
  cannot accept the Ha).
• Ho contains the claim, and you
• Reject Ho Theres enough evidence to reject the
  claim that ..
• FTR Ho There is not enough evidence to reject
  the claim that
• Ha contains the claim, and you
• Reject Ho There is enough evidence to support
  the claim that.
• FTR Ho There is not enough evidence to support
  the claim that

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Wording of Final Conclusion
Figure 7-7
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Note about Forming Your Own Claims (Hypotheses)

• If you are conducting a study and want to use
  a hypothesis test to support your claim, the
  claim must be worded so that it becomes the
  alternative hypothesis.

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How to Make a Decision about the Ho

• We start with tests about the population mean, µ.
  
• State Ho, Ha. Assume H0 is true µ µ0
• Collect sample data. Compute the test statistic.
  (For tests about µ, use ).
• Recall For xN or ngt30 and s known

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How to Make a Decision about the Ho

• Determine the Tails of the Test
• Left-tailed Test
• Right-Tailed Test
• Two-Tailed Test

• Basic Idea Assuming the Ho is true, (i.e, µ µ0
  ) where would the sample mean have to fall to
  convince you to reject the Ho and accept the Ha?
  

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Left-tailed Test

• H0 µ µ0
• Ha µ lt µ0

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Right-tailed Test

• H0 µ µ0
• Ha µ gt µ0

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Two-tailed Test

• H0 µ µ0
• Ha µ ? µ0

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How to Make a Decision about the Ho

• Once you have determined the tails of the test,
  use your sample mean to make a decision about the
  Ho using either
• The Traditional Method
• The P-Value Method

Main Idea How unusual would your sample mean
have to be for you to reject the Ho and accept
the Ha??
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The Traditional Method

• Critical Region contains values of the test
  statistic that would be considered unusual,
  assuming the Ho is true.
• If the test statistic falls in the critical
  region, that would be considered unusual,
  assuming the Ho was true. DECISION REJECT HO.
• If the test statistic falls in the non-critical
  regions, then that would not be considered
  unusual. DECISION FTR Ho.

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Finding the Critical Region

• Critical Region contains values of the test
  statistic that would be considered unusual,
  assuming the Ho is true.
• Area of the Critical Region a (a is also known
  as the significance level of the test). It is
  also the probability of a Type I error.
• Critical Values values that mark the boundaries
  of the critical region, za

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Section 7-4 Testing a Claim About a Mean ? Known
Created by Erin Hodgess, Houston, Texas
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Assumptions for Testing Claims About Population
Means

• 1) The sample is a simple random   sample.
• 2) The value of the population standard
  deviation ? is known.
• 3) Either or both of these conditions is
  satisfied The population is normally distributed
  or n gt 30.

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H.T about a Mean (with ? known)

1. Use the claim to write Ho, Ha. Assume the Ho is
   true.
2. Determine whether youre doing a left-tailed,
   right-tailed, or 2-tailed test.
3. Note a, the significance level of the test.
4. Get sample mean, x.
   Sketch the sampling distribution.

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H.T about a Mean (with ? Known)

• Convert the sample mean to a z-score
• Make a decision about Ho using either
• Traditional Method
• P-value Method
• Write a conclusion in terms of the original
  claim.

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Traditional Method

• Shade the critical region
• Find the critical values
• Make a decision
• If z is in the critical region, reject Ho.
• If z is not in the critical region, FTR the Ho.

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P-Value Method for Making Decisions
Instead of determining whether your sample mean
is unusual just by whether or not it falls in a
critical region, be more precise Find
the P-value
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P-Value Method for Making Decisions
P-value probability of getting a test statistic
as extreme or more as the one from the sample
data (assuming the Ho is true to begin with).
If P-value a (unusual), Reject Ho. If
Pvalue gt a (not unusual), FTR HO.
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Example Finding P-values.
Figure 7-6
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Section 7-5 Testing a Claim About a Mean ? Not
Known
Created by Erin Hodgess, Houston, Texas
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H.T about Mean (with ? unknown)

• Follow the same procedure as 7.3.
• For population normal or n gt 30 and s unknown,
  the sampling distribution of x has a
  t-distribution with n-1 degrees of freedom.
• Convert sample mean x to a standard t-score

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Choosing between the Normal and Student t
Distributions when Testing a Claim about a
Population Mean µ
Use the Student t distribution when ? is not
known and either or both of these conditions is
satisfied The population is normally distributed
or n gt 30.
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Section 7-3 Testing a Claim About a Proportion
Created by Erin Hodgess, Houston, Texas
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Hypotheses About p

• Note The original claim may be in the null or
  the alternative hypothesis.

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Assumptions about a Hypotheses Test for a
Proportion

1. To conduct a hypothesis test about a population
   proportion p, use a sample proportion, p.
2. We assume the sample was a simple random sample.
   
3. If np 5, nq 5, the sampling distribution of p
   is
4. Follow the procedures of 7.3

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Hypothesis Test about a Population Proportion

1. Use the claim to write Ho, Ha. Assume the Ho is
   true.
2. Determine whether youre doing a left-tailed,
   right-tailed, or 2-tailed test.
3. Note a, the significance level of the test.
4. Get sample proportion, p.
   Sketch the sampling distribution.

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Hypothesis Test about a Proportion

• Convert the sample proportion to a z-score
• Make a decision about Ho using either
• Traditional Method
• P-value Method
• Write a conclusion in terms of the original
  claim.

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CAUTION

• When the calculation of p results in a
  decimal with many places, store the number on
  your calculator and use all the decimals when
  evaluating the z test statistic.
• Large errors can result from rounding p too
  much.

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Section 7-6 Testing a Claim About a Standard
Deviation or Variance
Created by Erin Hodgess, Houston, Texas
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Assumptions for Testing Claims About ? or ?2

• 1. The sample is a simple random sample.
• 2) The population has values that are normally
  distributed (a strict requirement).

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Chi-Square Distribution
Test Statistic
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Chi-Square Distribution
Test Statistic

• n sample size
• s 2 sample variance
• ??2 population variance
• (given in null hypothesis)

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P-values and Critical Values for Chi-Square
Distribution

• Use Table A-4.
• The degrees of freedom n 1.

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Properties of Chi-Square Distribution

• All values of ?2 are nonnegative, and the
  distribution is not symmetric (see Figure 7-12).
• There is a different distribution for each number
  of degrees of freedom (see Figure 7-13).
• The critical values are found in Table A-4 using
  n 1 degrees of freedom.

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Properties of Chi-Square Distribution
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Example For a simple random sample of adults, IQ
scores are normally distributed with a mean of
100 and a standard deviation of 15. A simple
random sample of 13 statistics professors yields
a standard deviation of s 7.2. Assume that IQ
scores of statistics professors are normally
distributed and use a 0.05 significance level to
test the claim that ? 15.
H0 ? 15 H1 ? ? 15 ? 0.05 n 13 s 7.2
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Example For a simple random sample of adults, IQ
scores are normally distributed with a mean of
100 and a standard deviation of 15. A simple
random sample of 13 statistics professors yields
a standard deviation of s 7.2. Assume that IQ
scores of statistics professors are normally
distributed and use a 0.05 significance level to
test the claim that ? 15.
?2 2.765
H0 ? 15 H1 ? ? 15 ? 0.05 n 13 s 7.2
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Example For a simple random sample of adults, IQ
scores are normally distributed with a mean of
100 and a standard deviation of 15. A simple
random sample of 13 statistics professors yields
a standard deviation of s 7.2. Assume that IQ
scores of statistics professors are normally
distributed and use a 0.05 significance level to
test the claim that ? 15.
?2 2.765
H0 ? 15 H1 ? ? 15 ? 0.05 n 13 s 7.2
The critical values of 4.404 and 23.337 are found
in Table A-4, in the 12th row (degrees of freedom
n 1) in the column corresponding to 0.975 and
0.025.
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Example For a simple random sample of adults, IQ
scores are normally distributed with a mean of
100 and a standard deviation of 15. A simple
random sample of 13 statistics professors yields
a standard deviation of s 7.2. Assume that IQ
scores of statistics professors are normally
distributed and use a 0.05 significance level to
test the claim that ? 15.
?2 2.765
H0 ? 15 H1 ? ? 15 ? 0.05 n 13 s 7.2
Because the test statistic is in the critical
region, we reject the null hypothesis. There is
sufficient evidence to warrant rejection of the
claim that the standard deviation is equal to 15.