One-Sample Tests of Hypothesis

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Chapter 10

• One-Sample Tests of Hypothesis

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Goals

1. Define a hypothesis and hypothesis testing
2. Describe the five step hypothesis testing
   procedure
3. Distinguish between a one-tailed and a two-tailed
   test of hypothesis

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Goals

1. Conduct a test of hypothesis about a population
   mean
2. Conduct a test of hypothesis about a population
   proportion

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Goals
Chapters 1 9 Everything we have learned so far,
we can use in this chapter! Chapters 10 We will
test peoples claims by running experiments and
then conclude whether the initial claims are
reasonable or not!
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Define A Hypothesis

• A Hypothesis is a statement about the value of a
  population parameter developed for the purpose of
  testing.
• Examples of hypotheses made about a population
  parameter are
• The mean monthly income for systems analysts is
  3,625
• Twenty percent of all customers at Bovines Chop
  House return for another meal within a month
• The mean yearly salary for a real estate agent is
  85,000

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Define Hypothesis Testing

• Hypothesis testing is a procedure, based on
  sample evidence and probability theory, used to
  determine whether
• The hypothesis is a reasonable statement and
  should not be rejectedor
• The hypothesis is unreasonable (not reasonable)
  and should be rejected
• Parallel Examples Legal System, Doctors

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Define Hypothesis Testing

• Illustration (Example 5)
• The hypothesized mean yearly salary earned by
  full-time realtors is 85,000 (sigma 12,549)
• The mean salary is not significantly different
  from 85,000
• If we take a sample and get a sample mean of
  88,595, we must make a decision about the
  difference of 3,595
• Is it a true differenceor
• Is it sampling error
• Five steps to hypothesis testing ?

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5 Steps In Hypothesis Testing Procedure

• Step 1 State Null Hypothesis (H0)and Alternate
  Hypothesis (H1).
• Step 2 Select a Level of Significance (alpha).
• Step 3 Select the Test Statistic (z or t). (Draw
  picture and calculate Critical Value).
• Step 4 Formulate the Decision Rule based on
  steps 1-3.
• Step 5 Make a decision about the Null Hypothesis
  based on sample information. (Take a random
  sample, compute the test statistic, compare it to
  critical value, and make decision to reject or
  not reject null hypotheses). Interpret the
  results of the test.

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Step 1 State Null Hypothesis (H0)and Alternate
Hypothesis (H1)

• Null Hypothesis H0
• H sub zero, or H sub not
• A statement about the value of a population
  parameter
• This is the value we test
• After the experiment, we will either
• Reject H0, and accept alternative hypothesis
• Fail to reject H0 (Does not prove that H0 is
  true)
• Example H0 µ 85,000

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Step 1 State Null Hypothesis (H0)and Alternate
Hypothesis (H1)

• Alternative Hypothesis H1
• H sub one
• A statement that is accepted if the sample data
  from the experiment rejects the null hypothesis
• If H1 taken
• Reject H0
• H1 replaces H0
• If H1 rejected
• Fail to reject H0
• Assumption, H0, holds
• Example H1 µ gt 85,000

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Step 1 State Null Hypothesis (H0)and Alternate
Hypothesis (H1)

• The trick to hypothesis testing is translating
  the words into ?, lt, gt for H1
• It is usually easier to state H1 first and then
  state H0 .
• H0 will always have the equal sign , ,
• H1 will never have the equal sign ?, gt, lt
• Step 1 for Example 5
• If someone claims The mean yearly salary earned
  by full-time realtors is more than 85,000
• Write H1 gt 85,000 first
• Then write H0 85,000

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Step 1 State Null Hypothesis (H0)and Alternate
Hypothesis (H1)

• H0 The mean yearly salary earned by full-time
  realtors is 85,000
• Second we would write ? H0 µ 85,000
• Why ?
• In our test, anything equal to or less than
  85,000 contradicts µ gt 85,000!
• H1 The mean yearly salary earned by full-time
  realtors is more than 85,000
• First we would write ? H1 µ gt 85,000

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Step 2 Select a Level of Significance (alpha)

• Level of Significance (?)
• The probability of rejecting the null hypothesis
  when it is actually true
• The risk we are willing to take in committing an
  error
• The lower the number, the lower the risk of
  committing an error
• Standards for Level of Significance
• ? .10 (political polls)
• ? .05 (consumer research project)
• ? .01 (quality assurance)

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Errors

• Type I Error ?
• Rejecting the null hypothesis when it is actually
  true
• H0 was true, but experiment rejected H0
• Innocent, but found guilty
• Type II Error ß
• Accepting the null hypothesis when it is actually
  false
• H0 was false, but experiment failed to reject H0
• Guilty, but found not guilty

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Step 3 Select the Test Statistic (z or t).
(Draw picture and calculate Critical Value)

• Is the critical value and the test statistic z or
  t?
• Given level of significance, look up critical
  value in table
• The dividing point between the region where the
  null hypothesis is rejected and the region where
  it is not rejected
• The actual test statistic will be calculated in
  step 5
• A value, determined from sample information, used
  to determine whether or not to reject the null
  hypothesis
• Before we can draw our picture, we must
  distinguish between a one-tailed and a two-tailed
  test of hypothesis ?

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One-tailed Tests Of Significance

• A test is one-tailed when the H1 uses lt or gt
• Step 2 for Example5
• H1 The mean yearly salary earned by full-time
  realtors is more than 85,000
• H0 µ 85,000
• H1 µ gt 85,000
• One tailed test to the right ?
• Another example
• H1 The mean speed of trucks traveling on I-5 in
  Washington is less than 60 miles per hour
• H0 µ 60
• H1 µ lt 60
• ? One tailed test to the left

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One Tailed Test To The Right H1 uses Greater
Than gt

• Step 3 for Example 5 H0 µ 85,000 H1 µ gt
  85,000

Critical value (z 1.65)
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One Tailed Test To The LeftH1 uses Less Than lt

• H0 µ some number
• H1 µ lt some number

Critical value (z or t)
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Two-tailed Tests Of Significance

• A test is two-tailed when H1 uses ?
• Examples
• H1 The mean amount spent by customers at the
  Wal-Mart in Georgetown is not equal to 25
• H0 µ 25
• H1 µ ? 25
• H1 The mean price for a gallon of gasoline is
  not equal to 3.55
• H0 µ 3.55
• H1 µ ? 3.55

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Two Tailed TestNot Equal ?

• H0 µ some number
• H1 µ ? some number

Critical value (z or t)
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Step 4 Formulate the Decision Rule based on
steps 1-3

• State decision rule as
• If H1 passes some test, we reject H0 and accept
  H1, otherwise H0 is not rejected
• Step 4 for Example 5
• ? .05, z 1.65
• H0 µ 85,000
• H1 µ gt 85,000
• Decision Rule If test statistic is greater than
  1.65, then we reject H0 and accept H1, otherwise
  we fail to reject H0
• Good statisticians go through all the steps up to
  writing down the decision rule before they look
  at the results

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Step 5 Make a decision about the Null
Hypothesis based on sample information. (Take a
random sample, compute the test statistic,
compare it to critical value, and make decision
to reject or not reject null hypotheses).
Interpret the results of the test.

• Take sample, calculate mean and standard
  deviation, then calculate the test statistic and
  come to a conclusion
• The two possible conclusions are
• Do not reject H0or
• Reject H0, accept H1
• State decision as
• The experiment supports the claim that H1
• The experiment does not support the claim that
  H1
• Evidence indicates
• Data shows

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Compute The Test StatisticTesting For The
Population Mean
s known
s unknown
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Step 5 Make a decision about the Null
Hypothesis based on sample information. (Take a
random sample, compute the test statistic,
compare it to critical value, and make decision
to reject or not reject null hypotheses).
Interpret the results of the test.

• Failing to reject H0 does not prove H0, it simply
  means that we have failed to disprove H0
• Rejecting H0 does not prove H1, it simply means
  that H0 is not reasonable

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Step 5 for Example 5
Because 1.72 is greater than 1.645 we reject H0
and accept H1. The evidence suggests that the
mean yearly salary is greater than 85,000.00. We
can be reasonably sure that the mean salary is
greater than 85,000.00. Because 0.0428 is less
than 0.05 we reject H0 and accept H1
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P-value In Hypothesis Testing

• Assuming that the null hypothesis is true, a
  p-Value is the probability of finding a value of
  the test statistic at least as extreme as the
  computed value for the test
• p-Value lt significance level
• H0 is rejected
• If H0 is very small, there is little likelihood
  that H0 is true
• p-Value gt significance level
• H0 is not rejected
• If H0 is very large, there is little likelihood
  that H0 is false

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Computation Of The P-value

• One-Tailed Test
• p-Value
• P( z absolute value of the computed test
  statistic value) and compare to alpha
• Two-Tailed Test
• For a two-tail test, use alpha divided by 2
  (alpha/2) and compare that to the p-value from
  one side of the two tails
• p-Value P( z absolute value of the computed
  test statistic value) and compare p to alpha/2
• OR
• For a two-tail test, use (2 p-value) and compare
  to alpha
• p-Value 2P( z absolute value of the computed
  test statistic value) and compare p to alpha

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Interpreting The Weight Of Evidence Against H0

• If the p-value is less than
• .10, we have some evidence that H0 is not true
• .05, we have strong evidence that H0 is not true
• .01, we have very strong evidence that H0 is not
  true
• .001, we have extremely strong evidence that H0
  is not true

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Conduct A Test Of Hypothesis About A Population
MeanHypothesis Testing Example 1

• The processors of Fries Catsup indicate on the
  label that the bottle contains 16 ounces of
  catsup
• The standard deviation of the process is 0.5
  ounces
• A sample of 36 bottles from last hours
  production revealed a mean weight of 16.12 ounces
  per bottle
• At the .05 significance level is the process out
  of control?
• Can we conclude that the mean amount per bottle
  is different from 16 ounces?

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Hypothesis Testing Example 1

• Step 1
• State the null and the alternative hypotheses
• H0 µ 16
• H1 µ ? 16
• Step 2
• Select a level of significance
• We select .05 significance level (.05/2 .025)
• Step 3
• Identify the test statistic and draw
• Because we know the population standard
  deviation and n 30, the test statistic is z
• . .025 ? .5 - .025 .475 ?leads us to /-1.96

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Hypothesis Testing Example 1

• Step 4
• Formulate the decision rule
• If our test statistic is outside the range -1.96
  to 1.96, Reject H0 and accept H1, otherwise we
  fail to reject H0

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Hypothesis Testing Example 1

• Step 5
• Sample, Compute Test Statistic And Compare To
  Critical Value, Reject Or Not Reject Ho
• Because 1.44 is less that 1.96 and greater than
  -1.96, we do not reject H0
• The evidence suggests that the mean is not
  different from 16 oz.
• The difference between 16.12 16.00 can be
  attributed to sampling variation
• The processors claim that the bottles contain 16
  ounces of catsup seems reasonable

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Computation Of The P-value

• Two tailed test from example 1Significance
  level .05
• z 1.44
• The p-Value 2P( z 1.44) 2(.5-.4251)
  .1498
• Because .1498 gt .05, do not reject H0
• Note, if you use Students t Distribution to
  estimate p-value, from textbook The usual
  practice is to report that the p-value is less
  than the larger of the two significance levels

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Hypothesis Testing Example 2

• Roders Discount Store chain issues its own
  credit card
• Lisa, the credit manager, wants to find out if
  the mean monthly unpaid balance is more than 400
• The level of significance is set at .05
• A random check of 172 unpaid balances revealed
• Sample mean 407
• Sample standard deviation 38
• Should Lisa conclude that the population mean is
  greater than 400, or is it reasonable to assume
  that the difference of 7 (407-400) is due to
  chance?

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Hypothesis Testing Example 2

• Step 1
• State the null and the alternative hypotheses
• H0 µ 400
• H1 µ gt 400
• Step 2
• Select a level of significance
• We select .05 significance level
• Step 3
• Identify the test statistic and draw
• Because sigma is not known, the test statistic is
  t
• .05 ? leads us to t 1.65 (one-tail test to
  right)

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Hypothesis Testing Example 2

• Step 4
• Formulate the decision rule
• If our test statistic is greater than 1.65 (t
  gt1.65), Reject H0 and accept H1, otherwise we
  fail to reject H0
• Step 5
• Sample, Compute Test Statistic And Compare To
  Critical Value, Reject Or Not Reject Ho
• Because 2.42 gt 1.65, H0 is rejected and H1 is
  accepted
• The evidence indicates that the mean unpaid gt
  400
• Lisa can conclude that the mean unpaid balance is
  greater than 400
• It is not reasonable to assume that a computed
  t-score of 2.42 is due to sampling variation

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Hypothesis Testing Example 3

• The current rate for producing 5 amp fuses at
  Neary Electric Co. is 250 per hour
• A new machine has been purchased and installed
  that, according to the supplier, will increase
  the production rate
• A sample of 10 randomly selected hours from last
  month revealed
• Mean hourly production for new machine 256
  units
• Sample standard deviation 6 per hour
• At the .05 significance level can Neary conclude
  that the new machine is faster?

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Hypothesis Testing Example 3

• Step 1
• State the null and the alternative hypotheses
• H0 µ 250
• H1 µ gt 250
• Step 2
• Select a level of significance
• We select .05 significance level
• Step 3
• Identify the test statistic and draw
• t distribution because s is unknown
• There are 10 1 9 degrees of freedom
• .05 and df 9 ? leads use to t 1.833

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Hypothesis Testing Example 3

• Step 4
• Formulate the decision rule
• If our test statistic is greater than 1.833 (t gt
  1.833), Reject H0 and accept H1, otherwise we
  fail to reject H0
• Step 5
• Sample, Compute Test Statistic And Compare To
  Critical Value, Reject Or Not Reject Ho
• Because 3.162 gt 1.833, we reject Ho and accept H1
  
• The evidence suggests that the mean number of
  amps produced per hour is greater than 250

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Hypothesis Testing Example 4

• In the past, 15 of the mail order solicitations
  for a certain charity resulted in a financial
  contribution
• A new solicitation letter that has been drafted
  is sent to a sample of 200 people and 35
  responded with a contribution
• Assume Experiment passes all the binomial tests
• At the .05 significance level can it be concluded
  that the new letter is more effective?
• Conduct A Test Of Hypothesis About A Population
  Proportion
• Test statistic for testing a single population
  proportion
• p Sample proportion
• ? Population proportion

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Hypothesis Testing Example 4

• Step 1
• State the null and the alternative hypotheses
• H0 ? .15
• H1 ? gt .15
• Step 2
• Select a level of significance
• We select .05 significance level (one-tail to
  right)
• Step 3
• Identify the test statistic and draw
• For Proportions that pass binomial test, we use
  z. .05 ? z 1.65
• Step 4
• Formulate the decision rule
• If our test statistic is greater than 1.65 (z
  gt1.65), Reject H0 and accept H1, otherwise we
  fail to reject H0

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Hypothesis Testing Example 4

• Step 5
• Sample, Compute Test Statistic And Compare To
  Critical Value, Reject Or Not Reject
  Ho
• Because .990148 in not greater than 1.65, we fail
  to reject Ho
• The evidence does not suggest that the new
  letters are more effective