Testing a Hypothesis about a Population Proportion Section 9'4

1
Testing a Hypothesis about a Population
Proportion Section 9.4

• Alan Craig
• 770-274-5242
• acraig_at_gpc.edu

2
Objectives 9.4

• Test a hypothesis about a population proportion
  using the classical approach
• Test a hypothesis about a population proportion
  using the P-value approach
• Test a hypothesis about a population proportion
  on a small sample

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Recall from Section 8.3

• Best point estimate of a population proportion p
  is
• Sample distribution of is approximately
  normal with
• mean and
• standard deviation

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Classical MethodTest Statistic

• The classical approach to testing a hypothesis
  about a population proportion p is the same as
  what we have already seen for the mean with s
  known except the test statistic is
• where is the assumed value of the population
  proportion

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Requirements for Testing Hypotheses about a
Population Proportion

• A simple random sample is obtained
• np0(1 - p0) 10
• n 0.05 N
• sample size is no more than 5 of population size

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Example 22, p. 428
A survey found that 85 of American adults eat
salad at least once a week. A nutritionist
believes the percentage is higher than this. She
conducts a survey of 200 American adults and
finds that 171 eat salad at least once a week.
Is there sufficient evidence to support the
claim at the a 0.10 level of significance?
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Example 22, p. 428

• Check whether requirements are met
• This is a simple random sample
• p0 0.85
• np0(1 - p0) (200)(.85)(.15) 25.5 10
• n 0.05 N
• n 200 vs all American adults
• Requirements are met.

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Example 22, p. 428Classical Method

• Step 1 State the hypotheses
• This is a right-tailed test.
• Step 2 The critical value at a 0.10 level of
  significance is z0.10 1.28
• (from Table II, find Z-score with left area 1
  0.10 0.90)

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Example 22, p. 428Classical Method

• Step 3 Compute the test statistic

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Example 22, p. 428Classical Method

• Step 4 Compare the test statistic with the
  critical value
• Step 5 There is not sufficient evidence to
  support the claim that more than 85 of Americans
  eat salad at least once a week.

Reject H0
z0.20 z1.28
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Using P-Values

• The P-value approach to testing a hypothesis
  about a population proportion p is the same as
  what we have already seen.
• The test statistic is
• The P-value is the probability of observing the
  test statistic assuming the null hypothesis is
  true.
• Reject H0 if P-value lt a

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Example 22, p. 428 Using P-values

• Step 1 State the hypotheses (same as before)
• This is a right-tailed test.
• Step 2 Compute the test statistic
• Same as before Z 0.20

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Example 22, p. 428Classical Method

• Step 3 Compute the P-value (for a right-tailed
  test, this is the probability of obtaining a
  sample proportion of or larger assuming H0 is
  true)

Area to right of z0 is P-value
z00.20
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Example 22, p. 428Using P-values

• Step 4 Compare the P-value with the level of
  significance, a
• Step 5 There is not sufficient evidence to
  support the claim that more than 85 of Americans
  eat salad at least once a week.

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Small-Sample Hypothesis Tests

• What if the requirement is not met that
• np0(1 - p0) 10 ?
• Then we can use the binomial probability formula
  to compute exact P-values.

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Example 26, p. 428Small-Sample

• Check whether requirement is met
• p0 0.015
• np0(1 - p0)
• (500)(0.015)(0.985)
• 7.4
• lt 10
• Requirement is NOT met.

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Example 28, p. 428 Small Sample

• Step 1 State the hypotheses
• This is a left-tailed test.
• Step 2 Compute the P-value
• Let the random variable X be the number of males
  who are teachers. For a left-tailed test, the
  P-value is the probability of obtaining x or
  fewer successes. For right-tailed, probability
  of x or more successes.

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Example 28, p. 428 Small Sample

• Step 2 Compute the P-value (continued)
• Let the random variable X be the number of males
  who are teachers. For a left-tailed test, the
  P-value is the probability of obtaining x or
  fewer successes.

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Example 28, p. 428 Small Sample

• Step 3 If P-value lt a, reject H0
• Let the random variable X be the number of males
  who are teachers. For a left-tailed test, the
  P-value is the probability of obtaining x or
  fewer successes.
• Reject H0. There is sufficient evidence to
  support the claim that fewer than 1.5 of males
  living in Colorado are teachers.

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Questions

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