Section%203.3:%20The%20Story%20of%20Statistical%20Inference%20Section%204.1:%20Testing%20Where%20a%20Proportion%20Is

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Section 3.3 The Story of Statistical
InferenceSection 4.1 Testing Where a
Proportion Is
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Statistical Inference Confidence Intervals vs.
Hypothesis Testing

• A confidence interval is used to estimate an
  unknown population parameter, e.g. p.
• A hypothesis test is used to test a claim about a
  population parameter, e.g. p.

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Determining the Childs Sex During Pregnancy

• Advances in medicine make it possible to
  determine the sex of a child early in a
  pregnancy. Because some cultures value male
  children more highly than female children,
  theres a fear that some parents may not carry
  pregnancies of girls to term.

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Punjab, India Study 1994

• 56.9 of the 550 live births that year were boys.
• Its a medical fact that male babies are slightly
  more common than female babies. The authors
  report a baseline for this region of 51.7 male
  live births.
• Question Is the sample proportion of 56.9
  evidence of a higher proportion of male births?

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The Nuts and Bolts

• Who is the population?
• What is the parameter of interest?

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Facts about the Null Hypothesis,

• Recall that is a statement about the
  population parameter, p, and not the sample
  statistic, .
• is the hypothesis of no difference.
• While performing the hypothesis test we assume
  the null hypothesis is true and see if
  we can find enough evidence to disprove this
  claim.

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Testing this Claim

• Collect a random sample from the population and
  compute the sample proportion.

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Statistical Significance

• To determine if our sample results are
  statistically significant, we need to determine
  the p-value
• Assuming the null hypothesis is correct, how
  likely is it to get a sample proportion as
  extreme or more extreme than the one that we
  observed?

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To answer this question, we need to know how
varies in repeated sampling.

• Easy question, right?
• Draw a well-labeled graph which describes how
  repeated random samples, each of size 550, would
  vary.

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Intuition Check

• Before doing any calculations and by looking at
  the graph you drew, do you think theres evidence
  to suggest that birth ratios of boys to girls is
  equal?
• Lets now test your intuition.

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How close is to p?

• The z-value provides us the answer since it is a
  measure of how many standard errors our sample
  statistic is from our population parameter.
• Compute the z-value and interpret it in the
  context of this problem.

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More on the z-value

• A sample proportions z-value indicates where, in
  the distribution of sample values, that
  proportion falls.
• What does a negative z-value tell you?
• What does a positive z-value tell you?
• What does a z-value of -0.85 tell you?
• What does a z-value of 5.7 tell you?

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Finding the P-value

• See Table 4.1.1. in the text.

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Conclusions

• Is there enough evidence to reject the null
  hypothesis or should we fail to reject the null
  hypothesis? (Note We do not ever accept the
  null hypothesis.)
• Legal system analogy.

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Example 2

• Suppose instead that 52.6 of the 550 live births
  were male. Would this sample proportion have
  been strong enough to reject the null hypothesis?
• Do the appropriate calculations.

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How small does the p-value need to be to reject
the null hypothesis?

• What if ?
  
• What if ?
• What if ?
• How about if ?

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Level of Significance,

• We can define rare event arbitrarily by setting
  a threshold ( -value) for our p-value.
• If the p-value falls below the threshold, well
  reject the null hypothesis and call the results
  statistically significant. If not we fail to
  reject the null hypothesis.