Statistical Inference

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Biostatistics Academic Preview Session 3
Statistical Inference
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Outline

• The what and why of statistical inference
• Statistical estimation and confidence intervals
• Statistical significance tests

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Statistical Inference
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Statistical Inference

• Estimation the process by which sample data are
  used to indicate the value of an unknown quantity
  in the population. Results can be expressed as
• Point estimate
• Confidence intervals
• Significance tests
• P-values

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Statistical Estimation
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Confidence intervals (cont)

• A range of values in which a population parameter
  may lie is a confidence interval.
• The probability that a particular value lies
  within this interval is called a level of
  confidence.

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90 and 95 confidence intervals for ?
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Confidence intervals (cont)

• Formula
• Estimate margin of error
• Steps
• Calculate the sample statistic to use as an
  estimate of the population parameter
• Calculate the margin of error based on the
  distribution of sample statistic
• Calculate the lower (LL) and the upper limits
  (UL) of the confidence interval
• Write out the confidence interval

LL lt population parameter lt UL
LL, UL
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Understanding Confidence Intervals

• A confidence interval for a population parameter
  is a set of plausible values of the parameter
  that could have generated the observed data as a
  likely outcome
• The level of confidence tells the probability the
  method produced an interval that includes the
  unknown parameter
• Narrow widths and high confidence levels are
  desirable, but these two things affect each other

Higher confidencewider confidence limits (all
else held constant)
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Case Study 1

• Construct a 95 confidence interval for the mean
  birth weight in grams
• Construct a 95 confidence interval for the
  percent of women who deliver low birth weight
  babies

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Case Study 1 Example 1

• 95 Confidence interval for the mean birth weight
  in grams
• Calculate
• Calculate the margin of error m104.6
• Calculate the lower and upper limit of the
  confidence interval
• LL2944.7-104.62840.1
• UL29447104.63049.3
• Confidence interval
• 2840.1, 3049.3
• 2840.1 lt µ lt 3049.3

I am 95 confident that the mean birth weight for
omen in Springfield Massachusetts in 1986 is
between 2840.1 and 3049.3
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Case Study 1 Example 2

• 95 confidence interval for the percent of women
  who deliver low birth weight babies
• Calculate
• Calculate the margin of error m0.066
• Calculate the lower and upper limit of the
  confidence interval
• LL0.312-0.0660.246
• UL0.3120.0660.378
• Confidence interval 0.246, 0.378

I am 95 confident that the proportion of women
giving birth to low birth weight babies in
Springfield Massachusetts in 1986 is between
0.246 and 0.378
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Case study 1

• In order to determine if
• There is a difference in the mean birth weight
  between smokers and non-smokers
• There is a difference in the proportion of low
  birth weight babies between white women and black
  women
• We need to calculate a confidence interval around
  the difference in the estimates between the two
  groups in the aforementioned examples

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Confidence intervals (cont)

• Formula
• Estimate (group1) estimate (group 2) margin
  of error
• Steps
• Calculate the sample statistics to use as an
  estimate of the population parameter
• Calculate the margin of error based on the
  distributions of the sample statistic
• Calculate the lower (LL) and the upper limits
  (UL) of the confidence interval
• Write out the confidence interval
• LL lt population parameter lt UL or LL, UL

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Case Study 1 Example 3

• 95 confidence interval for the difference in the
  percent of women who deliver low birth weight
  babies between whites and blacks
• Calculate
• Calculate the margin of error m.2085
• Calculate the lower and upper limit of the
  confidence interval
• LL-.1835-.2085-.392
• UL-.1835.2085.025
• Confidence interval -.392, .025

I am 95 confident that the difference in the
proportion of white women and black women giving
birth to low birth weight babies in Springfield
Massachusetts in 1986 is between -.392 and .025
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Significance /Hypothesis Testing

• Statistical inference that allows one to test a
  claim about a population parameter.
• Using information from your study sample you can
  test any desired claim

Next set of slides drafted from the following
reference Elementary Statistics by Larson and
Farber, 2nd edition
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A statistical hypothesis is a claim about a
population.
Alternative hypothesis Ha contains a statement
of inequality such as lt , ¹ or gt
Null hypothesis H0 contains a statement of
equality such as ³ , or .
If I am false, you are true
If I am false, you are true
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Writing Hypotheses
Write the claim about the population. Then, write
its complement. Either hypothesis, the null or
the alternative, can represent the claim.

• A hospital claims its ambulance response time is
  less than 10 minutes.

H0
Ha

• A consumer magazine claims the proportion of
  cell phone calls made during evenings and
  weekends is at most 60.

H0
Ha
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Hypothesis Test Strategy

• Begin by assuming the equality condition in the
  null hypothesis is true. This is regardless of
  whether the claim is represented by the null
  hypothesis or by the alternative hypothesis,

• Collect data from a random sample taken from the
  population and calculate the necessary sample
  statistics.

• If the sample statistic has a low probability of
  being drawn
• from a population in which the null hypothesis is
  true, you will
• reject H0 . (As a consequence, you will support
  the alternative
• hypothesis.)
• If the probability is not low enough, fail to
  reject H0 .

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Underlying Rationale of Hypotheses Testing

• If, under a given observed assumption, the
  probability of getting the sample is
  exceptionally small, we conclude that the
  assumption is probably not correct.
• When testing a claim, we make an assumption
  (null hypothesis) that contains equality. We then
  compare the assumption and the sample results and
  we form one of the following conclusions

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Underlying Rationale of Hypotheses Testing

• If the sample results can easily occur when the
  assumption (null hypothesis) is true, we
  attribute the relatively small discrepancy
  between the assumption and the sample results to
  chance.
• If the sample results cannot easily occur when
  that assumption (null hypothesis) is true, we
  explain the relatively large discrepancy between
  the assumption and the sample by concluding that
  the assumption is not true.

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Errors and Level of Significance
Actual Truth of H0
H0 True
H0 False
Do not reject H0
Type II Error
Decision
Type I Error
Reject H0
A type I error Null hypothesis is actually true
but the decision is to reject it.
Level of significance, a Maximum probability of
committing a type I error.
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Types of Hypothesis Tests
Right-tail test Ha m gt value
Left-tail test Ha m lt value
Two-tail test Ha m ¹ value
Hypothesis tests can also be one or two sampled
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P-Values
The P-value is the probability of obtaining a
sample statistic with a value as extreme or more
extreme than the one determined by the sample
data.
P-value indicated area
If z is negative, twice the area in the left tail
If z is positive twice the area in the right tail
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Test Decisions with P-values
The decision about whether there is enough
evidence to reject the null hypothesis can be
made by comparing the P-value to the value of
a, the level of significance of the test.
If P ? a reject the null hypothesis
If P gt a fail to reject the null hypothesis
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Understanding p-values

• A p-value tells you the chance of getting a
  statistics as extreme or more extreme than the
  one calculated for the sample
• measures the strength of evidence against the
  null hypothesis
• The smaller the p-value, the more convincing the
  evidence is against the null hypothesis
• A p-value is not
• the probability that the null hypothesis is
  true/false
• The effect size
• The significance of results

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Interpreting the Decision
Claim
Claim is H0
Claim is Ha
There is enough evidence to reject the claim.
There is enough evidence to support the claim.
Reject H0
Decision
There is not enough evidence to reject the claim.
There is not enough evidence to support the claim.
Fail to reject H0
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Steps in a Hypothesis Test
1. Write the null and alternative hypothesis
Write H0 and Ha as mathematical statements.
Remember H0 always contains the symbol.
2. State the level of significance
This is the maximum probability of rejecting the
null hypothesis when it is actually true. (Making
a type I error.)
3. Identify the sampling distribution
The sampling distribution is the distribution for
the test statistic assuming that the equality
condition in H0 is true and that the experiment
is repeated an infinite number of times.
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4. Find the test statistic
Perform the calculations to standardize your
sample statistic.
5. Calculate the P-value for the test statistic
This is the probability of obtaining your test
statistic or one that is more extreme from the
sampling distribution.
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6. Make your decision
If the P-value is less than ? (the level of
significance) reject H0. If the P value is
greater ?, fail to reject H0.
7. Interpret your decision

• If the claim is the null hypothesis you will
  either reject
• the claim or determine there is not enough
  evidence to
• reject the claim.

• If the claim is the alternative hypothesis you
  will either support the claim or determine there
  is not enough
• evidence to support the claim.

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Case study 1

• Perform a hypothesis test to determine if the
  proportion of women giving birth to low birth
  weight babies is less than .50.
• Perform a hypothesis test to determine if there
  is a difference in the mean birth weight between
  smokers and non-smokers

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Case Study 1 Example 4

• Step 1 Write out the null and alternative
  hypothesis
• Ho p.05
• Ha plt.05
• Step 2 level of significance a.05
• Step 3 skip (normal distribution)
• Step 4 test statistic z.5.16
• Step 5 p-valuelt.0001
• Step 6 Decision-we reject Ho because
  p-valueltalpha
• Step 7 Conclusion There is strong evidence to
  suggest that the percent of mothers giving birth
  to low birth weight babies in Springfield
  Massachusetts in 1986 is less than .50

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Case Study 1 Example 5

• Step 1 Write out the null and alternative
  hypothesis
• Ho µsmokers- µnon-smokers0
• Ha µsmokers- µnon-smokers ?0
• Step 2 level of significance a.05
• Step 3 skip (t-distribution)
• Step 4 test statistic t-2.634
• Step 5 p-value.0092
• Step 6 Decision-we reject Ho because
  p-valueltalpha
• Step 7 Conclusion There is strong evidence to
  suggest that there is a difference in the mean
  birth weight between smokers and non-smokers for
  women in Springfield Massachusetts in 1986.

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Next session

• Results of our in class survey