Basis of statistical Inference 1

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Basis of statistical Inference 1

• Epidemiology in Human Populations 202.251
• A/Prof. Cord Heuer
• EpiCentreMassey University

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Data types

• Qualitative data nominal scale
• Gender, location, socio-economic criteria
• Quantitative data
• Continuous and ratio any value (weight, height,
  time)
• Binary data yes/no (pregnant, exposed, diseased,
  dead)
• Discrete data no subdivision (number of disease
  events, number of babies, number of traffic
  violations)
• Ordinal there is a rank order (mild moderate
  severe negative suspicious positive)

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Simple statistics

• Measures of central tendency
• Mean sum of all values / number of observations
• Median

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Skewed distribution
Mean
Median
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Simple statistics

• Measures of variation
• Standard deviation

-x1
x1
-x1
x1
-x1
x1
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Simple statistics of a sample

• Example
• Values 3,4,6,7
• Median 5
• Standard deviation

-1
1
2
-2
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Normal bell curve
Probability
Mean 5 Std 1.83
- ? mean, and ? standard deviation - the
equation describes an infinite number of bell
curves
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Standard normal curve

• Convert all values to a single standard
• Centre around zero
• Scale to 1 stdev

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Standard normal curve

• Properties
• Symmetrical about y-axis
• Area under the curve 1
• Standard deviation z 1
• Curve asymptotically approaches
• the x-axis
• Extends to infinity in both directions
• Highest point
• We can now use a table to convert any observed
  value x to a z, and z to the probability of
  coming from a specific population!

0.4
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Standard normal curve

• Example
• A chicken dealer at the market grabs a bird of
  2.2kg and offers it to you at the given price
  is the bird really an average bird of the flock?
• What is the probability the bird has the given
  weight or less
• Mean 3.5kg, stdev 0.8kg

2.2
Only 5 chance ? you are 95 sure he was
cheating! Does this mean he is really cheating??
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Inference on continuous data
Population parameters ? true population mean ?
population standard deviation
Sample n ltlt N X-bar sample mean S standard
deviation
Use sample statistics to make inferences
about population parameters
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Sampling distribution of mean

• Describes the distribution of means of repeated
  samples

population
Sample1 mean1, s1
Sample2 mean2, s2
Sample3 mean3, s3
..
Samplek meank, sk
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Distributions for statistics

• Population ?, ?2
• 3 sample parameters
• 1. sample mean
• 2. standard deviation SD
• Standard deviation of the mean standard error
  SE

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Our example

• Mean
• Standard deviation
• Sample size n4
• Standard error

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Central limit theorem

• The set of means from all possible samples of
  size n sampling distribution
• mean of the means ?
• SD of this mean of means SE
• If population is normal, sampling dist is normal
• Even if pop is not normal, sampling dist is
  normal as long as n is large enough (30)

GOTO
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Sample mean

• A random sample of n 150 adult weights has a
  sample mean 80.7, SD 9.2
• What is the mean of the entire population?
• ? somewhere near

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How close is to ? ?

• What is the difference between the sample and the
  true value of the mean (?)?
• - ? deviation from the true mean
• Scale this deviation to units of SEmean
• Solve for ?
• Z is the standard normal distribution of all
  sampling distributions with mean 0 and SE 1

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Our sample

• A random sample of n 150 adult weights has a
  sample mean 80.7, SD 9.2
• The mean of the entire population is
• -zSD/sqrt(150) lt mean lt zSD/sqrt(150)
• The value of z depends on the level of
  confidence
• So, choose your z for sufficient confidence

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Confidence Interval

• 95 data points between 1.96 and 1.96

Probability
Z
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Confidence interval

• 95 data points between z -1.96 and z1.96

Confidence interval
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Our example

• For z 1.96 and z 1.96 the 95 CI is
• SE 9.2/sqrt(150) 0.75
• mean 80.7
• 95 confidence interval
• 80.70 1.960.75 lt mean lt 80.70 1.960.75
• 79.23 lt 80.70 lt 82.17
• Write as (79.23, 82.17)
• About 95 of 100 sample means will be between
  these limits, provided sampling was unbiased

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µ 75
Up to 5 of 100 samples will have 95 confidence
intervals that do not include 75, hence ? 0.05