Mean Analysis

1
Mean Analysis
2
Introduction

• If we use sample mean (the mean of the sample) to
  approximate the population mean (the mean of the
  population), errors will be introduced.
• Two questions
• How good is the approximation?
• If the error is too large, what should we do to
  reduce it?

3
Confidence interval

• Confidence interval gives us the interval in
  which the population mean would most likely fall.
• The confidence level tells us (intuitively) the
  possibility the population mean will be in the
  confidence interval.
• The smaller the confidence interval and the
  higher the confidence level, the better the
  approximation.

4
Calculation of the confidence interval

• The equation for confidence interval is
• The half-width of the C.I.
• The point estimate is

5
Some z values

• Confidence level z
• 90 1.65
• 95 2
• 99 2.6
• 99.7 3

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How to find z value

• Given a confidence level, say 80, we need to
  find the Area value from the normal table that is
  closest to 80 and then find the corresponding z
  value, which will be the z value for the
  confidence interval formula.

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Properties

• The larger the sample size n, the narrower the
  confidence interval.
• The higher the confidence level, the larger z and
  thus the wider the confidence interval.
• The population size has nothing to do with the
  confidence interval, as long as it is large
  enough.

8
Distribution of the sample mean

• If the sample size is large enough, say larger
  than 30, the histogram of the sample means is
  very close to a Normal distribution.
• The mean of the sample means equals to the
  population mean.
• The standard deviation of the sample means equals
  to the population standard deviation divided by
  the squared root of n.

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Conditions for Confidence Interval

• The sampling method should be unbiased
• The sample size n should be large enough, say
  larger than 30.
• The population size N should be much larger than
  the sample size n.

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Determine the correct sample size n

• 1. Take a random sample of size n.
• 2. Calculate the confidence interval based on
  your sample.
• 3. Check if the confidence interval is small
  enough. If it is too wide, decide a proper
  width.
• 4. Assuming the sample standard deviation will
  remain the same, use the equation for confidence
  interval to estimate the required sample size.
• 5. Take a new sample and calculate the confidence
  interval.

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When n is small

• When n is small, say, n5, a formula for the
  confidence interval can be derived only when the
  population distribution is normal.
• The formula for confidence interval is very
  similar to the one we have used, the difference
  is we will have to employ the so-called Student t
  distribution instead of normal distribution.

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The Confidence interval for population proportion

• The equation is
• n is the sample size,
• z is determined by the confidence level, e.g., if
  the confidence level is 95 z is equal to 2, and
• p is the proportion of yes answers in the sample.
  

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Conditions for Confidence Interval

• The sampling method should be unbiased
• The sample size n should be large enough. In
  particular, both np and n(1-p) should be larger
  than 5.
• The population size N should be much larger than
  the sample size n.

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Comparing the means of two populations

• The formula is
• y2 and y1 are the sample means
• s1 and s2 are the standard deviations of the two
  samples
• n1 and n2 are the sample sizes of the two
  samples and
• z is selected to provide the desired confidence
  level.

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Conditions for Confidence Interval

• The two samples should be independently and
  randomly selected
• The sample sizes should be large enough at
  least 30
• The population sizes should be much larger than
  the corresponding sample sizes.

16
Compare proportions of two means

• The formula is the following