Statistical tests

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Statistical tests

• We can use the properties of probability density
  functions to make probability statements about
  the likelihood of events occurring.
• The standard normal curve provides us with a
  scale or benchmark for the likelihood of being at
  (or above or below) any point on the scale

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Standard normal values

• Note for instance that if we look at the value
  1.5 under the standard normal table, we find the
  value .4332.
• This means that the probability of having a
  standard normal value greater than 1.5 is .5 -
  .4332 .0668

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In Applied Terms

• If IQ has a mean of 100, and a standard deviation
  of 20, what is the probability that any given
  individuals IQ will be greater than or equal to
  130.
• Standardize the score of 130
• Look up 1.5 in the standard normal table

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Two-tailed hypotheses

• In general our hypothesis is
• Did the sample come from some particular
  population?
• If the sample mean is too high or too low, we
  suspect that it did not.
• Thus, we must check to see if the sample mean is
  either significantly higher, or significantly
  lower.
• This is called a two-tailed test.
• When in doubt, most tests are best done as
  two-tailed ones

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The One Tailed Hypothesis

• Sometimes we suspect, or hypothesize, direction
• e.g. The average income for West Virginia will be
  significantly lower than the country as a whole.
• HA Xbar lt ?
• This is a one-tailed test
• We ignore the tail in the direction not
  hypothesized

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The Z-test

• The z-test is based upon the standard normal
  distribution.
• It uses the standard normal distribution in the
  same way. In this case we are making statements
  about the sample mean, instead of the actual data
  values

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The Z-test (cont.)

• Note that the Z-test is based upon two parts.
• The standard normal transformation
• The standard deviation of the sampling
  distribution.

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The Z-test an example

• Suppose that you took a sample of people off the
  street in Morgantown and found that their
  personal income is 19,362
• And you have information that the national
  average for personal income per capita is 26,412
  in 1998.
• Should you conclude that West Virginia is lower
  than the national average?
• Is it significantly lower?
• Could it simple be a bad sample
• How do you decide?

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Example (cont.)

• We will hypothesize that WV income is lower than
  the national average.
• HA WVInc lt USInc (Alternate Hypothesis)
• H0 WVInc USInc (Null Hypothesis)
• Since we know the national average (26412) and
  standard deviation (4234), we can use the z-test
  to make decide if WV is indeed significantly
  lower than the nation

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Example (cont.)

• Using the z-test, we get

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The Probability of a Type I error

• We would like to not make mistakes.
• We know we will.
• With statistical inference, we have the ability
  to decide how often we find it acceptable to be
  wrong by random chance.
• Thus we set the probability of making a Type I
  error.
• P(Type I error) ? ?
• By convention ?.05

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The Critical Value of Z (cont)

• Ok, now we know z
• We know that we can make probability statements
  about z, since it is from the standard normal
  distribution
• We know that if z 1.96 then the area out in the
  tail past 1.96 is equal to .025
• This means that the likelihood of obtaining a
  value of z gt 1.96 by random chance in any given
  sample is less than .025.

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The Critical Values of Z to memorize

• Two tailed hypothesis
• Reject the null (H0) if z ? 1.96, or z ? -1.96
• One tailed hypothesis
• If HA is Xbar gt ?, then reject H0 if z ? 1.645
• If HA is Xbar lt ?, then reject H0 if z ? -1.645

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Z test example (cont.)

• Suppose we decided to look at a different state,
  say Oregon with a mean of 24,766, and had a much
  smaller sample, say 16.
• Using the z-test, we get
• What would we conclude?
• What if n25? 100?

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The t test

• We frequently run into a problem with trying to
  do a z test.
• While the population mean (?) may be frequently
  available, the population standard deviation (?)
  frequently is not.
• Thus we use our best estimate of the population
  standard deviation the sample standard
  deviation (s).

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The t-test (cont.)

• The t-test is a very similar formula.
• Note the two differences
• using s instead of ?
• The resultant is a value that has a
  t-distribution instead of a standard normal one.

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The t distribution

• The t distribution

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Two-sample t-test

• Frequently we need to compare the means of two
  different samples.
• Is one group higher/lower than some other group?
• e.g. is the Income of blacks significantly lower
  than whites?
• The two-sample t difference of means test is the
  typical way to address this question.

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The Difference of means Test

• The standard two-sample t-test is

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The equal Variance test

• If the variances from the two samples are the
  same we may use a more powerful variation
• Where

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Contingency Tables

• Often we have limited measurement of our data.
• Contingency Tables are a means of looking at the
  impact of nominal and ordinal measures on each
  other.
• They are called contingency tables because one
  variables value is contingent upon the other.
• Also called cross-tabulation or crosstabs.

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Contingency Tables

• The procedure is quite simple and intuitively
  appealing
• Construct a table with the independent variable
  across the top and the dependent variable on the
  side
• This works fairly well for low numbers of
  categories (r,c lt 6 or so)

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Contingency Tables An example

• Presidents are often suspected of using military
  force to enhance their popularity.
• What do you suppose the data actually look like?
  
• Any conjectures
• Lets categorize presidents as using force,or
  not, and as having popularity above and below 50
• Are their definition problems here?
• Which is independent and which is dependent?

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Contingency Tables

Presidential Approval Presidential Approval Presidential Approval Presidential Approval Presidential Approval
Use of Military Force lt 50 gt 50
Use of Military Force Not Used 16 70 28 41 44 48
Use of Military Force Used 7 30 40 59 52
Use of Military Force Total 23 100 68 100 91 100
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Measures of Independence

• Are the variables actually contingent upon each
  other?
• Is the use of force contingent upon the
  presidents level of popularity?
• We would like to know if these variables are
  independent of each other, or does the use of
  force actually depend upon the level of approval
  that the president have at that time?

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?2 Test of Independence

• The ?2 Test of Independence gives us a test of
  statistical significance.
• It is accomplished by comparing the actual
  observed values to those you would expect to see
  if the two variables are independent.

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?2 Test of Independence

• Formula
• Where