Statistical Inference Statistical Inference • by ewghwehws


									                          Statistical Inference

• Basic Principle Underlying all inferential statistics…
    samples are representative of the population from which they are drawn
• Hypotheses Testing
• Estimation
    – The inferential process of using sample statistics to estimate population
      parameters is called estimation
• Accuracy / Precision
    – Small error
• Unbias-ness
    – Expected Value of the Estimator is the population parameter
            When to make Estimation ?

• Goal: Estimate the value of an unknown population parameter, usually the
  value for an unknown population mean

• Estimate effect after the null hypothesis is rejected
• Already know an effect and wish to estimate for it
• Population parameter is unknown and wish to gain info
                 Two Ways of making Estimates

• Point estimation
    – Use a single number to estimate an unknown population parameter

• Interval estimation
    – Because the estimate is associated with confidence, it is also called
      confidence interval
    – Use a range of values as an estimate
    – Advantage: increase confidence
    – Disadvantage: decrease precision
          Procedure for Interval Estimation of
                 Population Mean µ

• Begin by calculating the sample mean. This provides a point estimate for µ.
• Best bet is to predict that the sample mean is located somewhere in the
  centre of the distribution
• Determine the amount of confidence required, say 95%
• Estimate that our sample mean is somewhere in the middle 95% of the
  sample means. This section of the distribution is bounded by
    – z scores of             -1.96 , +1.96            or
    – t scores of            - t 0.025, df , + t 0.025, df
• Compute the limits of the 95% Confidence Interval for µ
    – Using z:      sample mean +/- z * standard error
    – Using t :     sample mean +/- t * standard error
                    Example 12.2 (page 381)

• Recent studies allowed psychologists to establish definite links between
  specific foods and specific brain functions. E.g. lecithin (found in soybeans,
  eggs, and liver) has been shown to increase the conc of certain brain
  chemicals that help regulate memory and motor coordination. This expt is
  designed to demo the importance of this particular food substance.

• Two independent samples of rats, 10 with normal diet with standard
  amounts of lecithin, 5 in another group are fed a special diet which contains
  almost no lecithin. After 6 months, each of the rats is tested on a specially
  designed learning problem that requires both memory and motor
  coordination. The purpose of expt is to demo the deficit in performance that
  results from lecithin deprivation. Score for each animal is number of errors
  it makes before it solves the learning problem.

• Data:           No-lecithin diet: n=5,     x1-bar=33, SS=140
                  Regular diet:     n=10,    x2-bar=25, SS=250
                         Example continued

• Note:         S22 =250/9=28
                S12 =140/4=35
• Estimate µ1-µ2 =33-25=8
• Sp2 = (250+140)/(9+4) =30
• S (x1-bar - x2-bar) = (30/10 +30/5) **.5 =3

• t .025, 13 =2.16
• A 95% CI for µ1-µ2 is 8 +/- 3*2.16 = 1.52, 14.48

• Note that the text book provides a 80% confidence interval (page 383) which
  is 3.95,12.05

• Compare the interval estimates on ‘confidence’ and ‘width’

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