Inference about the mean of a population of measurements (m) is based on the standardized value of the sample mean (Xbar).

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• Inference about the mean of a population of
  measurements (m) is based on the standardized
  value of the sample mean (Xbar).
• The standardization involves subtracting the mean
  of Xbar and dividing by the standard deviation of
  Xbar recall that
• Mean of Xbar is m and
• Standard deviation of Xbar is s/sqrt(n)
• Thus we have (Xbar - m )/(s/sqrt(n)) which has a
  Z distribution if
• Population is normal and s is known or if
• n is large so CLT takes over

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• But what if s is unknown?? Then this standardized
  Xbar doesnt have a Z distribution anymore, but a
  so-called t-distribution with n-1 degrees of
  freedom
• Since s is unknown, the standard deviation of
  Xbar, s/sqrt(n), is unknown. We estimate it by
  the so-called standard error of Xbar, s/sqrt(n),
  where sthe sample standard deviation.
• There is a t-distribution for every value of the
  sample size well use t(k) to stand for the
  particular t-distribution with k degrees of
  freedom. There are some properties of these
  t-distributions that we should note

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• Every t-distribution looks like a N(0,1)
  distribution i.e., it is centered and symmetric
  around 0 and has the same characteristic bell
  shape however, the standard deviation of t(k)
  sqrt(k/(k-2)) is greater than 1, the s.d. of Z
  so the t-distribution density curve is more
  spread out than Z. Probabilities involving r.v.s
  that have the t(k) distributions are given by
  areas under the t(k) density curve Table D in
  the back of our book gives us the probabilities
  we need

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• The good news is that everything weve already
  learned about constructing confidence intervals
  and testing hypotheses about m carries through
  under the assumption of unknown s
• So e.g., a 95 confidence interval for m based
  on a SRS from a population with unknown s is
• Xbar /- t(s.e.(Xbar))
• Recall that s.e.(Xbar) s/sqrt(n). Here t
  is the appropriate tabulated value from Table D
  so that the area between t and t is .95
• As we did before, if we change the level of
  confidence then the value of t must change
  appropriately

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• Similarly, we may test hypotheses using this
  t-distributed standardized Xbar e.g., to test
  the H0 m m0 against Ha m gtm0 we use
• (Xbar - m0)/(s/sqrt(n)) which has a
  t-distribution with n-1 df, assuming the null
  hypothesis is true. See page 422 (7.1, 3/7) for
  a complete summary of hypothesis testing in the
  case of the one-sample t-test
• HW Read section 7.1 thru p. 433 go over all the
  examples carefully and answer the HW questions
  following them 7.1-7.9 Work on the following
  problems (p.441 ff) (use software as needed)
  7.15-7.22, 7.25, 7.32, 7.35-7.37, 7.41.

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Is there a difference in aggressive behavior of
patients on "moon days" compared with "non-moon
days"?

• To summarize the analysis
• when the data comes in matched pairs, the
  analysis is performed on the differences between
  the paired measurements
• then use the t-statistic with n-1 d.f. (n of
  pairs) to construct confidence intervals and test
  hypotheses on the true mean difference.

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• In a matched pairs design, subjects are matched
  in pairs and the outcomes are compared within
  each matched pair. A coin toss could determine
  which of the two subjects gets the treatment and
  which gets the control One special kind of
  matched pairs design is when a subject acts as
  his/her own control, as in a before/after study
  See example 7.7 on page 428ff (7.1, 4/7). Note
  that the paired observations ( of agressive
  behaviors) are subtracted and the difference in
  scores becomes the single number analyzed with a
  one-sample t-statistic with n-1 df, where nthe
  number of pairs see the top of page 431 and the
  next page for a summary of the process.
• HW Read through p.433. Go over Example 7.7
• then do 7.32, 7.35, 7.41.

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• Read the section on Robustness of the t
  procedures (starting p.432 (7.1, 5/7)) note the
  definition of the statistical term robust
  essentially, a statistic is robust if it is
  insensitive to violations of the assumptions made
  when the statistic is used. For example, the
  t-statistic requires normality of the population
  how sensitive is the t-statistic to violations of
  normality?? Look at the practical guidelines for
  inference on a single mean at bottom of p.432
• If the sample size is lt 15, use the t procedures
  if the data are close to normal.
• If the sample size is gt 15 then unless there is
  strong non-normality or outliers, t procedures
  are OK
• If the sample size is large (say n gt 40) then
  even if the distribution is skewed, t procedures
  are OK