Statistics: Lesson 12Hypothesis testing and conf' intervals in regression

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Statistics Lesson 12---Hypothesis testing and
conf. intervals in regression

• In engineering analysis, the slopes and
  intercepts from our regression analysis often
  have physical meaning.
• For this and other reasons, we are interested in
  testing various hypotheses about the regression
  coefficients.

Reference MRH 3rd ed., section 6-2.
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• For example, suppose we were asked whether an
  existing heat exchanger could be used to handle a
  heat duty, Q Btu/h for a condensing vapor. We
  measure the mass flow rate of the stream it is to
  supposed to condense as wt. of condensate versus
  time.

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• The slope of that line (wt. vs. time) gives us
  the design flow rate, but it is only an estimate
  of the population value. Hypothesis testing on
  this measured value would tell us whether, with a
  specified significance level, we can use the
  exchanger.

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t-Tests

• Assignment test the hypothesis that the
  regression slope equals some constant

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• Because we showed last time that the values of Y
  are normally distributed (because the errors are
  randomly distributed), the t-statistic is
  normally distributed with n-2 dof.

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• The t-statistic can be calculated for the value
  of the constant

standard error of
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• We can do the same thing for the intercept

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• We reject Ho if the absolute value of the
  calculated t-value was greater than the t-value
  for the significance level we choose. (Remember
  we need both the significance level and dof to
  find t).

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• In engineering applications, we often want to
  know if the slope (which usually has some
  physical meaning) is significantly different from
  zero.

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• If we reject Ho we are saying that there
  IS---i.e., the regression model does suggest a
  relationship between Y and x

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• If we do not reject Ho we are saying that there
  IS NOT a linear relationship between Y and x.
• In this case the mean of all the values of Y is
  the best estimator of Y

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When we do not reject Ho

• there are two cases to examine. First, suppose
  the plot of Y vs. x looked like this

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• But suppose our plot looked like this

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• Despite the fact that we come to the same
  conclusion (do not reject the null hypothesis
  Ho?10), there is a clear relationship between Y
  and x that is NOT described by the linear
  regression model.
• We would then need to formulate another model
  (maybe a quadratic or higher order equation in
  this case).

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If we do reject Ho

• there are two similar cases to look at. Suppose
  the relationship looked like this

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• We would say that the linear model adequately
  explains the relationship between Y and x.

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• But suppose it looked like this

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• We still reject Ho, but it is clear that the
  model weve chosen does not adequately represent
  the data.

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Example

• We measure the concentration, C, of a reactant in
  a stirred tank, and plot lnC vs time. (This gives
  us a linear relationship). We calculate the
  regression coefficients for 20 data points and
  find

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• The Excel spreadsheet looks like this

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• and the data look like this

lnC
t
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• We conclude that a linear regression model
  adequately represents the data.

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• In this case, the t-statistic is calculated as
  follows

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• Lets say we want a level of significance in this
  case of 1. The t-value for 18 degrees of freedom
  and this significance is

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• Because our calculated t-value (18.77) is much
  greater than the critical value of 2.88 (even for
  a relatively small significance level), there is
  very little probability that

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• An Excel output of this data contains the
  following at the bottom

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• the t-stat is the value of t computed from the
  equation we developed earlier
• for a value of

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• This number tells us the t-value corresponding to
  probability that
• The higher this number, the more likely it is
  that we can reject the null hypothesis

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• In other words, if we look at the t-Table, we see
  that we have to go to smaller and smaller levels
  of significance (larger and larger t-values and
  confidence intervals) at a given dof to be able
  to include larger values of t. (go to Table II).

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• Another way to think about it
• The larger the value of t in the Excel
  spreadsheet, the more likely it is that the
  calculated value of the regression coefficient is
  significantly different from zero.