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Tests of Significance

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Tests of Significance In this section we deal with two tests used for comparing two analytical methods, one is a new or proposed method and the other is a standard ... – PowerPoint PPT presentation

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Title: Tests of Significance


1
Tests of Significance
  • In this section we deal with two tests used for
    comparing two analytical methods, one is a new or
    proposed method and the other is a standard
    method. The two methods are compared in terms of
    whether they provide comparable precision ( the F
    test ), based on their standard deviations or
    variances. The other test ( t test ) tells
    whether there is a statistical difference between
    results obtained by the two methods.

2
The F Test
  • The precision of two methods could be compared
    based on their standard deviations using the F
    test which can be defined as the ratio between
    the variances ( the variance is the standard
    deviation squared ) of the two methods. The ratio
    should always be larger than unity. That is, the
    larger variance of either method is placed in the
    nominator.
  •  
  • F S12/S22

3
  • Where, S12 gt S22
  •  
  • Values of F ( a statistical factor ) at
    different confidence levels which can be obtained
    from statistical F tables. When Fcalculated lt
    Ftabulated this is an indication of no
    statistical difference between precision or
    variances of the two methods.

4
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5
Example
  • In the analysis of glucose using a new developed
    procedure and a standard procedure, the variances
    of the two procedures were 4.8 and 8.3. If the
    tabulated F value at 95 confidence level at the
    number of degrees of freedom used was 4.95.
    Determine whether the variance of the new
    procedure differs significantly from that of the
    standard method

6
  • F S12/S22
  •  
  • F 8.8/4.8 1.73 (the subscript is because the
    answer is less than the key number)
  • Since Fcalculated lt Ftabulated there is no
    significant statistical difference between the
    variances of the two methods (i.e. there is no
    significant statistical difference between the
    precision of the two methods).

7
The Student t Test
  • To check whether there is a significant
    statistical difference between the results of a
    new or proposed procedure and a standard one, the
    t test is used. As we did above, we calculate t
    and compare it to the tabulated value at the
    required confidence level and at the used degrees
    of freedom. There is no significant statistical
    difference between the results of the two methods
    when tcalculated lt ttabulated .
  • There are three situations where the t test is
    applied

8
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9
a.      When an Accepted Value is Known
  • The tcalc is calculated from the relation below
    and compared to ttab
  •  
  • m x ts/N1/2 or more conveniently,
  •  
  • t (x - m) N1/2/s

10
Example
  • A new procedure for determining copper was used
    for the determination of copper in a sample. The
    procedure was repeated 5 times giving an average
    of 10.8 ppm and a standard deviation of 0.7 ppm.
    If the true value for this analysis was 11.7 ppm,
    does the new procedure give a statistically
    correct value at the 95 confidence level? ttab
    2.776
  •  

11
  • Substitution into equation below, we get
  •   t (x - m) N1/2/s
  • t (10.8-11.7) 51/2/0.7
  • t 2.9
  •  
  • the tcalc is larger than the ttab. Therefore,
    there is a significant statistical difference
    between the two results which also means that it
    is NOT acceptable to use the new procedure for
    copper determination.

12
b. Comparison between two means
  • When an accepted value is not known and the
    sample is analyzed using the new procedure and a
    standard procedure. Here, we have two sets of
    data, a standard deviation for each set of data
    and a number of data points or results in each
    set. Under these conditions, we use the pooled
    standard deviation for the two sets. The same
    equation in a is used but with some
    modifications. The t value is calculated from the
    relation
  •  

13
  • Where, x1 and xs are means of measurements
    using the new and standard methods. N1 and Ns
    are number of replicates done using the new and
    standard methods, respectively. Sp is the pooled
    standard deviation.
  • In such calculations it is wise to apply the F
    test first, and if it passes the t test is then
    applied.

14
Example
  • Nickel in a sample was determined using a new
    procedure where six replicate samples resulted in
    a mean of 19.65 and a variance of 0.4524. Five
    replicate analyses where conducted using a
    standard procedure resulting in a mean of 19.24
    and a variance of 0.105. If the pooled standard
    deviation was 0.546, is there a significant
    difference between the two methods?

15
  • First, let us find whether there is a significant
    difference in precision between the two
    procedures, by applying the F test
  •  
  • F 0.4524/0.105 4.31
  •  
  • The tabulated F value is 6.26. Since Fcalculated
    lt Ftabulated , then there is no significant
    statistical difference between the precision of
    the two procedures. Therefore, we continue with
    calculation of t test.

16
  •  
  • t 1.23
  •  
  • The tabulated t value is 2.262. Since
    tcalculated lt ttabulated for nine degrees of
    freedom at 95 confidence level, we conclude that
    there is no significant statistical difference
    between the results of the two methods.
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