1. 4.1 Hypothesis Testing z-test for a single value
double-sided and single-sided z-test for one average
z-test for two averages
double-sided and single-sided t-test for one average
the F-parameter and F-table
the F-test
the t-test for two averages
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2. z-Test for a Single Value
Suppose that an analytical technique is done sufficiently often that s is known. Also assume that only a single measured value is available.
The following hypothesis test involves the situation where m is specified or obtained by theory.
Null Hypothesis: the measured value, x, comes from a normal pdf having m as its mean.
Alternative Hypothesis: the measured value, x, comes from a normal pdf not having m as its mean.
To test the hypothesis compute, zcalc = |x - m|/s.
If zcalc  1.96, accept the null hypothesis.
If zcalc > 1.96, accept the alternative hypothesis.
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3. Graphical Interpretation of Test
The null hypothesis will be accepted for any pdf having a mean over the range m = x s (blue curve) to m = x s (red curve). Accepting the null hypothesis does not guarantee that x comes from a pdf having m as its mean. Acceptance infers that the pdf resulting in x is statistically indistinguishable from one having m as its mean.
Rejection of the null hypothesis states with 95% confidence that x comes from a pdf not having m as its mean.
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4. z-Test for One Average
Suppose that an analytical technique is done sufficiently often that s is known. Also assume that an average, , has been obtained using N replicate measurements.
The following hypothesis test involves the situation where m is specified or obtained by theory.
Null Hypothesis: the average, , comes from a normal pdf having m as its mean.
Alternative Hypothesis: the average, , comes from a normal pdf not having m as its mean.
To test the hypothesis compute,
If zcalc  1.96, accept the null hypothesis.
If zcalc > 1.96, accept the alternative hypothesis.
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5. Single-Sided z-Test for One Average
Confidence limits can also be used to guarantee that the measured average comes from a pdf with a mean greater than or equal to a specified value, C0. The entire 5% uncertainty is put on the low-value side of the pdf. For a normal pdf F(x) occurs at = m s. Now, zcalc = (C )/s, and can be negative.
Null hypothesis: z  1.64, the measured average is greater than or equal to the specified value.
Alternative hypothesis:
z > 1.64, the measured average is less than the specified value.
A similar strategy can be used to test if the measured average is less than a specified value.
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6. z-Test for Two Averages
It is possible to statistically test whether two experimental averages come from the same pdf. Let have the normal pdf, n(m,s/N11/2), and have the normal pdf, n(m,s/N21/2). To test the hypothesis calculate the difference, D, and test whether it is statistically indistinguishable from 0.
For zcalc  1.96, the averages are not statistically distinguishable.
For zcalc > 1.96, the averages come from pdfs with different means.
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7. The t-Test for One Average
When the value of s is not known and is estimated by s, the confidence limits are determined by the t-parameter. The hypothesis test uses tcalc instead of zcalc.
To perform the hypothesis test, an appropriate value of ttable is found by choosing the confidence level and the degrees of freedom, f = N-1.
Null hypothesis: tcalc  ttable, the average comes from a pdf with a mean indistinguishable from m.
Alternative hypothesis: tcalc > ttable, the average comes from a pdf with a mean different than m.
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8. t-Test Examples
A NIST nickel standard known to contain 6.15 mmol is analyzed by a gravimetric method. Three replicate measurements were obtained: {5.88, 5.68, 6.16 mmol}. The average is 5.91 mmol and the standard deviation is 0.24 mmol.
Since tcalc  ttable, the average is indistinguishable from a pdf having a mean of 6.15 mmol. An A-grade! A second student ran 10 replicates and obtained an average of 5.46 mmol and a standard deviation of 0.29 mmol.
Since tcalc > ttable, the average does not come from a pdf having 6.15 mmol as its mean. The student has to repeat the determination and identify the determinate error.
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9. Single-Sided t-Test
A pollution regulation requires that the concentration of airborne SO2 be less than 10 ppm. Three measurements are made {12.64, 11.04,14.57} with an average of and a standard deviation of By analogy with the single-sided z-test on slide we can write the following.
A single-sided t-test requires that we use a 90% double-sided table. f 2 3 4 9 19 29 39 ∞
t(0.90)
2.92
2.35
2.13
1.83
1.73
1.70
1.69
1.64
Since tcalc  ttable(0.90,2), the average of is statistically indistinguishable from the 10 ppm regulation.
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10. The F-Parameter(see note)
The ratio of two experimental variances is called the F-parameter, where F = s12/s22. The pdf, f(F), is asymmetric and has to be integrated numerically.
The red line is F(3,3), the blue F(5,5), the green F(10,10), the magenta F(20,20), and the cyan F(50,50).
As both N1 and N2 increase, the pdf becomes symmetric and the mean approaches 1.
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11. The F-Table
By convention the larger variance is placed into the numerator so that 1  F  +∞.
The value of F yielding any particular confidence level depends upon the degrees of freedom used to compute both variances. The numerator f1 is in the top row, the denominator f2 in the left column.
f2 | f1 2 3 4 9 19 29 39 ∞
19.00
19.16
19.25
19.38
19.44
19.46
19.47
19.50
9.55
9.28
9.12
8.81
8.67
8.62
8.60
8.53
6.94
6.59
6.39
6.00
5.81
5.75
5.72
5.63
4.26
3.86
3.63
3.18
2.95
2.87
2.83
2.71
3.52
3.13
2.90
2.42
2.17
2.08
2.03
1.88
3.33
2.93
2.70
2.22
1.96
1.86
1.81
1.64
3.24
2.85
2.61
2.13
1.76
1.70
1.52
3.00
2.37
1.59
1.47
1.40
1.00
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12. Example F-Test
Two students analyzed a steel sample for nickel. The first used a gravimetric method and obtained 3 values, {9.87, 9.95, and mmol}, with s = 0.07 mmol. The second used a titrimetric method and obtained 5 values, {9.98, 9.99, 9.99, 10.06, 9.97 mmol}, with
s = 0.04 mmol. Although the two standard deviations differ by a factor of ~2, are they statistically different?
Since Fcalc  Ftable(0.95,flarger=2,fsmaller = 4) = 6.94, the two standard deviations are statistically indistinguishable.
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13. t-Test for Two Averages
Two experimental averages can be compared as long as they both come from pdfs having the same standard deviation. Ordinarily the two experimental standard deviations are first checked using the F-test. If the F-test is passed, then the t-test for two averages can proceed.
First compute a pooled variance using the following equation, then calculate t.
Why is this one wrong?
The ttable value depends upon the confidence level and N1+N2 -2 degrees of freedom.
If tcalc  ttable, accept the null hypothesis; if tcalc > ttable, accept the alternative hypothesis.
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14. Example t-Test for Two Averages
Use the previous example where two students analyzed a steel sample for nickel. The first used a gravimetric method and obtained 3 values with = 9.94 and s1 = 0.07 mmol. The second used a titrimetric method and obtained 5 values with = and
s2 = 0.04 mmol. The two variances have already been shown to be statistically indistinguishable.
Since tcalc  ttable, the null hypothesis is accepted. The two means are statistically indistinguishable.
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