9 Glycerol is a major by-product of ethanol fermentation in wine production and con-

Example 8.9 Glycerol is a major by-product of ethanol fermentation in wine production and con-

  tributes to the sweetness, body, and fullness of wines. The article “A Rapid and Simple Method for Simultaneous Determination of Glycerol, Fructose, and Glucose in Wine” (American J. of Enology and Viticulture, 2007: 279–283) includes the following observations on glycerol concentration (mgmL) for samples of standard-quality (uncertified) white wines: 2.67, 4.62, 4.14, 3.81, 3.83. Suppose the desired concentration value is 4. Does the sample data suggest that true average concentration is something other than the desired value? The accompanying normal probability plot from Minitab provides strong support for assuming that the popu- lation distribution of glycerol concentration is normal. Let’s carry out a test of appropriate hypotheses using the one-sample t test with a significance level of .05.

  Mean 3.814 StDev 0.7185

  RJ 0.947 P-Value >0.100

  Glycerol conc

  Figure 8.4 Normal probability plot for the data of Example 8.9

  1. m 5 true average glycerol concentration

  2. H 0 :m54

  3. H a :m24

  CHAPTER 8 Tests of Hypotheses Based on a Single Sample

  x24

  4. t5 s 1n

  5. The inequality in H a implies that a two-tailed test is appropriate, which requires t a2,n21 5t 025,4 5 2.776 . Thus H 0 will be rejected if either t 2.776 or

  t 22.776 .

  5 19.07, and g x i 5 74.7979 , from which x 5 3.814 , s 5 .718, and the estimated standard error of the mean is s 1n 5 .321. The test statistic value is then . t 5 (3.814 2 4).321 5 2.58

  6. gx

  i

  7. Clearly t 5 2.58 does not lie in the rejection region for a significance level of .05. It is still plausible that m54 . The deviation of the sample mean 3.814 from

  its expected value 4 when H 0 is true can be attributed just to sampling variability

  rather than to H 0 being false.

  The accompanying Minitab output from a request to perform a two-tailed one- sample t test shows identical calculated values to those just obtained. The fact that the last number on output, the “P-value,” exceeds .05 (and any other reason- able significance level) implies that the null hypothesis can’t be rejected. This is discussed in detail in Section 8.4.

  Test of mu 4 vs not 4

  Variable

  N

  Mean StDev SE Mean

  glyc conc 5 3.814 0.718

  ■

  B and Sample Size Determination The calculation of b at the alternative value m in case I was carried out by expressing the rejection region in terms of (e.g., x

  xm 1z 0 a s 1n ) and then subtracting m to standardize correctly. An equiva-

  lent approach involves noting that when m 5 mr , the test statistic

  Z 5 (X 2 m 0 )(s 1n) still has a normal distribution with variance 1, but now the

  mean value of Z is given by (mr 2 m 0 )(s 1n) . That is, when m 5 mr , the test sta-

  tistic still has a normal distribution though not the standard normal distribution. Because of this, b(mr) is an area under the normal curve corresponding to mean

  value (mr 2 m 0 )(s 1n) and variance 1. Both a and b involve working with nor-

  mally distributed variables.

  The calculation of b(mr) for the t test is much less straightforward. This is

  because the distribution of the test statistic T 5 (X 2 m 0 )(S 1n) is quite compli- cated when H 0 is false and H a is true. Thus, for an upper-tailed test, determining

  b(mr) 5 P(T , t a,n21 when m 5 mr rather than m 0 )

  involves integrating a very unpleasant density function. This must be done numeri- cally. The results are summarized in graphs of b that appear in Appendix Table A.17. There are four sets of graphs, corresponding to one-tailed tests at level .05 and level .01 and two-tailed tests at the same levels.

  To understand how these graphs are used, note first that both b and the nec- essary sample size n in case I are functions not just of the absolute difference

  um 0 2 mr u but of d5um 0 2 mr u s . Suppose, for example, that um 0 2 mr u 5 10. This departure from H 0 will be much easier to detect (smaller b) when s52 , in which case m 0 and m are 5 population standard deviations apart, than when

  s 5 10 . The fact that b for the t test depends on d rather than just um 0 2 mr u is

  unfortunate, since to use the graphs one must have some idea of the true value of s. A conservative (large) guess for s will yield a conservative (large) value of b(mr) and a conservative estimate of the sample size necessary for prescribed a and . b(mr)

  8.2 Tests About a Population Mean

  Once the alternative mr and value of s are selected, d is calculated and its value located on the horizontal axis of the relevant set of curves. The value of b is the height of the

  df curve above the value of d (visual interpolation is nec-

  essary if n21 is not a value for which the corresponding curve appears), as illus- trated in Figure 8.5.

  curve for n

  0 d Value of d corresponding to specified alternative '

  Figure 8.5

  A typical b curve for the t test

  Rather than fixing n (i.e.,

  , and thus the particular curve from which b is

  read), one might prescribe both a (.05 or .01 here) and a value of b for the chosen m and s. After computing d, the point (d, b) is located on the relevant set of graphs. The curve below and closest to this point gives n21 and thus n (again, interpola- tion is often necessary).