Large-Sample Tests

Case II: Large-Sample Tests

  When the sample size is large, the z tests for case I are easily modified to yield valid test procedures without requiring either a normal population distribution or known s. The key result was used in Chapter 7 to justify large-sample confidence intervals:

  A large n implies that the standardized variable X2m

  Z5 S 1n

  has approximately a standard normal distribution. Substitution of the null value m 0 in place of m yields the test statistic

  which has approximately a standard normal distribution when H 0 is true. The use of rejection regions given previously for case I (e.g., zz a when the alternative hypothesis is H a :m.m 0 ) then results in test procedures for which the significance

  8.2 Tests About a Population Mean

  level is approximately (rather than exactly) a. The rule of thumb n . 40 will again

  be used to characterize a large sample size.

  Example 8.8

  A dynamic cone penetrometer (DCP) is used for measuring material resistance to penetration (mmblow) as a cone is driven into pavement or subgrade. Suppose that for a particular application it is required that the true average DCP value for a cer- tain type of pavement be less than 30. The pavement will not be used unless there is conclusive evidence that the specification has been met. Let’s state and test the appropriate hypotheses using the following data (“Probabilistic Model for the Analysis of Dynamic Cone Penetrometer Test Values in Pavement Structure Evaluation,” J. of Testing and Evaluation, 1999: 7–14):

  Figure 8.3 shows a descriptive summary obtained from Minitab. The sample mean DCP is less than 30. However, there is a substantial amount of variation in the data

  (sample coefficient of variation 5 s x 5 .4265 ), so the fact that the mean is less

  than the design specification cutoff may be a consequence just of sampling variabil- ity. Notice that the histogram does not resemble at all a normal curve (and a normal probability plot does not exhibit a linear pattern), but the large-sample z tests do not require a normal population distribution.

  Descriptive Statistics

  Variable: DCP

  Anderson-Darling Normality Test

  –3.9E–01

  14.1000 1st Quartile

  27.5000 3rd Quartile

  Median

  57.0000 95 Confidence Interval for Mu

  Maximum

  3.21761 20 25 30 95 Confidence Interval for Sigma

  15.2098 95 Confidence Interval for Median

  Figure 8.3 Minitab descriptive summary for the DCP data of Example 8.8

  1. m5 true average DCP value

  2. H 0 : m 5 30

  CHAPTER 8 Tests of Hypotheses Based on a Single Sample

  3. H a : m , 30 (so the pavement will not be used unless the null hypothesis is rejected)

  x 2 30

  4. z5 s 1n

  5. A test with significance level .05 rejects H 0 when (a z 21.645 lower-tailed

  test).

  6. With , n 5 52, x 5 28.76 and s 5 12.2647 ,

  7. Since , 2.73 . 21.645 H 0 cannot be rejected. We do not have compelling evi-

  dence for concluding that m , 30 ; use of the pavement is not justified.

  ■

  Determination of b and the necessary sample size for these large-sample tests can be based either on specifying a plausible value of s and using the case I formu- las (even though s is used in the test) or on using the methodology to be introduced shortly in connection with case III.