Choosing Sample Sizes for Inferences about ␮

EXAMPLE 6.10 One of the crucial factors in the construction of large buildings is the amount of time it takes for poured concrete to reach a solid state, called the “set-up” time. Researchers are attempting to develop additives that will accelerate the set-up time without diminishing any of the strength properties of the concrete. A study is being designed to compare the most promising of these additives to concrete with- out the additive. The research hypothesis is that the concrete with the additive will have a smaller mean set-up time than the concrete without the additive. The researchers have decided to have the same number of test samples for both the concrete with and without the additive. For an a ⫽ .05 test, determine the appro- priate number of test samples needed if we want the probability of a Type II error to be less than or equal to .10 whenever the concrete with the additive has a mean set-up time of 1.5 hours less than the concrete without the additive. From previous experiments, the standard deviation in set-up time is 2.4 hours. Solution Let m 1 be the mean set-up time for concrete without the additive and m 2 be the mean set-up time for concrete with the additive. From the description of the problem, we have ● One-sided research hypothesis: m 1 ⬎ m 2 ● ● a ⫽ .05 ● b ⱕ .10 whenever m 1 ⫺ m 2 ⱖ 1.5 ⫽ ⌬ ● n 1 ⫽ n 2 ⫽ n From Table 1 in the Appendix, z a ⫽ z .05 ⫽ 1.645 and z b ⫽ z .10 ⫽ 1.28. Substituting into the formula, we have Thus, we need 44 test samples of concrete with the additive and 44 test samples of concrete without the additive. n ⬇ 2s 2 z a ⫹ z b 2 ∆ 2 ⫽ 22.4 2 1.645 ⫹ 1.28 2 1.5 2 ⫽ 43.8, or 44 s ⬇ 2.4 Sample Sizes for Testing ␮ 1 ⴚ ␮ 2 , Independent Samples One-sided test: Two-sided test: where n 1 ⫽ n 2 ⫽ n and the probability of a Type II error is to be ⱕ b when the true difference |m 1 ⫺ m 2 | ⱖ ∆. Note: If s is unknown, substitute an estimated value to obtain an approximate sample size. n ⫽ 2s 2 z a 兾2 ⫹ z b 2 ∆ 2 n ⫽ 2s 2 z a ⫹ z b 2 ∆ 2 The sample sizes obtained using this formula are usually approximate because we have to substitute an estimated value of s, the common population standard deviation. This estimate will probably be based on an educated guess from infor- mation on a previous study or on the range of population values. Corresponding sample sizes for one- and two-sided tests of m 1 ⫺ m 2 based on specified values of a and b, where we desire a level a test having the probability of a Type II error bm 1 ⫺ m 2 ⱕ b whenever |m 1 ⫺ m 2 | ⱖ ⌬, are shown here. Sample-size calculations can also be performed when the desired sample sizes are unequal, n 1 ⫽ n 2 . Let n 2 be some multiple m of n 1 ; that is, n 2 ⫽ mn 1 . For example, we may want n 1 three times as large as n 2 ; hence, . The displayed formulas can still be used, but we must substitute m ⫹ 1 兾m for 2 and n 1 for n in the sample-size formulas. After solving for n 1 , we have n 2 ⫽ mn 1 . EXAMPLE 6.11 Refer to Example 6.10. Because the set-up time for concrete without the additive has been thoroughly documented, the experimenters wanted more information about the concrete with the additive than about the concrete without the additive. In particular, the experimenters wanted three times more test samples of concrete with the additive than without the additive; that is, n 2 ⫽ mn 1 ⫽ 3n 1 . All other specifica- tions are as given in Example 6.10. Determine the appropriate values for n 1 and n 2 . Solution In the sample size formula, we have m ⫽ 3. Thus, replace 2 with . We then have Thus, we need n 1 ⫽ 30 test samples of concrete without the additive and n 2 ⫽ mn 1 ⫽ 330 ⫽ 90 test samples with the additive. Sample sizes for estimating m d and conducting a statistical test for m d based on paired data differences are found using the formulas of Chapter 5 for m. The only change is that we are working with a single sample of differences rather than a single sample of y-values. For convenience, the appropriate formulas are shown here. n 1 ⬇ 冢 m ⫹ 1 m 冣 s 2 z a ⫹ z b 2 ∆ 2 ⫽ 冢 4 3 冣 2.4 2 1.645 ⫹ 1.28 2 1.5 2 ⫽ 29.2, or 30 m ⫹ 1 m ⫽ 4 3 n 2 ⫽ 1 3 n 1 Sample Sizes for Testing ␮ 1 ⴚ ␮ 2 , Paired Samples One-sided test: Two-sided test: where the probability of a Type II error is b or less if the true difference m d ⱖ ⌬ . Note: If s d is unknown, substitute an estimated value to obtain an approxi- mate sample size. n 艑 s 2 d z a 兾2 ⫹ z b 2 ∆ 2 n ⫽ s 2 d z a ⫹ z b 2 ∆ 2 Sample Size for a 1001 ⴚ ␣ Confidence Interval for ␮ 1 ⴚ ␮ 2 of the Form d ⴚ ⴞ E , Paired Samples Note: If s d is unknown, substitute an estimated value to obtain approximate sample size. n ⫽ z 2 a 兾2 s 2 d E 2

6.7 Research Study: Effects of Oil Spill on Plant Growth

The oil company, responsible for the oil spill described in the abstract at the begin- ning of this chapter, implemented a plan to restore the marsh to prespill condition. To evaluate the effectiveness of the cleanup process, and in particular to study the residual effects of the oil spill on the flora, researchers designed a study of plant growth 1 year after the burning. In an unpublished Texas AM University disser- tation, Newman 1997 describes the researchers’ plan for evaluating the effect of the oil spill on Distichlis spicata, a flora of particular importance to the area of the spill. We will now describe a hypothetical set of steps that the researchers may have implemented in order to successfully design their research study. Defining the Problem The researchers needed to determine the important characteristics of the flora that may be affected by the spill. Some of the questions that needed to be answered prior to starting the study included the following: 1. What are the factors that determine the viability of the flora? 2. How did the oil spill affect these factors?

3.

Are there data on the important flora factors prior to the spill? 4. How should the researchers measure the flora factors in the oil-spill region? 5. How many observations are necessary to confirm that the flora has undergone a change after the oil spill? 6. What type of experimental design or study is needed? 7. What statistical procedures are valid for making inferences about the change in flora parameters after the oil spill? 8. What types of information should be included in a final report to docu- ment the changes observed if any in the flora parameters? Collecting the Data The researchers determined that there was no specific information on the flora in this region prior to the oil spill. Since there was no relevant information on flora density in the spill region prior to the spill, it was necessary to evaluate the flora den- sity in unaffected areas of the marsh to determine whether the plant density had changed after the oil spill. The researchers located several regions that had not been contaminated by the oil spill. The researchers needed to determine how many tracts would be required in order that their study yield viable conclusions. To determine how many tracts must be sampled, we would have to determine how accurately the researchers want to estimate the difference in the mean flora density in the spilled and unaffected regions. The researchers specified that they wanted the estimator of the difference in the two means to be within 8 units of the true difference in the means. That is, the researchers wanted to estimate the difference in mean flora den- sity with a 95 confidence interval having the form y Con ⫺ y Spill ⫾ 8. In previous studies on similar sites, the flora density ranged from 0 to 73 plants per tract. The number of tracts the researchers needed to sample in order to achieve their specifi- cations would involve the following calculations. We want a 95 confidence interval on m Con ⫺ m Spill with E ⫽ 8 and z a 2 ⫽ z .025 ⫽ 1.96. Our estimate of s is ⫽ range4 ⫽ 73 ⫺ 04 ⫽ 18.25. Substituting into the sample size formula, we have Thus, a random sample of 40 tracts should give a 95 confidence interval for m Con ⫺ m Spill with the desired tolerance of 8 plants provided 18.25 is a reasonable estimate of s. The spill region and the unaffected regions were divided into tracts of nearly the same size. From the above calculations, it was decided that 40 tracts from both the spill and unaffected areas would be used in the study. Forty tracts of exactly the same n ⫽ 2z a 兾2 2 s ˆ 2 E 2 ⫽ 21.96 2 18.25 2 8 2 ⫽ 39.98 ⬇ 40 s ˆ