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(c) What is the type II error probability of this test if 4.5 15-17. Suppose that we take a sample of n 10 measurements from a normal distribution with 1. We wish to test 0 against H1: 0. The normal test statistic is H0: Z0 X ( 1n), and we decide to use a critical region of 1.96 (that is, reject H0 if z0 1.96). (a) What is for this test (b) What is for this test, if 1 (c) If a sign test is used, specify the critical region that gives an value consistent with for the normal test. (d) What is the value for the sign test, if 1 Compare this with the result obtained in part (b). 15-18. Consider the test statistic for the sign test in Exercise 15-9. Find the P-value for this statistic. 15-19. Consider the test statistic for the sign test in Exercise 15-10. Find the P-value for this statistic. Compare it to the P-value for the normal approximation test statistic computed in Exercise 15-11.

15-3

= Ox8bf37fa2, = Ox63247e24,

The sign test makes use only of the plus and minus signs of the differences between the observations and the median 0 (or the plus and minus signs of the differences between the observations in the paired case). It does not take into account the size or magnitude of these differences. Frank Wilcoxon devised a test procedure that uses both direction (sign) and magnitude. This procedure, now called the Wilcoxon signed-rank test, is discussed and illustrated in this section. The Wilcoxon signed-rank test applies to the case of symmetric continuous distributions. Under these assumptions, the mean equals the median, and we can use this procedure to test the null hypothesis that m 5 m0. We now show how to do this.

15-3.1

Negative or left skew (a)

= 0~1089fc26, = Oxld630daf,

x x Symmetric (b)

Positive or right skew (c)

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When a scene is too dark, the G11 s autofocus system cannot see enough detail to work properly. When these situations arise, the Red-Eye Reduction lamp turns on to

X s = 0x4e4f430a1 = 0x43415254, X7 = Ox410aOa54, = 0x68742074, Xg = 0x72702065, = 0x20656369, Xi1 = 0 ~ 2 4 2 0 6 6 6 f ,

distribution is skewed to the right, whereas mode median mean if the distribution is skewed to the left. Frequency distributions and histograms can also be used with qualitative or categorical data. In some applications there will be a natural ordering of the categories (such as freshman, sophomore, junior, and senior), whereas in others the order of the categories will be arbitrary (such as male and female). When using categorical data, the bins should have equal width. EXAMPLE 6-6 Figure 6-12 presents the production of transport aircraft by the Boeing Company in 1985. Notice that the 737 was the most popular model, followed by the 757, 747, 767, and 707. A chart of occurrences by category (in which the categories are ordered by the number of occurrences) is sometimes referred to as a Pareto chart. See Exercise 6-41. In this section we have concentrated on descriptive methods for the situation in which each observation in a data set is a single number or belongs to one category. In many cases, we work with data in which each observation consists of several measurements. For example, in a gasoline mileage study, each observation might consist of a measurement of miles per gallon, the size of the engine in the vehicle, engine horsepower, vehicle weight, and vehicle length. This is an example of multivariate data. In later chapters, we will discuss analyzing this type of data. EXERCISES FOR SECTION 6-4

6-30. Construct a frequency distribution and histogram for the motor fuel octane data from Exercise 6-14. Use eight bins. 6-31. Construct a frequency distribution and histogram using the failure data from Exercise 6-15. 6-32. Construct a frequency distribution and histogram for the cotton content data in Exercise 6-16. 6-33. Construct a frequency distribution and histogram for the yield data in Exercise 6-17.

X = 0 ~ 2 ~ 3 6 3 7 3 1 , i 3 = 0x20353934, Xi5 = 0 ~ 7 7 6 f 6 ~ 4 2 . = 0~20666~41,

Figure 6-12 Airplane production in 1985. (Source: Boeing Company.)

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