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A physical interpretation of the sample mean as a measure of location is shown in the dot diagram of the pull-off force data. See Figure 6-1. Notice that the sample mean x 13.0 can be thought of as a balance point. That is, if each observation represents 1 pound of mass placed at the point on the x-axis, a fulcrum located at x would exactly balance this system of weights. The sample mean is the average value of all the observations in the data set. Usually, these data are a sample of observations that have been selected from some larger population of observations. Here the population might consist of all the connectors that will be manufactured and sold to customers. Recall that this type of population is called a conceptual or

11-63. Show that, for the simple linear regression model, the following statements are true: (a) a 1 yi

(3.13)

11-64. An article in the IEEE Transactions on Instrumentation and Measurement ( Direct, Fast, and Accurate Measurement of VT and K of MOS Transistor Using VT-Sift Circuit, Vol. 40, 1991, pp. 951 955) described the use of a simple linear regression model to express drain current y (in milliamperes) as a function of ground-to-source voltage x (in volts). The data are as follows:

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y x 23.1 32.8 31.8 32.0 30.4 24.0 39.5 24.2 52.5 y 10.5 16.7 18.2 17.0 16.3 10.5 23.1 12.4 24.9 x 37.9 30.5 25.1 12.4 35.1 31.5 21.1 27.6 y 22.8 14.1 12.9 8.8 17.4 14.9 10.5 16.1 0.734 0.886 1.04 1.19 1.35

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x 1.1 1.2 1.3 1.4 1.5

Of course. thc initialization does not stop after the i = 4 step, but, a s in th: K:$ (xw. the chanco that (3.13) holds is h o i i t 0.05. Chsoquently. with a sufficient riiimbcr of I s of the forni (3.12), I rudy can detcrmiric K;1. Y Coiitiiiiiirig in this way, any nwribcr of key hytes call hc recovered, provided enough IVs of the correct form (about 60 for cacti key byto) arc availablr and I rudy knows tho first keystreani hytc of oach corresponding packet. This same tcchnique can h!extended to rccovcr additional key bytes, K s , Ktj,. . .. In fact, if a sufficient riiiinber of packets arc availablc, ti key of any lcrigth can t)c rccovered with a trivial amoiirit of work. This is one reason why WEP is said to be uiisaf< at any koy size [154]. Coiisider oiicc again the attilck to rewvor thr: first uiikrion.n key byte Ks. It is worth notirig that soine IVs that arc riot of the form (3, 255, V ) will be uscful bo Trudy. For examplc. supposc the I V is (2,253,O). Then after the i : initialization step. the array S is 3

y 1.50 1.66 1.81 1.97 2.12

x 1.6 1.7 1.8 1.9 2.0

0 1 0 2

(a) Estimate the correlation between Y and X. (b) Test the hypothesis that 0, using 0.05. (c) Fit a simple linear regression model and test for signi cance of regression using 0.05. What conclusions can you draw How is the test for signi cance of regression related to the test on in part (b) (d) Test the hypothesis H0: 0 0 versus H1: 0 0 and draw conclusions. Use 0.05. (e) Analyze the residuals and comment on model adequacy. 11-62. A random sample of n 25 observations was made on the time to failure of an electronic component and the temperature in the application environment in which the component was used. (a) Given that r 0.83, test the hypothesis that 0, using 0.05. What is the P-value for this test (b) Find a 95% con dence interval on .

(a) Draw a scatter diagram of these data. Does a straight-line relationship seem plausible (b) Fit a simple linear regression model to these data. (c) Test for signi cance of regression using 0.05. What is the P-value for this test (d) Find a 95% con dence interval estimate on the slope. (e) Test the hypothesis H0: 0 0 versus H1: 0 0 using 0.05. What conclusions can you draw 11-65. The strength of paper used in the manufacture of cardboard boxes ( y) is related to the percentage of hardwood concentration in the original pulp (x). Under controlled conditions, a pilot plant manufactures 16 samples, each from a different batch of pulp, and measures the tensile strength. The data are shown in the table that follows: (a) Fit a simple linear regression model to the data. (b) Test for signi cance of regression using 0.05. (c) Construct a 90% con dence interval on the slope 1. (d) Construct a 90% con dence interval on the intercept 0. (e) Construct a 95% con dence interval on the mean strength at x 2.5. (f) Analyze the residuals and comment on model adequacy.

y 0.31 4.20 4.88 3.48 7.58 2.63 4.99 0.59 8.19 4.79 0.51 1.74 4.10 3.94 0.96 3.29 0.44 3.24 2.14 5.71 4.12 1.90 0.51 8.33 14.94

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