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The relative conductivity of a semiconductor device is determined by the amount of impurity "doped" into the device during its manufacture. A silicon diode to be used for a specific purpose requires an average cut-on voltage of .60 V, and if this is not achieved, the amount of impurity must be adjusted. A sample of diodes was selected and the cut-on voltage was determined. The accompanying SAS output resulted from a request to test the appropriate hypotheses. N Mean Std Dev T Prob. > T 15 0.0453333 0.0899100 1.9527887 0 . 0711 [Note: SAS explicitly tests Ho: u = 0, so to test H: u = .60, the null value .60 must be subtracted from each x ; the reported mean is then the average of the (x; - .60) values. Also, SAS's P-value is always for a two-tailed test.] What would be con- cluded for a significance level of .01? .05? .10? [Note: SAS explicitly tests Ho : p = 0, so to test Ho : p = .60, the null value .60 must be subtracted from each xi; the reported mean is then the average of the (x - .60)values. Also, SAS's P-value is always for a two-tailed test.] What would be concluded for a significance level of .01? .05? .10?The National Health Statistics Reports dated Oct. 22, 2003, stated that for a sample size of 2?? 13yearold Ameri can males, the sample mean waist circumference was 85.3 cm. A. somewhat complicated method was used to estimate various population percentiles, resulting in the following values: 5lil It)" 25" 50:Ill T5\" 90" 95" 69.6 'l'iflslall 735.2 31.3 95.4 lll 1 16.4 a. Is it plausible that the waist size distribution is at least approximately normal?\" Explain your reasoning. lfyour answer is no, conjecture the shape of the population distribution. b. Suppose that the population mean waist size is 85 crnand that the population standard deviation is 15 cm. How likely is it that a random sample of 2?? individuals will result in a sample mean waist size of at least 86.3 cm? c. Referring back to {b}, suppose now that the population mean waist size in 82 cm. Now what is the {approximate} probability that the sample mean will be at least 85.3 cm? In light of this calculation, do you think that 82 cm is a reasonable value for p'? A binary communication channel transmits a sequence of "bits" (0s and 1s). Suppose that for any particular bit transmitted, there is a 10% chance of a transmission error (a 0 becoming a 1 or a 1 becoming a 0). Assume that bit errors occur independently of one another. a. Consider transmitting 1000 bits. What is the approximate probability that at most 125 transmission errors occur? b. Suppose the same 1000-bit message is sent two different times independently of one another. What is the approximate probability that the number of errors in the first transmission is within 50 of the number of errors in the second?Consider a small ferry that can accommodate cars and buses. The toll for cars is $3, and the toll for buses is $10. Let X and Y denote the number of cars and buses, respectively, carried on a single trip. Suppose the joint distribution of X and Yis as given in the table of Exercise 7. Compute the expected revenue from a single trip. Reference Exercise 7 The joint probability distribution of the number X of cars and the number Y of buses per signal cycle at a proposed left-turn lane is displayed in the accompanying joint probability table. p(x, y) 0 .025 .015 010 .050 .030 020 .125 .075 050 .150 .090 .060 .100 .060 040 .050 .030 .020 a. What is the probability that there is exactly one car and exactly one bus during a cycle? b. What is the probability that there is at most one car and at most one bus during a cycle? c. What is the probability that there is exactly one car during a cycle? Exactly one bus? d. Suppose the left-turn lane is to have a capacity of five cars, and that one bus is equivalent to three cars. What is the probability of an overflow during a cycle? e. Are X and Y independent rv's? Explain.In Exercise 66, the weight of the beam itself contributes to the bending moment. Assume that the beam is of uniform thickness and density so that the resulting load is uniformly distributed on the beam. If the weight of the beam is random, the resulting load from the weight is also random; denote this load by W (kip-ft). a. If the beam is 12 ft long, Whas mean 1.5 and standard deviation .25, and the fixed loads are as described in part (a) of Exercise 66, what are the expected value and variance of the bending moment? [Hint: If the load due to the beam were w kip-ft, the contribution to the bending moment would be w xdx.] ] b. If all three variables (X1, X2, and W) are normally distributed, what is the probability that the bending moment will be at most200 kip-ft? Reference Exercise 66 If two loads are applied to a cantilever beam as shown in the accompanying drawing, the bending moment at 0 due to the loads is a, X, + a,X2. (0, 1)- (x, 1 - x) a. Suppose that X1 and X2 are independent rv's with means 2 and 4 kips, respectively, and standard deviations .5 and 1.0 kip, respectively. If a, = 5 ft and a2 _ 10 ft, what is the expected bending moment and what is the standard deviation of the bending moment? b. If X1 and X2 are normally distributed, what is the probability that the bending moment will exceed 75 kip-ft? c. Suppose the positions of the two loads are random variables. Denoting them by A, and A2, assume that these variables have means of 5 and 10 ft, respectively, that each has a standard deviation of .5, and that all Aj's and Xi's are independent of one another. What is the expected moment now? d. For the situation of part (c), what is the variance of the bending moment? e. If the situation is as described in part (a) except that Corr(X,, X2) = .5 (so that the two loads are not independent), what is the variance of the bending moment?According to the article "Fatigue Testing of Condoms" (Polymer Testing, 2009: 567-571), "tests currently used for condoms are surrogates for the challenges they face in use," including a test for holes, an inflation test, a package seal test, and tests of dimensions and lubricant quality (all fertile territory for the use of statistical methodology!). The investigators developed a new test that adds cyclic strain to a level well below breakage and determines the number of cycles to break. A sample of 20 condoms of one particular type resulted in a sample mean number of 1584 and a sample standard deviation of 607. Calculate and interpret a confidence interval at the 99% confidence level for the true average number of cycles to break. [Note: The article presented the results of hypothesis tests based on the t distribution; the validity of these depends on assuming normal population distributions.]Two different companies have applied to provide cable television service in a certain region. Let p denote the proportion of all potential subscribers who favor the first company over the second. Consider testing H.: p = .5 versus H,: p # .5 based on a random sample of 25 individuals. Let the test statistic X be the number in the sample who favor the first company and x represent the observed value of X. a. Describe type I and II errors in the context of this problem situation. b. Suppose that x = 6. Which values of X are at least as contradictory to H. as this one? c. What is the probability distribution of the test statistic X when H, is true? Use it to compute the P-value when x = 6. d. If Ho is to be rejected when P-values .044, compute the probability of a type II error when p = 4, again when p = .3, and also when p = .6 and p = .7. [Hint: P-value > .044 is equivalent to what inequalities involving x (see Example 8.4)?] e. Using the test procedure of (d), what would you conclude if 6 of the 25 queried favored company 1
