Probability and Statistics for Engineering and the Sciences (with Student Suite Online)

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Unformatted text preview: IEng/Math 381 Probability & Statistics I Name: jDL-‘J‘WDHS Summer 2004 Exam IV RWJ Instructions: 0 Please read questions carefully 0 Show work/reasoning for @artial) credit 0 Conduct all hypothesis tests at significance level a = 0.05. 0 Note point values on questions 0 Make sure your exam has 4 questions PLEASANT FoLk, THOSE STATI- STlaANss Yes, EVEN we MEAN STA'I'ISTICIAN is QUITE Nice. l. /28 2. /28 3. /28 4. /16 / 100 1. (28 pts.) Industries such as refineries, steel mills, and food processing plants release wastewater into the environment. This wastewater should be neutralized before it is released. Treated wastewater pH values are recorded for a sample from an industry using a particular treatment process‘: 6.2, 6.5, 7.6, 7.7, 7.0, 7.0, 7.2, 6.8, 7.5, 8.1, 7.1, 7.0, 7.1, 7.8, 8.5 (There are 15 observations with a sample average of 7.273 and a sample standard deviation of 0.602.) Based upon these data, is there evidence that the treatment process does not yield an average pH of 7 as desired? Conduct the appropriate test by answering the questions below: a. State the null and alternative hypotheses for this test ' Ho: M37 Ha= A167 where ,u is the true mean treated wastewater pH. b. Draw the density curve the test statistic follows if the null hypothesis is true. Correctly add, along the hoiizontal axis, the labels "Accept Ho" and "Reject Ho" regions. Label the horizontal axis with the numerical value(s) that separate the accept and reject regions. _ ¢L g 0015 - W" ’ “ 43H- pnéh’ 0" - '0” a... q I t W .. .H MJW'M 5 fl-é-‘J'ecr Ho . . . . N‘flo’r /4 c. Compute the test statistic(1t’s either a t or a z). b {2:- f“ 2 “B”? 9/75 ' 5/“; a. (DZ/fl; (1. Should the null hypothesis or rejected? ‘ Gray, David and Marshall, Jeff (1992), “How to choose a pH measurement system”, Pollution Engineering, pp. 45- 47. e. Give a 95% confidence interval for p. “4:24 5” :73?) 52.115 Y 4 v: VT; 2‘- (6.4%, '14: ) f. What must we assume about the population of treated wastewater pH reading values . . . m i. . . . in the hypothesis test conducted above (if anything)? , 45 Fal/w; Jam- mar-m! [n < 4‘) ii. . . . in the confidence interval consu'ucted above (if anything)? Fa//pwJ Jam'w chaff») 2. (28 pts) The package of GE 60 Watt light bulbs I have at home claims they last 1250 hours (on average). Suppose I take a sample of 50 light bulbs and they last an average of 1230 hours with standard deviation of 80 hours. I Is the true mean duration of the bulbs less than 1250 hours? Conduct the appropriate test by answering the questions below: a. State the null and alternative hypotheses for this test . Ho: Ag/be Ha: M < {7.50 where p is the true mean lifetime. , b. Draw the density curve the test statistic follows if the null hypothesis is true. Correctly add, along the horizontal axis, the labels "Accept Ho" and "Reject Ho" regions. Label the horizontal axis with the numerical value(s) that separate the accept and reject regions. 05 ., . 1 M59, d réhbm‘e L W W flEs‘ticr 14, ’l‘ 61'5 flcaew'r H, c. Compute the test statistic (it’s either a t or a z). ._ “2 ~ 2: Z/u‘: l23c> l60¢__/.77 . 54/; 8041?; (1. Should the null hypothesis be accepted or e. Give a 95% confidence interval for y. 2' 9° ~ ’ 74; 136 E = (23° 4.4.74 if: =Clw‘7.£,/252.z) D f. What must we assume about the population of light bulb lifetimes . . . i. . . . in the hypothesis test conducted above (if anything)? ' 4; Napalm, C h z 4») ii. . . . in the confidence interval constructed above (if anything)? “.5 WWW ("2 7°) 3. (28 pts.) Devor, Chang, and Sutherland (1992) examined a process for manufacturing electrical resistors2 that have a nominal resistance of 100 ohms with a specification of i20hms. Suppose management has expressed a concern that the true proportion of resistors with resistances outside the specifications has increased from the historical level of 10%. A random sample of 180 resistors yielded 46 with resistances outside the specifications. Conduct the appropriate test by answering the questions below: a. State the null and alternative hypotheses for this test H 0 Z P 2 g (0 H a : p 7 , ( o in terms of p, the proportion of resistors that don’t meet the specification. b. Draw the density curve the test statistic follows if the null hypothesis is true. Correctly add, along the horizontal axis, the labels "Accept Ho" and "RejectHo" regions. Label the horizontal axis with the numerical value(s) that separate the accept and reject regions. . 1 An£fl=d=.05 / I (I I W, Aafip’i‘ Hg WC 1. Ha c. Compute the appropriate test statistic. d. Should the null hypothesis be accepted oEeJected?) 2 Devor, K, Chang, T., and Sutherland, J. (1992), Statistical Quality Design and Control: Contemporary Concepts and Methods, Macmillan, New York. e. Give an approximate 95% confidence interval for p. « ?: 1.7: 190—?) ; (new?) h f. What must we assume about the population of resistor resistance values . . . ’- .——. i. . . . in the hypothesis test conducted above (if anything)? 45 ND’W’H" (a a 3D) 11. . . . in the confidence interval constructed above (if anything)? a: woman (n2 3») 4. (16 pts.) Method of Moments a. If a particle moves in a plane so that its horizontal and vertical distances from the origin are each, independently, normal with mean zero with the same variance 9’, then the Rayleigh density is the density of the particle’s distance from the origin. The Rayleigh density, given by 0 x<0 has mean u = J; 9. Suppose we observe 20 distances from the origin for such a particle with an average of 24.1 1. Determine the method of moments estimate of 19; * faraw- a ‘ f fl 2*le ~ if "" J15 ‘1-1533 ' b. Suppose runners in a race are wearing numbers (“bibs”) running from 1 through N, with N unknown. For such a population of numbers it can be readily shown that ,u = You watch this race seeing just / the following numbers: 126,33,213, 172,289 E ._.' (16 4- m4. 17:; Give the method of moments estimate of the number of runners, N. 5 = 456,; ...
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