groupactivity2solutions - Group Activity 2Solutions 1. (8...

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Unformatted text preview: Group Activity 2Solutions 1. (8 pts) Determine which probability distribution best describes each of the following scenarios: (Binomial/Poisson/Normal/Exponential/Uniform/None we have learnt) a. A manufacturer of light bulbs claims that 90% of its bulbs have a lifetime of 2000 hrs. or more. A consumer takes a sample of size 15 independently and with replacement. She records X= the number of light bulbs in the sample that have more than 2000 hrs. _________Binomial___________ b. Let Y = the number of arrivals at a supermarket in a two minute period. Ten people arrive every 2 minutes. _________Poisson_______ c. A manufacturer of light bulbs claims that 90% of its bulbs have a lifetime of 2000 hrs. or more. She records Z= the number of light bulbs selected until you get a light bulb with a lifetime of more than 2000 hrs. ______None_you have learnt________ d. Let W= the waiting time between arrivals of each person at the supermarket. __Exponential___________ 2. (5 pts) Which of the following is NOT an assumption of the Binomial distribution? (a) All trials must be identical. (b) All trials must be independent. (c) Each trial must be classified as a success or a failure. (d) The number of successes in the trials is counted. (e) The probability of success is equal to 0.5 in all trials. 3. (5 pts) In a triangle test a tester is presented with three food samples, two of which are alike, and is asked to pick out the odd one by testing. If a tester has no well developed sense and can pick the odd one only, by chance, what is the probability that in five trials he will make four or more correct decisions? Let X: number of correct decisions X~Binomial(5,1/3) P(X≥4)=11/243 4. (5 pts) Suppose 60% of a herd of cattle is infected with a particular disease. Let Y = the number of non-diseased cattle in a random sample of size 5. The distribution of Y is (a) binomial with n = 5 and p = 0.6 (b) binomial with n = 5 and p = 0.4 (c) binomial with n = 5 and p = 0.5 (d) the same as the distribution of X, the number of infected cattle. (e) Poisson with λ=0 .6 5. (5 pts) Suppose flaws (cracks, chips, specks, etc.) occur on the surface of glass with an average of 3 flaws per square metre. What is the probability of there being exactly 4 flaws on a sheet of glass of area 0.5 square metre? Let X: no. of flaws on 0.5 square metre X~Poisson(1.5) P(X=4)=0.047 6. (5 pts) Let X be the weight (in grams) of a miniature candy bar. Assume that E(X)=24.43 and Var(X)=2.2. What is the probability that the mean weight of a random sample of 31 candy bars lies between 24.17 grams and 24.82 grams? Since n≥30, we can apply CLT. By CLT, X ~N(24.43,2.2/31) P(24.17< X <24.82)= P( 24.17 24.43 24.82 24.43 Z ) 2.2 / 31 2.2 / 31 =P(Z<1.46)-P(Z<-0.98) =0.9278-0.1635=0.7643 7. ( 7 pts) Suppose life of a certain type of electronic component has an exponential distribution with a mean life of 500 hours. Let X denotes the life of the component. (a) What is the probability that that component lasts for at least 300 hours? P(X>300)= 300 500e 1 x /500 dx =exp(-300/500)=0.5488 (b) What is the probability that the component lasts for at least another 600 hours given that that it lasts 300 hours? P(X>900|X>300)=P(X>600+300|X>300)=P(X>600) (by memoryless property) =exp(-600/500)=0.3012 8. (10 pts) I drew random samples of size 25 from a normal distribution with mean 0 and variance 1. Following are the sample means for 15 samples, each of size 25: 0.045,-0.265,-0.089, 0.28,-0.288, 0.244, 0.453,-0.003,-0.278,-0.122,0.132,-0.374, 0.306, 0.000, -0.028. (a) Make a histogram of the above data. Consider class intervals of size 0.1 (e.g.,-0.40 to -0.30,0.30 to -0.20 etc) Comment on the shape of the distribution. (b) Calculate the mean and variance for the above data. Mean=0.0009 Variance=0.0607 (c) Based on the theory you have learnt in class what would be the mean of the sampling distribution of the sample mean. What would be the variance of the distribution? What would be name of the distribution of the sample mean? X follows a normal distribution with mean 0 and variance 1/25 ...
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This note was uploaded on 01/25/2011 for the course MATH 380 taught by Professor Staff during the Fall '08 term at UNL.

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