HOMEWORK – 4 CHAPTER – 8 1) Given the discrete uniform population f(x)= 1/3, X= 2,4,6 0, elsewhere find the probability that a random sample of size 54, selected with replacement, will yield a sample mean greater than 4.1 but less than 4.4. Assume the means are measured to the nearest tenth. 2) If a certain machine makes electrical resistors having a mean resistance of 40 ohms and a standard deviation of 2 ohms, what is the probability that a random sample of 36 of these resistors will have a combined resistance of more than 1458 ohms? 3) A random sample of size 25 is taken from a nor- mal population having a mean of 80 and a standard deviation of 5. A second random sample of size 36 is taken from a different normal population having a mean of 75 and a standard deviation of 3. Find the probability that the sample mean computed from the 25 measurements will exceed the sample mean computed from the 36 measurements by at least 3.4 but less than 5.9. Assume the difference of the means to be measured to the nearest tenth. 4) Two different box-filling machines are used to fill cereal boxes on an assembly line. The critical measure- ment influenced by these machines is the weight of the product in the boxes. Engineers are quite certain that the variance of the weight of product is σ2 = 1 ounce. Experiments are conducted using both machines with sample sizes of 36 each. The sample averages for ma- chines A and B are x Ā = 4.5 ounces and x B̄ = 4.7 ounces. Engineers are surprised that the two sample averages for the filling machines are so different. 1. (a) Use the Central Limit Theorem to determine P ( X ̄ B − X ̄ A ≥ 0 . 2 ) under the condition that μA = μB . 2. (b) Do the aforementioned experiments seem to, in any way, strongly support a conjecture that the popu- lation means for the two machines are different? Explain using your answer in (a). 5) Let X1,X2,…,Xn be a random sample from a distribution that can take on only positive values. Use the Central Limit Theorem to produce an argument that if n is sufficiently large, then Y = X1X2 · · · Xn has approximately a lognormal distribution. 6) For a chi-squared distribution, find χ2α such that (a) P(X2 >χ2α)=0.01 whenv=21; (b) P(X2
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