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Numerade Educator

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Problem 43 Hard Difficulty

Three firms carry inventories that differ in size. Firm A's inventory contains 2000 items, firm $B^{\prime}$ s inventory contains 5000 items, and firm $C$ 's inventory contains $10,000$ items. The population standard deviation for the cost of the items in each firm's inventory is $\sigma=144$ . A statistical consultant recommends that each firm take a sample of 50 items from its inventory to provide statistically valid estimates of the average cost per item. Managers of the small firm state that because it has the smallest population, it should be able to make the estimate from a much smaller sample than that required by the larger firms. However, the consultant states that to obtain the same standard error and thus the same precision in the sample results, all firms should use the same sample size regardless of population size.
a. Using the finite population correction factor, compute the standard error for each of the three firms given a sample of size $50 .$
b. What is the probability that for each firm the sample mean $\overline{x}$ will be within $\pm 25$ of the population mean $\mu ?$

Answer

a. $20.1135,20.2646,20.3147$
b. $0.7850,0.7814,0.7814$

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Video Transcript

all right. So were given some data about three firms that have different inventories and different size inventories were taking simple random samples of size 50 to show the average cost for item, uh, and were given the standard deviation for each cost. So we're given that the inventory of from A has 2000 inventory for B s 5000 and inventory of terms. Uh, not term firm see has 10,000. All three of their standard deviations is $144 in terms of cost to make, and then we're taking sample sizes of fifties. So part eh wants us to find standard errors for the sampling distributions of each firm. So we could start with a, uh, using the formula for a finite population that's gonna be an A minus lower and a over capital and a minus one square. To that, multiply that by our standard deviation of a all over the square root, that's not a square root symbol. There we go. And a when you plug that end, that's gonna be 2000 minus 50 over 2000. Just one. Take the square root. Multiply that by 1 44 over the square root of 50. That equals 21 point aside, 20.1135 party. Not party firm. Be same thing except over a sub. Scripts are now bees. All right, plug this end. You get plug in everything from up here, you get that. The standard error is 20.2646 and then would see it's gonna be N C minus. Lower end. See over capital N C minus one, huh? Go over the entire thing. Multiply that by the standard error over the square root of little and see you get that. This equals 20 point 3147 All right. Part B wants us to find the probability for each of these that the sample mean is within 25 population means. So let's do that for each one things. First firm, eh? Working a Z score Negative. And positive. 25. This will be our lower Z score in our upper Z score. It's gonna be negative. 25 over. Standard error of a which it was negative. 25 over 20.1135 This is about negative. 1.24 Same thing over here. So 25 over 20.1135 So that's 1.24 Looking at what probability this is this is a lower probability. 0.0 Sorry. 0.1075 and then our upper probability. Looking at our table zero point 89 to 5, you find a probability of a girl's probability of a human. It's a probability of a l. What do you subtract that? That's 0.7850 All right, let's move this up a little bit because I need some space. Furby C B L on the z bu Right. So that's and 25 over center deviation sampling of B Uh, this is equal to negative 1.24 and then over here. Oh, sorry, not 1.24 I was looking at my notes for a That's 1.23 There we go than same thing down here. This becomes positive. 1.23 This becomes PBL of here equals 0.1093 and P v. U equals 0.8907 So our p B equals P v u rips PB you minus PBL which is when you subtract it out. 0.7814 Okay, Finally for firm see every Z score, it's gonna be a little squished, but that's okay, because it turns out that work for this is very simple. Because when you divide this out Oh, man, hold on. When you divide this out, this is negative. 25 over Sigma sub s C. And this equals negative 1.23 when you round it. And then this also 25 over, which is 1.23 These were the same values as firm. Be so same work has B you get that your PC will equal 0.7814 and there you have it.