Three firms carry inventories that differ in size. Firm A's inventory contains 2,000 items, firm B's inventory contains 5,000 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 σ = 121. 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. (Round your answers to two decimal places.) (b) What is the probability that for each firm the sample mean x will be within ±25 of the population mean μ? (Round your answers to four decimal places.)
Three firms carry inventories that differ in size. Firm A's inventory contains 2,000 items, firm B's inventory contains 5,000 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 σ = 121. 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. (Round your answers to two decimal places.) (b) What is the probability that for each firm the sample mean x will be within ±25 of the population mean μ? (Round your answers to four decimal places.)
Three firms carry inventories that differ in size. Firm A's inventory contains 2,000 items, firm B's inventory contains 5,000 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 σ = 121. 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. (Round your answers to two decimal places.) (b) What is the probability that for each firm the sample mean x will be within ±25 of the population mean μ? (Round your answers to four decimal places.)
Three firms carry inventories that differ in size. Firm A's inventory contains 2,000 items, firm B's inventory contains 5,000 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
σ = 121.
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. (Round your answers to two decimal places.)
(b)
What is the probability that for each firm the sample mean
x
will be within ±25 of the population mean μ? (Round your answers to four decimal places.)
Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...
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