Three firms carry inventories that differ in size. Firm A's inventory contains 2000 items, firm B'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 = 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.
Using the finite population correction factor, compute the standard error for each of the three firms given a sample of size 50 (to 2 decimals).
SEm = SD/√n
The formula to compute the standard error for each firm, using the finite population correction factor, is:
Standard Error = (Population Standard Deviation) / sqrt(Sample Size) * sqrt((Population Size - Sample Size) / (Population Size - 1))
Let's calculate the standard error for each firm.
For Firm A:
Population Size = 2000
Sample Size = 50
Population Standard Deviation = 144
Standard Error for Firm A = 144 / sqrt(50) * sqrt((2000 - 50) / (2000 - 1))
= 144 / sqrt(50) * sqrt(1950 / 1999)
≈ 20.29
For Firm B:
Population Size = 5000
Sample Size = 50
Population Standard Deviation = 144
Standard Error for Firm B = 144 / sqrt(50) * sqrt((5000 - 50) / (5000 - 1))
= 144 / sqrt(50) * sqrt(4950 / 4999)
≈ 19.78
For Firm C:
Population Size = 10000
Sample Size = 50
Population Standard Deviation = 144
Standard Error for Firm C = 144 / sqrt(50) * sqrt((10000 - 50) / (10000 - 1))
= 144 / sqrt(50) * sqrt(9950 / 9999)
≈ 19.75
Therefore, the standard errors for Firm A, Firm B, and Firm C are approximately 20.29, 19.78, and 19.75 respectively.
To compute the standard error for each firm, we need to apply the finite population correction factor. The formula for the standard error is:
Standard Error = Population Standard Deviation / sqrt(Sample Size) * sqrt((Population Size - Sample Size) / (Population Size - 1))
Let's calculate the standard error for each firm:
For Firm A:
Population Size (N) = 2000
Population Standard Deviation (σ) = 144
Sample Size (n) = 50
Standard Error = 144 / sqrt(50) * sqrt((2000 - 50) / (2000 - 1))
Standard Error = 144 / 7.07 * sqrt(1950 / 1999)
Standard Error ≈ 144 / 7.07 * 0.9864
Standard Error ≈ 20.58 (rounded to 2 decimals)
For Firm B:
Population Size (N) = 5000
Population Standard Deviation (σ) = 144
Sample Size (n) = 50
Standard Error = 144 / sqrt(50) * sqrt((5000 - 50) / (5000 - 1))
Standard Error = 144 / 7.07 * sqrt(4950 / 4999)
Standard Error ≈ 144 / 7.07 * 0.9934
Standard Error ≈ 20.66 (rounded to 2 decimals)
For Firm C:
Population Size (N) = 10000
Population Standard Deviation (σ) = 144
Sample Size (n) = 50
Standard Error = 144 / sqrt(50) * sqrt((10000 - 50) / (10000 - 1))
Standard Error = 144 / 7.07 * sqrt(9950 / 9999)
Standard Error ≈ 144 / 7.07 * 0.9975
Standard Error ≈ 20.79 (rounded to 2 decimals)
Therefore, the standard errors for Firm A, Firm B, and Firm C given a sample size of 50 are approximately 20.58, 20.66, and 20.79, respectively.