Finite Population Correction Factor In Exercises 57 and 58, use the information below.
In this section, you studied the construction of a confidence interval to estimate a population mean. In each case, the underlying assumption was that the sample size n was small in comparison to the population size N. When n ≥ 0.05N however, the formula that determines the standard error of the mean needs to be adjusted, as shown below.
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Recall from the Section 5.4 exercises that the expression sqrt[(N-n)/(n-1)] is called a finite population correction factor. The margin of error is
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Use the finite population correction factor to construct each confidence interval for the population mean.
a. c = 0.99, xbar = 8.6, σ = 4.9, N = 200, n = 25.

Verified step by step guidance

1. Step 1: Identify the given values from the problem. Here, the confidence level c = 0.99, sample mean x̄ = 8.6, population standard deviation σ = 4.9, population size N = 200, and sample size n = 25.
2. Step 2: Check if the finite population correction factor is needed. The condition for using the correction factor is n ≥ 0.05N. Calculate 0.05N = 0.05 × 200 = 10. Since n = 25, which is greater than 10, the finite population correction factor is required.
3. Step 3: Compute the finite population correction factor using the formula: sqrt((N - n) / (N - 1)). Substitute the values N = 200 and n = 25 into the formula: sqrt((200 - 25) / (200 - 1)).
4. Step 4: Adjust the standard error of the mean using the finite population correction factor. The formula for the adjusted standard error is: SE = (σ / sqrt(n)) × sqrt((N - n) / (N - 1)). Substitute the values σ = 4.9, n = 25, N = 200, and the correction factor computed in Step 3.
5. Step 5: Use the adjusted standard error to calculate the margin of error (E) for the confidence interval. The formula for the margin of error is: E = z * SE, where z is the critical value corresponding to the confidence level c = 0.99. Look up the z-value for a 99% confidence level (z ≈ 2.576) and substitute it along with the adjusted SE from Step 4. Finally, construct the confidence interval as: (x̄ - E, x̄ + E).

Key Concepts

Finite Population Correction Factor (FPC)

The Finite Population Correction Factor (FPC) is a statistical adjustment used when sampling without replacement from a finite population. It accounts for the fact that the sample size is a significant fraction of the total population, which can lead to a smaller standard error than what would be estimated using the standard formula. The FPC is calculated as sqrt[(N-n)/(N-1)], where N is the population size and n is the sample size.

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter with a specified level of confidence, typically expressed as a percentage (e.g., 95% or 99%). It provides an estimate of uncertainty around the sample mean and is calculated using the sample mean, the standard error, and a critical value from the relevant statistical distribution. The wider the interval, the more uncertainty there is about the population parameter.

Standard Error of the Mean (SEM)

The Standard Error of the Mean (SEM) quantifies the amount of variability in sample means that you would expect if you were to take multiple samples from the same population. It is calculated as the standard deviation of the sample divided by the square root of the sample size (σ/√n). When the sample size is large relative to the population size, the SEM must be adjusted using the FPC to provide a more accurate estimate of the population mean's variability.