Sample Size Determination for Weight Estimation

1. (b) Determine the required sample size for estimating the true weight cereal container from a population of N = 5000.

Based on the past records variance of weight is 121 grams and this estimate should be within 2.5 grams of the true average weight with 99% confidence.

Will there be a change in the size of the sample if we assume population to be infinite? If so, explain by how much? (Table enclosed)

How to Approach

The question requires calculating the sample size for a finite population and then comparing it with the sample size for an infinite population. The approach involves using the appropriate sample size formulas for both scenarios, identifying the given parameters (population size, variance, margin of error, confidence level), and finding the corresponding Z-score. The finite population correction factor will be crucial for the first part. Finally, a clear explanation of the difference and its magnitude will be provided.

1. Determining Sample Size for a Finite Population

To determine the required sample size for estimating the true weight of cereal containers from a finite population, we use the formula that incorporates the finite population correction factor (FPC). Given parameters:

Population size (N) = 5000
Variance of weight ($\sigma^2$) = 121 grams
Standard Deviation ($\sigma$) = $\sqrt{121}$ = 11 grams
Estimate should be within (Margin of Error, E) = 2.5 grams
Confidence Level = 99%

Step 1: Find the Z-score for 99% Confidence Level

For a 99% confidence level, the alpha ($\alpha$) value is 1 - 0.99 = 0.01. We need to find the Z-score that corresponds to an area of $\frac{\alpha}{2}$ in each tail of the normal distribution. This means we look for the Z-score corresponding to a cumulative probability of $1 - \frac{0.01}{2} = 1 - 0.005 = 0.995$. From standard Z-tables, the Z-score for a 99% confidence level is approximately 2.576. (Note: Some tables might approximate it as 2.58).

Step 2: Calculate the Initial Sample Size (for infinite population assumption)

First, we calculate the sample size assuming an infinite population. This is often denoted as $n_0$. The formula for sample size when estimating a mean for an infinite population is: $n_0 = \frac{Z^2 \sigma^2}{E^2}$ Where:

Z = Z-score (2.576)
$\sigma$ = Standard Deviation (11 grams)
E = Margin of Error (2.5 grams)

Plugging in the values: $n_0 = \frac{(2.576)^2 \times (11)^2}{(2.5)^2}$ $n_0 = \frac{6.635776 \times 121}{6.25}$ $n_0 = \frac{8029.28896}{6.25}$ $n_0 \approx 1284.686$ Rounding up to the nearest whole number, the initial sample size $n_0 = 1285$.

Step 3: Apply the Finite Population Correction Factor

Since the population is finite (N = 5000), we need to adjust the initial sample size using the finite population correction (FPC) formula. The formula for sample size with finite population correction is: $n = \frac{n_0}{1 + \frac{n_0 - 1}{N}}$ Where:

$n_0$ = Initial sample size (1285)
N = Population size (5000)

Plugging in the values: $n = \frac{1285}{1 + \frac{1285 - 1}{5000}}$ $n = \frac{1285}{1 + \frac{1284}{5000}}$ $n = \frac{1285}{1 + 0.2568}$ $n = \frac{1285}{1.2568}$ $n \approx 1022.43$ Rounding up, the required sample size for the finite population is $n = 1023$.

2. Change in Sample Size if Population is Assumed to be Infinite

If we assume the population to be infinite, the sample size required would be the initial sample size calculated in Step 2, which is $n_0 = 1285$. Explanation of Change: Yes, there will be a change in the size of the sample if we assume the population to be infinite. When dealing with a finite population, especially when the sample size is a significant proportion (typically more than 5%) of the total population, the finite population correction factor (FPC) is applied. The FPC reduces the calculated sample size because, in a finite population, sampling without replacement means that the probability of selecting subsequent items changes, and the variability in the sample estimates is actually lower than what would be assumed in an infinite population. Essentially, knowing more about a larger portion of the population reduces the uncertainty. Calculation of the Difference: Difference in sample size = Sample size (Infinite Population) - Sample size (Finite Population) Difference = $n_0 - n$ Difference = $1285 - 1023$ Difference = $262$ The sample size decreases by 262 units when accounting for the finite population of 5000 cereal containers, compared to assuming an infinite population. This reduction reflects the increased precision gained by sampling from a limited pool, where each selected item provides relatively more information about the overall population.

The calculations are summarized in the table below:

Parameter	Value
Population Size (N)	5000
Variance ($\sigma^2$)	121 grams
Standard Deviation ($\sigma$)	11 grams
Margin of Error (E)	2.5 grams
Confidence Level	99%
Z-score (Z)	2.576
Initial Sample Size ($n_0$, infinite population)	1285
Adjusted Sample Size (n, finite population)	1023
Change in Sample Size	262 (decrease)

Additional Resources

Key Definitions

Sample Size

The number of observations or replicates included in a sample, which is a subset of a population. It is a crucial statistical concept that determines the representativeness and reliability of inferences drawn about the entire population.

Finite Population Correction Factor (FPC)

A statistical adjustment applied to the standard error of a statistic (and consequently to sample size calculations) when sampling without replacement from a finite population. It is typically used when the sample size exceeds 5% of the population size, reducing the standard error as the sample provides a significant proportion of the total population information.

Key Statistics

A common practice in industry is to ensure quality control processes achieve a 99.73% confidence level (corresponding to +/- 3 standard deviations, or '3-sigma'), especially in manufacturing sectors like pharmaceuticals and food production, to minimize defects and ensure product consistency.

Source: General Quality Control and Six Sigma Methodologies

While ideal, achieving a 99% confidence level in surveys and polls often requires larger sample sizes, which can increase costs and logistical challenges. For instance, national political polls often aim for a 95% confidence level with a margin of error around +/- 3-4% with sample sizes typically between 1,000 to 1,500 respondents.

Source: Pew Research Center, Gallup

Examples

Pharmaceutical Quality Control

In pharmaceutical manufacturing, strict quality control demands high precision. For a batch of 10,000 tablets, a sample size calculation similar to the cereal container example would be performed to determine how many tablets to test to ensure the active ingredient dosage is within a specified margin of error with a high confidence level, like 99.9%. This minimizes the risk of releasing ineffective or harmful medication.

Environmental Sampling for Pollution

When assessing pollution levels in a lake with a known number of distinct sections (a finite population), environmental scientists use sample size calculations to determine how many water samples to take from different sections. This ensures that the estimated average pollutant concentration is accurate within a certain margin and confidence, without having to test every single liter of water.

Frequently Asked Questions

▶Why does the sample size decrease when considering a finite population?

The sample size decreases for a finite population because when you sample without replacement from a limited population, the probability of selecting subsequent items changes, and each item sampled provides relatively more information about the remaining population. This reduces the uncertainty and thus allows for a smaller sample to achieve the same level of precision and confidence compared to an infinite population where sampling an item doesn't significantly impact the overall pool.

▶What is the impact of a higher confidence level on sample size?

A higher confidence level (e.g., 99% vs. 95%) requires a larger sample size, assuming all other factors (variance, margin of error) remain constant. This is because a higher confidence level demands a wider interval to be more certain that the true population parameter falls within the estimated range, which necessitates collecting more data to reduce the sampling error.