Question

We examine the effect of different inputs on determining the sample size needed. Find the sample size needed to give a margin of error within ±3 with \(99 \%\) confidence. With \(95 \%\) confidence. With \(90 \%\) confidence. Assume that we use \(\tilde{\sigma}=30\) as our estimate of the standard deviation in each case. Comment on the relationship between the sample size and the confidence level desired.

Short answer

The required sample sizes for 99%, 95%, and 90% confidence levels are 237, 139, and 101 respectively. There exists an inverse relationship between the confidence level and the required sample size.

Step-by-step solution

Calculate for 99% confidence level

Firstly, one must find the Z-value corresponding to a 99% confidence interval, which is 2.576. Substituting the values into the sample size formula results in \( n =\left(\frac{2.576*30}{3}\right)^2= 236.25\). However, since we cannot have a fraction of a sample, round up to the nearest whole number, giving a sample size of 237.

Calculate for 95% confidence level

Now, find the Z-value corresponding to a 95% confidence interval, which is 1.96. Substituting into the formula gives \( n = \left(\frac{1.96*30}{3}\right)^2 = 138.04 \). Again, since we cannot have a fraction of a sample, round up to the nearest whole number, giving a sample size of 139.

Calculate for 90% confidence level

Lastly, find the Z-value corresponding to a 90% confidence interval, which is 1.645. Substituting into the formula gives \( n = \left(\frac{1.645*30}{3}\right)^2 = 100.23 \). We round up to the nearest whole number, giving a sample size of 101.

Comment on relationship

As observed, as the confidence level decreases, the required sample size decreases as well. This implies an inverse relationship between confidence level and sample size.

Key concepts

Margin of Error

Understanding the margin of error is crucial when conducting surveys or studies. Essentially, it is an expression of the amount of random sampling error in a survey's results. Think of it as a measurement of the potential discrepancy between the sample's estimate of a population parameter and the actual population parameter. It is represented by the letter 'E' in many mathematical formulas.

For example, if you're seeing a margin of error of ±3, this means that the true population parameter is expected to lie within 3 units above or below the survey's reported value. In our exercise, the goal was to determine the sample size needed to achieve this specified margin of error, which requires other statistical components like the confidence level and the standard deviation to be factored in. The smaller the margin of error, the closer we can expect our sample's estimate to reflect the true population parameter, but this often requires a larger sample size.

Confidence Level

A confidence level signifies the degree of certainty with which we can expect the results from our sample to reflect the true population parameters. Expressed as a percentage, common confidence levels include 90%, 95%, and 99%.

Choosing a higher confidence level implies greater assurance in our results, but it also necessitates a larger sample size to maintain a constant margin of error. In practical terms, if you choose a 99% confidence level, as in the given exercise, you are saying that if the same population were sampled multiple times, 99 out of 100 sample results would fall within the margin of error of the true population parameter. The confidence level is directly tied to the Z-value, which adjusts the sample size calculation accordingly.

Standard Deviation

Standard deviation is a measure that quantifies the amount of variation or dispersion in a set of data values. A lower standard deviation indicates that the data points are closer to the mean or average value, while a higher standard deviation indicates a wider spread of values.

In the context of sample size determination, standard deviation (often estimated as \( \tilde{\sigma} \) in the absence of true population statistics) helps to capture the variability we expect within our target population. The greater the variability or standard deviation, the larger the sample size required to achieve a specific margin of error at a given confidence level. This relationship is pivotal in calculating the right sample size for reliable statistical inference.

Z-value

The Z-value, also known as the Z-score, is a numerical measurement that describes a value's relationship to the mean of a group of values, measured in terms of standard deviations. In the realm of determining sample size, the Z-value corresponds to the chosen confidence level and can be obtained from a standard normal distribution table or a Z-table.

As the Z-value increases, which happens with higher confidence levels, the sample size needs to be larger to maintain the same margin of error. In our exercise, varying Z-values for different confidence levels (99%, 95%, and 90%) were provided, and each critical Z-value resulted in different sample size requirements: the higher the Z-value, the larger the sample size that's needed. This showcases how the Z-value influences the sample size calculation to ensure the study's accuracy and reliability.