Question

When we want \(95 \%\) confidence and use the conservative estimate of \(p=0.5,\) we can use the simple formula \(n=1 /(M E)^{2}\) to estimate the sample size needed for a given margin of error ME. In Exercises 6.40 to 6.43, use this formula to determine the sample size needed for the given margin of error. A margin of error of 0.04 .

Short answer

The sample size needed for a margin of error of 0.04 with 95% confidence level and an estimated p of 0.5 is roughly 625.

Step-by-step solution

Identifying the Knowns

From the problem, we can identify the known values: 1. The value for Margin of Error (ME) is 0.04. 2. Confidence level is 95%, however this value does not feature in the formula given, hence it's not crucial for solving the problem. 3. The estimated value of probability of success 'p' is 0.5 but again it's not involved directly in the formula provided in the problem.

Apply the formula

To find the sample size 'n', plug the Margin of Error (ME) value into the formula \(n = 1 / (ME)^{2}\). Thus, replace ME with 0.04 in the formula: \(n = 1 / (0.04)^{2}\).

Calculating Sample Size

Carry out the calculation to determine the sample size 'n'. Since \(0.04^{2}\) equals 0.0016, when you divide one by 0.0016, you get a result of roughly 625.

Rounding Off

Since the sample size cannot be a fraction, we round up to the next whole number, that is, 625.

Key concepts

Margin of Error

Understanding the margin of error (ME) is crucial for grasping the precision of survey or experiment results. It essentially indicates the range within which the true population parameter lies, based on the sample findings. For example, if a poll shows that 50% of people favor a candidate with a margin of error of 4%, we can be fairly certain that the actual percentage in the entire population is between 46% and 54%.

The smaller the margin of error, the closer we can expect the sample's estimate to reflect the true population value. However, achieving a smaller margin of error often requires a larger sample size. As shown in the textbook exercise, to calculate the needed sample size for a 0.04 margin of error, you'd use the formula \(n = 1 / (ME)^{2}\) to determine that you need approximately 625 people in your sample to attain that level of precision.

Confidence Level

The confidence level is a measure of how certain we can be that the population parameter (such as the mean or proportion) lies within the margins indicated by our sample statistics. Commonly used confidence levels are 90%, 95%, and 99%. A 95% confidence level, which is used in our textbook problem, means that if we were to take 100 different samples and compute 100 different confidence intervals, we would expect that approximately 95 out of the 100 confidence intervals would contain the true population parameter.

However, the confidence level does not directly impact the formula given in this particular case. This doesn't reduce its importance, though; it's essential for interpreting the margin of error within a probabilistic context. It implies that, under the assumption of a high confidence level, our margin of error enables us to make robust predictions about the entire population.

Probability of Success

In many research scenarios, the 'probability of success' refers to the likelihood of observing a particular outcome of interest. For instance, it could be the proportion of people who would vote for a certain candidate in an election. Typically denoted by \(p\) in statistics, this probability is important when determining the sample size for a study with a binary outcome (success or failure).

In the context of sample size calculation, the conservative estimate of \(p=0.5\) is often used because it maximizes the product of \(p(1-p)\), and thereby, according to the formula for sample size in a proportion scenario, yields the largest sample size required. This ensures that the sample is adequately sized to reflect the population, regardless of whether the actual probability of success is higher or lower than 0.5.