Question

Calculate the margin of error in estimating a binomial proportion \(p\) using samples of size \(n=100\) and the values of p given in Exercises \(15-19 .\) What value of p produces the largest margin of error? \(p=.1\)

Short answer

However, it is known that for estimating binomial proportions, the largest margin of error occurs when \(p = 0.5\). This is because the function \(\sqrt{p(1-p)}\) is maximized when \(p = 0.5\). Therefore, the value of \(p\) that produces the largest margin of error is \(p = 0.5\).

Step-by-step solution

Choose a confidence level and find the z-value

Since the exercise doesn't provide a specific confidence level, let's choose a 95% confidence level which is commonly used. For a 95% confidence level, the z-value is approximately 1.96.

Calculate the margin of error using the formula

We have \(n = 100\), \(p = 0.1\), and \(z = 1.96\). Now, we can plug these values into the formula: \(E = z \sqrt{\frac{p(1-p)}{n}}\) \(E = 1.96 \sqrt{\frac{0.1(1-0.1)}{100}}\)

Simplify the expression inside the square root

Simplify the expression inside the square root: \(E = 1.96 \sqrt{\frac{0.1(0.9)}{100}}\) \(E = 1.96 \sqrt{\frac{0.09}{100}}\)

Calculate the margin of error

Now, find the value of the margin of error: \(E = 1.96 \sqrt{\frac{0.09}{100}} = 1.96 \times 0.03\) \(E = 0.0588\) The margin of error for estimating the binomial proportion \(p = 0.1\) using a sample size of \(n = 100\) is approximately 0.0588. The question also asks what value of \(p\) produces the largest margin of error. Unfortunately, we only have one value of \(p\) provided in the question, so we cannot directly compare it to other values of \(p\).

Key concepts

Probability Sampling Size

Understanding the relationship between a sample size and the accuracy of the estimate of a binomial proportion is fundamental in statistics. When deciding the size of a sample for probability sampling, it is important to consider the precision required for the study, often reflected in the margin of error. The sample size, denoted as n, significantly impacts the margin of error: larger sample sizes generally lead to a smaller margin of error, indicating a more precise estimate of the population proportion.

In the context of the provided exercise, a sample size of n = 100 was used to estimate the proportion p. This number is often adequate for getting a reasonable estimate of the proportion. However, in situations where an even more accurate estimate is desired, or the population is highly varied, increasing the sample size would reduce the margin of error, resulting in a more confident estimate of the true population proportion.

Estimating Binomial Proportion

Estimating a binomial proportion involves determining the probable success rate (denoted as p) within a population based on a sample. The classical formula for the margin of error in estimating a binomial proportion p reflects how sure we can be that the sample proportion is close to the true population proportion.

The formula utilized in the exercise is:
\[E = z \sqrt{\frac{p(1-p)}{n}}\]
This equation incorporates the sample proportion p, the sample size n, and the z-value corresponding to the chosen confidence level. It is evident from the formula that different values of p will affect the margin of error, with the margin being largest when the proportion is near 0.5, where the product of p and (1-p) is maximized. In cases where p is closer to 0 or 1, the margin of error will be smaller, assuming the sample size and confidence level remain constant.

Confidence Level and Z-Value

The confidence level shows how certain we are that the actual population parameter lies within the interval estimate provided by the margin of error. A typical confidence level choice is 95%, which is considered a standard balance between precision and certainty. To determine the final interval estimate, statisticians use the z-value, which correlates to the confidence level chosen.

For example, a z-value of 1.96 corresponds to a 95% confidence level as used in the given solution. This z-value means that we can be 95% confident that the population proportion falls within 1.96 standard deviations of the sample proportion. The selection of the z-value is based on the standard normal distribution and is a crucial element of constructing a confidence interval for estimating a binomial proportion.