Question

Assume a binomial model for a certain random variable. If we desire a 90 percent confidence interval for \(p\) that is at most \(0.02\) in length, find \(n\). Hint: Note that \(\sqrt{(y / n)(1-y / n)} \leq \sqrt{\left(\frac{1}{2}\right)\left(1-\frac{1}{2}\right)}\).

Short answer

With the values substituted into the formula, we calculate \(n ≥ 4242.5625\). Since \(n\) cannot be a fraction, the sample size of the binomial model should be at least 4243.

Step-by-step solution

Define the Problem

We are trying to find the minimum sample size \(n\) where the 90% confidence interval for \(p\) is at most \(0.02\) in length.

Define the confidence interval formula

We know that the formula for the confidence interval at 90% confidence level (Z = 1.645 for 90% confidence level) is \(\[ p ± Z * \sqrt{\frac{p(1-p)}{n}}\]\). The length of this interval is \(2* Z * \sqrt{\frac{p(1-p)}{n}}\). We set this to be less than or equal to \(0.02\).

Apply inequality

As per the hint, we will use the inequality \(\sqrt{\frac{p(1-p)}{n}} ≤ \sqrt{\frac{1}{4n}}\). Substituting this into our inequality, we obtain \(2* Z * \sqrt{\frac{1}{4n}} ≤ 0.02\). Squaring both side to get rid off the square root gives us \((2 * Z)^2 * \frac{1}{4n} ≤ 0.02^2\).

Rearrange the equation to solve for n

Rearranging the equation \(n ≥ \frac{(2 * Z)^2}{4* 0.02^2}\) gives the minimum sample size needed to achieve a 90% confidence interval at most 0.02 in length.

Substitute the Z value into the formula

Finally, replacing the Z-value for the 90% confidence level (Z=1.645) into our formula, we get \(n ≥ \frac{(2 * 1.645)^2}{4* 0.02^2}\). Calculate this value to get the minimum sample size \(n\) required.

Key concepts

Sample Size Calculation

When conducting statistical analysis, determining the correct sample size is crucial. The sample size, often symbolized as \(n\), is the number of observations or data points to be collected. It plays a significant role in obtaining meaningful and accurate results. For a binomial model, the sample size calculation is particularly important, as it directly affects the confidence interval of the estimated probability \(p\).
The goal is to find the smallest sample size \(n\) that results in a confidence interval of a desired maximum length. In the current exercise, for example, we need a 90% confidence interval that is no longer than 0.02. Larger sample sizes typically provide more precise estimates, but also require more resources. Therefore, balancing precision with practicality is key.
To find \(n\) efficiently, we utilize statistical formulas and inequalities, often involving variables like the Z-score (a measure from the normal distribution) and the probability \(p\). Developing a clear understanding of these calculations helps you design statistically valid experiments that can inform decision-making or predict outcomes.

Confidence Interval Formula

The confidence interval formula is a key element in statistical analysis. It estimates the range of values within which a population parameter, like the probability \(p\) in a binomial distribution, is expected to lie.
In the context of a binomial model, the formula for a confidence interval becomes:

• \[ p \pm Z \times \sqrt{\frac{p(1-p)}{n}} \]

Here, \(p\) is the estimated probability, \(Z\) is the Z-score related to the desired confidence level (1.645 for a 90% confidence interval), and \(n\) is the sample size. The term \(\sqrt{\frac{p(1-p)}{n}}\) represents the standard error of the probability estimate.
This formula helps calculate the interval's length - essentially \(2 \times Z \times \sqrt{\frac{p(1-p)}{n}}\). Importantly, during calculations, such as the one in the exercise, we manipulate these elements to ensure the interval meets specific requirements like maximum length constraints.

Binomial Distribution Analysis

The binomial distribution is a fundamental concept in statistics used to model the number of successes in a fixed number of independent trials, each with the same probability of success \(p\). It applies to situations with two possible outcomes, often termed 'success' and 'failure'.
In a binomial distribution, the probability of obtaining exactly \(k\) successes in \(n\) trials is given by the binomial probability formula:

• \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]

where \(\binom{n}{k}\) is the binomial coefficient.
In analyzing binomial distributions, we often seek to understand the variability and reliability of our estimate for \(p\). Confidence intervals, like the one discussed earlier, provide insights into where the true probability \(p\) likely falls. These analyses enable us to make informed predictions about future trials and assess the uncertainty in our estimates, which is crucial for statistical inference and decision-making processes.