Question

Suppose you wish to estimate a binomial parameter \(p\) correct to within .04, with probability equal to .95. If you suspect that \(p\) is equal to some value between 1 and .3 and you want to be certain that your sample is large enough, how large should \(n\) be? (HINT: When calculating the standard error, use the value of \(p\) in the interval \(.1

Short answer

Answer: Approximately 730.

Step-by-step solution

Determine the standard error

The standard error of a binomial proportion is given by the formula \(\textit{SE} = \sqrt{\frac{p(1-p)}{n}}\). We will need to find the value of \(p\) that maximizes the standard error within the range (0.1, 0.3). Since the denominator is \(n\), the standard error will be maximized when the numerator \(p(1-p)\) is maximized. Let's find the maximum value of \(p(1-p)\) using calculus.

Find the maximum value of the function \(p(1-p)\)

To find the critical points of this function, we will find its derivative and set that equal to 0. First, let's find the derivative of the function \(f(p) = p(1-p)\): \(f'(p) = \frac{d}{dp}(p(1-p))\) This derivative can be found using the product rule: \(f'(p) = (1)(1-p) + (p)(-1)\) So, \(f'(p) = 1 - 2p\) Now, let's set \(f'(p)\) equal to 0 and solve for \(p\): \(1 - 2p = 0\) \(2p = 1\) \(p = 0.5\) However, since this value of \(p\) is outside of the given range for \(p\), we will instead consider the endpoints of the range (0.1, 0.3). The value that will give the largest standard error is \(p=0.3\).

Calculate the standard error

Now that we have found the value of \(p\) that maximizes the standard error, we can find the standard error value. To determine the sample size \(n\), we will use the Z-score for a 95% probability, which is 1.96. \(\textit{SE} = \frac{1.96}{0.04}\) \(\textit{SE} = 49\)

Solve for the sample size \(n\)

Now that we have found the standard error value, we can solve for \(n\) using the standard error formula: \(\textit{SE} = \sqrt{\frac{p(1-p)}{n}}\) Substituting the values of \(p\) and \(\textit{SE}\): \(49 = \sqrt{\frac{0.3(1-0.3)}{n}}\) Squaring both sides: \(2401 = \frac{0.3(0.7)}{n}\) Now, solve for \(n\): \(n = \frac{0.3(0.7)}{2401}\) \(n \approx 730\) Therefore, the sample size should be approximately 730 to estimate the binomial parameter \(p\) with a precision of 0.04 and a probability of 0.95.

Key concepts

Standard Error Calculation

When dealing with binomial proportions, it's crucial to understand what the standard error represents. The standard error (SE) measures the variability or uncertainty in the estimated proportion, given a specific sample size. It provides insight into how much the sample proportion is expected to fluctuate around the true population proportion.

The formula for the standard error in the context of a binomial proportion is:

• SE = \( \sqrt{\frac{p(1-p)}{n}} \)

Here, \( p \) represents the estimated proportion of success, and \( n \) is the sample size. This formula allows us to understand the amount of expected error in proportion estimates.

In our original exercise, the task was to maximize the standard error by finding a value of \( p \) that provided the largest possible numerator, \( p(1-p) \), within a given range. By using calculus, we determined that this maximal condition occurs when \( p = 0.5 \). However, since this value was not within our range, we used \( p = 0.3 \) for our calculations, as it provided the largest value within the spectrum of .1 to .3.

Binomial Proportion

Binomial proportion is the probability of success in a series of Bernoulli trials, where each trial has only two possible outcomes such as success or failure.

To estimate it accurately, we define a binomial experiment by its two elements:

• The number of trials, denoted as \( n \)
• The probability of success in each trial, denoted as \( p \)

In practice, the binomial proportion is often estimated using sample data by dividing the number of successes by the total number of trials.

To ensure statistical validity, it's important to understand that as \( n \) increases, the sample proportion \( \hat{p} \) becomes an increasingly accurate estimator of the true population proportion \( p \). This occurs because larger samples tend to better reflect the overall population characteristics.

Confidence Interval

A confidence interval is a range of values that is likely to contain the true value of an unknown population parameter. In statistics, the confidence interval provides an estimated range that's calculated from the sample data and is associated with a specific level of confidence, such as 95%.

To calculate the confidence interval for a binomial proportion, we rely upon the point estimate (usually the sample proportion) and the standard error. The confidence interval is determined by taking the point estimate and adding and subtracting a margin that includes the uncertainty quantified by the standard error.

This margin is multiplied by a factor called the 'critical value.' For a 95% confidence interval, this factor is typically 1.96, assuming a normal distribution.

• Formula for 95% CI: \( \hat{p} \pm 1.96 \times \text{SE} \)

In the original exercise, achieving this confidence involved finding the correct sample size, \( n \), so that this interval was narrow enough to meet specified precision requirements, such as within 0.04. This effectively means that we are 95% confident that our estimated interval contains the true value of the binomial proportion.