Question

Calculate the margin of error in estimating a binomial proportion \(p\) for the sample sizes given in Exercises \(11-14\). Use \(p=.5\) to calculate the standard error of the estimator, and comment on how an increased sample size affects the margin of error. \(n=1000\)

Short answer

How does increasing the sample size affect the margin of error? Answer: The margin of error (ME) for estimating a binomial proportion with a sample size of 1000 and p = 0.5 is 0.031. As the sample size increases, the margin of error decreases, providing us with a more accurate estimate of the true population proportion.

Step-by-step solution

Calculate the Standard Error (SE)

To calculate the SE for a binomial proportion, we use the formula: SE = \(\sqrt{\frac{p(1-p)}{n}}\) Here, \(p = 0.5\) and \(n = 1000\). Plugging these values into the formula, we get SE = \(\sqrt{\frac{0.5(1-0.5)}{1000}} = \sqrt{\frac{0.25}{1000}}\)

Calculate the Margin of Error

To find the margin of error (ME), we can use this formula: ME = \(Z * SE\) For a 95% confidence interval, the value of \(Z\) is 1.96. Using the calculated SE from Step 1, we get ME = \( 1.96 * \sqrt{\frac{0.25}{1000}}\)

Calculate the Margin of Error (ME)

Now, simply calculate the ME using the values: ME = \(1.96 * \sqrt{\frac{0.25}{1000}} = 1.96 * 0.0158 = 0.031\)

Comment on the effect of an increased sample size

The margin of error is affected by both the standard error and the confidence level (through the \(Z\) value). As the sample size (\(n\)) increases, the standard error becomes smaller, since it is inversely proportional to the square root of the sample size. This means that the margin of error will decrease as the sample size increases, providing us with a more accurate estimate of the true population proportion.

Key concepts

Binomial Proportion

When dealing with categorical data, especially in scenarios involving success or failure, we often come across the term 'binomial proportion'. This refers to the probability of a success in a binomial experiment. A classic example is flipping a coin, where the success could be landing on heads. In statistical terms, the binomial proportion, denoted as \( p \), is the ratio of the number of successful outcomes to the total number of trials.

Understanding binomial proportion is critical because it serves as a foundation for computing other statistics, such as the standard error and the margin of error. It provides us with insights into the 'typical' behavior of an observation, which is key when we are trying to make inferences about a larger population based on sample data. The value of \( p \) is essential when we're estimating the likelihood of an event or calculating confidence intervals around our estimates.

Standard Error

The standard error (SE) quantifies the variability or uncertainty in an estimate of a population parameter based on sample data. It essentially informs us about the precision of our sample mean as an estimate of the true population mean.

When you're working with a binomial proportion, the standard error can be calculated using the formula: \[ SE = \sqrt{\frac{p(1-p)}{n}} \]. Here, \(n\) represents the sample size, and \(p\) is the estimated probability of success in the population. For our example with \(p = 0.5\) and \(n = 1000\), the standard error helps gauge the extent to which our sample proportion might differ from the actual population proportion. A smaller SE indicates that our sample proportion is more likely to be close to the true population proportion.

Confidence Interval

A confidence interval offers a range within which we expect the true population parameter to fall, with a certain level of confidence. It takes into account the point estimate, like a sample mean or proportion, and adds and subtracts a margin of error to create a range.

The formula to calculate the margin of error for a confidence interval around a binomial proportion is: \[ ME = Z * SE \], where \(Z\) is the Z-score that corresponds to the desired confidence level (e.g., 1.96 for 95% confidence), and \(SE\) is the standard error. So, if you were to declare, 'I am 95% confident that the true population proportion falls within this interval,' you would be saying that if we were to take many samples and build confidence intervals in the same way, 95% of them would contain the true population proportion.

Sample Size

The sample size, \(n\), plays a pivotal role in the precision of our estimates from a statistical sample. An increase in sample size generally leads to a more accurate estimate of the population parameter and thus a smaller margin of error. Larger sample sizes reduce the standard error, which tightens the confidence interval, offering a clearer picture of where the true parameter lies.

Since most statistical measures, such as the margin of error, are inversely related to the square root of the sample size, increasing the number of observations significantly enhances the reliability of our conclusions. However, it's important to balance the benefits of a larger sample size with the costs and practicality of collecting data.