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. \(n=30\)

Short answer

Question: Based on the given value of p = 0.5 and a sample size of n = 30, calculate the margin of error for estimating a binomial proportion, and analyze the effect of increased sample size on the margin of error. Answer: With p = 0.5 and n = 30, the margin of error for estimating a binomial proportion is approximately 0.179. As the sample size increases, the margin of error decreases, providing a more precise estimation of the population proportion. This is because the margin of error is inversely related to the square root of the sample size.

Step-by-step solution

Calculate the Standard Error of the Estimator

In this case, we are given the value of \(p = 0.5\). The standard error of the estimator can be calculated using the formula: \(SE(p) = \sqrt{\frac{p(1-p)}{n}}\) Plug in the given values, and solve for SE(p).

Compute the Margin of Error

Once the standard error has been calculated, we can compute the margin of error using the formula: \(ME = 1.96 * SE(p)\) This equation multiplies the standard error by the z-value (1.96) which corresponds to a 95% confidence interval. Calculate the margin of error using the standard error from Step 1.

Analyze the Effect of Increased Sample Size on the Margin of Error

After computing the margin of error for the given sample size \(n = 30\), consider how an increased sample size would affect the value of the margin of error. Remember that the formula for the standard error contains the term \(\frac{1}{\sqrt{n}}\), which means that as the sample size \(n\) increases, the overall standard error decreases. This, in turn, will result in a decrease in the margin of error. Provide a brief discussion on the relationship between sample size and margin of error. Now we can go ahead and solve each step with the given \(n=30\).

Calculate the Standard Error of the Estimator (Solution)

Given, \(p = 0.5\) and \(n = 30\). \(SE(p) = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.5(1-0.5)}{30}} = \sqrt{\frac{0.25}{30}} \approx 0.0913\)

Compute the Margin of Error (Solution)

Using the standard error calculated in Step 1: \(ME = 1.96 * SE(p) = 1.96 * 0.0913 \approx 0.179\)

Analyze the Effect of Increased Sample Size on the Margin of Error (Solution)

As we can see from the formulas, as the sample size \(n\) increases, the standard error and hence the margin of error decrease. This is because a larger sample size provides more information about the population, making the estimation more precise. The margin of error is inversely related to the square root of the sample size, which means that if we want to decrease the margin of error, we need to increase the sample size. An increased sample size generally leads to better confidence in our estimation, assuming the sample is unbiased and representative of the population.

Key concepts

Binomial Proportion Estimation

Understanding the process of estimating a binomial proportion is crucial for comprehending margin of error and its implications. A binomial proportion refers to the probability of success, denoted by the symbol \(p\), in a Binomial distribution scenario where there are only two possible outcomes, typically referred to as 'success' and 'failure'.

In practical terms, if we conduct a survey to determine the percentage of people who prefer coffee over tea, the proportion of coffee lovers (\(p\)) in our sample would be an estimate of the overall coffee preference in the population. Estimating this proportion accurately requires a sample that is both random and representative of the whole population.

The formula \(SE(p) = \sqrt{\frac{p(1-p)}{n}}\) gives the standard error of this binomial proportion estimate, indicating the expected deviation from the true population proportion. Keep in mind that the value of \(p\) is often unknown and might be estimated as 0.5 to maximize the standard error when the true proportion is unclear, which results in a more conservative estimate of the margin of error.

Standard Error of Estimator

The standard error of an estimator reflects the level of uncertainty associated with the estimate of a population parameter. Specifically, for a proportion (\(p\)), the standard error measures how much the proportion estimated from different samples is expected to fluctuate around the true population proportion.

It is calculated with the formula \(SE(p) = \sqrt{\frac{p(1-p)}{n}}\), where \(n\) is the sample size. This formula tells us that the standard error is dependent both on the sample size and the estimated proportion. A smaller standard error implies a more reliable estimate of the population parameter.

In our exercise, using a 95% confidence interval requires multiplying the standard error by 1.96, which is the z-value indicating that the true binomial proportion is expected to lie within this margin of error 95% of the time. Thus, the smaller the standard error, the narrower the confidence interval, and the more precise our estimate becomes.

Sample Size Effects

The size of the sample is a significant factor in determining the accuracy of statistical estimates. As the sample size increases, the standard error decreases, which subsequently reduces the margin of error. This is clearly highlighted in the formula for standard error, \(SE(p) = \sqrt{\frac{p(1-p)}{n}}\), where the denominator includes the square root of the sample size (\(n\)).

A larger sample size thus impacts the margin of error by shrinking it, which allows for a more precise estimate of the population parameter. However, it is essential to recognize the law of diminishing returns in this context: doubling the sample size does not halve the margin of error but rather reduces it by a factor of \(\sqrt{2}\).

In summary, increasing the sample size is a powerful way to improve the reliability of your estimate, but as sample size grows, each addition has a smaller effect on reducing the margin of error. This balance between the cost of data collection and the level of precision required for an estimate is an important consideration in research design and statistical analysis.