Question

Use the information given to find the necessary confidence interval for the binomial proportion \(p .\) Interpret the interval that you have constructed. A \(95 \%\) confidence interval for \(p,\) based on a random sample of 500 trials of a binomial experiment which produced 27 successes.

Short answer

Answer: The 95% confidence interval for the binomial proportion p is [0.0359, 0.0721].

Step-by-step solution

Calculate the sample proportion

The sample proportion is denoted as \(\hat{p}\) and is calculated by dividing the number of successes by the total number of trials. In this problem, the number of successes is 27, and the number of trials is 500. Thus: $$\hat{p} = \frac{27}{500} = 0.054$$

Calculate the critical value (z-score) for a 95% confidence interval

We need to find the z-score that corresponds to a 95% confidence interval. This value can be found using a standard normal distribution table or a calculator with a function for finding z-scores. The z-score for a 95% confidence interval is 1.96.

Calculate the standard error

The standard error for a binomial proportion is calculated as follows: $$SE(\hat{p}) = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$ Where \(n\) is the number of trials. Plug in the sample proportion and the number of trials: $$SE(\hat{p}) = \sqrt{\frac{0.054(1-0.054)}{500}} = 0.00923$$

Calculate the margin of error

To calculate the margin of error, multiply the standard error by the z-score: $$ME = 1.96 \times 0.00923 = 0.0181$$

Construct the confidence interval

Now that we have the margin of error, we can construct the confidence interval using the following formula: $$CI = \hat{p} \pm ME$$ Plug in the sample proportion and margin of error: $$CI = 0.054 \pm 0.0181$$ Which results in the following confidence interval: $$CI = [0.0359, 0.0721]$$

Interpret the confidence interval

We can interpret the confidence interval as follows: we are 95% confident that the true proportion of successes in the population (p) lies between 3.59% and 7.21%.

Key concepts

Sample Proportion

Understanding the sample proportion is crucial when analyzing statistical data from experiments. The sample proportion, denoted as \(\hat{p}\), represents the ratio of the number of successful outcomes to the total number of trials in a study or an experiment.

For instance, if an experiment involves flipping a coin 100 times and the coin lands on heads 55 times, the sample proportion of heads is \(\hat{p} = \frac{55}{100} = 0.55\). This value serves as an estimate for the true proportion of the entire population. As sample sizes increase, the sample proportion tends to give a more accurate estimate of the population proportion.

Critical Value (Z-score)

In statistical analysis, the critical value, often referred to as the z-score, plays a critical role in determining the reliability of an estimate. The z-score is a numerical measurement that describes a value's relationship to the mean of a group of values, measured in terms of standard deviations.

For a given confidence interval, the z-score signifies the number of standard deviations a data point is from the mean. In a standard normal distribution, common critical values correspond to confidence levels: for example, a 95% confidence interval typically has a z-score of approximately 1.96. This means that if the data is normally distributed, 95% of the time, the true proportion would lie within 1.96 standard deviations of the sample proportion.

Standard Error

The standard error is a measure that represents how far the sample statistic, like the sample proportion, might be from the actual population parameter it estimates. It’s essentially the standard deviation of the sampling distribution of that statistic.

The standard error for a binomial proportion can be calculated using the formula \(SE(\hat{p}) = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\), where \(\hat{p}\) stands for the sample proportion and \(n\) for the number of trials. This gives us an estimate of the variability or the precision of the sample proportion in relation to the true population proportion. A smaller standard error indicates a more precise estimate.

Margin of Error

When reporting the results of a poll or study, the margin of error helps us to understand the range within which the true population parameter likely falls. The margin of error is calculated by multiplying the standard error by the z-score and reflects the maximum expected difference between the sample statistic and the population parameter due to sampling variability.

As the margin of error increases, the confidence interval widens, indicating less precision in our estimate. Conversely, a smaller margin of error suggests that the sample statistic is a more precise estimate of the population parameter. It is a crucial component in reporting the results as it allows for a cushion of error in the estimation.

Binomial Experiment

A binomial experiment is a statistical experiment that satisfies four conditions: there must be a fixed number of trials, each trial must be independent of others, there can only be two outcomes for each trial (usually categorized as 'success' or 'failure'), and the probability of success must be the same for each trial.

Examples of binomial experiments include flipping a coin multiple times, where each flip is independent and has two possible outcomes—heads or tails—with the probability of heads remaining constant at each flip. Binomial experiments provide a foundational framework for understanding probability and are frequently analyzed using the concepts of sample proportion, standard error, and confidence intervals.