Question

Independent random samples were selected from two binomial populations, with sample sizes and the number of successes given in Exercises \(1-2 .\) Construct a \(95 \%\) and a \(99 \%\) confidence interval for the difference in the population proportions. What does the phrase "95\% confident" or "99\% confident" mean? $$\begin{array}{lcc}\hline & \multicolumn{2}{c} {\text { Population }} \\\\\cline { 2 - 3 } & 1 & 2 \\\\\hline \text { Sample Size } & 500 & 500 \\\\\text { Number of Successes } & 120 & 147 \\\\\hline\end{array}$$

Short answer

Question: Construct the 95% and 99% confidence intervals for the difference in population proportions of two independent random samples, given the following information: Population 1 has 500 samples with 120 successes; Population 2 has 500 samples with 147 successes.

Step-by-step solution

Find the sample proportions

For each population, divide the number of successes by the sample size: Population 1: \(p_1 = \frac{120}{500}\) Population 2: \(p_2 = \frac{147}{500}\)

Calculate the standard error

For the difference in proportions, use the formula: \(\sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}}\) Standard error(SE) \(= \sqrt{\frac{\frac{120}{500}(1-\frac{120}{500})}{500} + \frac{\frac{147}{500}(1-\frac{147}{500})}{500}}\)

Determine the critical values

Consult the standard normal distribution (often referred to as the z-table) to find the critical values \(z_{\frac{\alpha}{2}}\) for the two confidence levels. For the 95% confidence level: \(\alpha = 0.05\) and \(z_{\frac{\alpha}{2}} = 1.96\) For the 99% confidence level: \(\alpha = 0.01\) and \(z_{\frac{\alpha}{2}} = 2.576\)

Compute the confidence intervals

Use the formula: \([\text{difference in sample proportions} \pm z_{\frac{\alpha}{2}} \times \text{standard error}]\) 95% confidence interval: \([(\frac{120}{500} - \frac{147}{500}) - 1.96 \times SE, (\frac{120}{500} - \frac{147}{500}) + 1.96 \times SE]\) 99% confidence interval: \([(\frac{120}{500} - \frac{147}{500}) - 2.576 \times SE, (\frac{120}{500} - \frac{147}{500}) + 2.576 \times SE]\)

Interpret the confidence intervals

We are 95% (or 99%) confident that the true difference in population proportions falls within the calculated 95% (or 99%) confidence interval. If we were to conduct this sampling repeatedly and calculate the confidence interval each time, 95% (or 99%) of these intervals would contain the true population difference.

Key concepts

Binomial Population

In statistics, a binomial population is one where each member of the population has exactly two possible outcomes in a given trial. These outcomes are often labeled as "success" and "failure." For example, when flipping a coin, the potential outcomes are heads or tails.

Key characteristics of a binomial population include:

• A fixed number of trials
• Each trial is independent
• The probability of success is the same in every trial
• Each outcome is a binary (success or failure)

In the exercise, we are comparing two binomial populations based on two different samples. Each sample has been evaluated in terms of the number of successes (e.g., 120 and 147 successes), over a specified sample size (500 for each population). This information helps us to understand the behavior of the binomial populations in question.

Difference in Proportions

When we talk about the difference in proportions, we refer to comparing the proportions of successes between two groups. Let's say, for example, we have two different treatments and wish to see if one is more effective than the other.

In the exercise, the objective was to find the difference in sample proportions between two binomial populations. Proportion measures are simply calculated by dividing the number of successes in each group by its total number of samples.

To find the difference in proportions, you subtract the proportion of the second sample from the proportion of the first. This gives insight into whether there is a significant statistical difference between the two groups. For instance, this can indicate if one population has a higher success rate than the other and help form hypotheses or conclusions in research.

Sample Proportions

Sample proportions are an estimate of the true proportion in the overall population, derived from a subset or sample. It is calculated by dividing the number of successful outcomes by the total number of trials in the sample.

For example:

• Population 1: Sample size = 500, Number of successes = 120. Thus, the sample proportion, denoted by \( p_1 \), is \( p_1 = \frac{120}{500} \approx 0.24 \).
• Population 2: Sample size = 500, Number of successes = 147. Thus, the sample proportion, denoted by \( p_2 \), is \( p_2 = \frac{147}{500} \approx 0.294 \).

Sample proportions play a crucial role in statistical analysis as they provide an indication of what the true population may look like. They are the foundation upon which confidence intervals and other inferential statistics techniques are built.

Standard Error

The standard error is a measure that describes the variability or dispersion of a sample statistic across repeated samples from the same population. When we calculate the difference in proportions, the standard error helps us assess how much the sample proportion differs from the actual population proportion.

The formula for standard error of the difference in proportions is:
\[ \text{SE} = \sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}} \]
Here, \( p_1 \) and \( p_2 \) are the sample proportions, and \( n_1 \) and \( n_2 \) are the sample sizes. This formula captures the combined variability of both sample proportions.

Understanding the standard error is crucial because it assists in constructing confidence intervals. These intervals give an estimate range where we expect the true proportion difference to lie. The smaller the standard error, the more precise the confidence interval will be, reflecting greater accuracy and reliability in the estimates.