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? $$\begin{array}{lcc}\hline & \text { Population } \\\\\cline { 2 - 3 } & 1 & 2 \\\\\hline \text { Sample Size } & 800 & 640 \\\\\text { Number of Successes } & 337 & 374 \\\\\hline\end{array}$$

Short answer

Answer: The 95% confidence interval for the difference in population proportions is (-0.230243, -0.096007), and the 99% confidence interval is (-0.251511, -0.074739).

Step-by-step solution

Given Information

We have the following information: Population 1: Sample Size (n1) = 800, Number of Successes (x1) = 337 Population 2: Sample Size (n2) = 640, Number of Successes (x2) = 374

Calculate sample proportions and their difference

First, we need to calculate the proportions for each population: Population 1: Proportion (p1) = x1 / n1 = 337 / 800 = 0.42125 Population 2: Proportion (p2) = x2 / n2 = 374 / 640 = 0.584375 Now calculate the difference in sample proportions: Difference (p1 - p2) = 0.42125 - 0.584375 = -0.163125

Calculate standard error

Next, we'll calculate the standard error for the difference in the proportions: Standard Error (SE) = sqrt{ [(p1 * (1 - p1)) / n1] + [(p2 * (1 - p2)) / n2] } SE = sqrt{ [(0.42125 * 0.57875) / 800] + [(0.584375 * 0.415625) / 640] } = 0.03415

Calculate critical values and confidence intervals

For 95% confidence interval, the critical value (z1) is 1.96. For 99% confidence interval, the critical value (z2) is 2.576. Now, we will calculate the confidence intervals: 95% Confidence Interval: Lower Limit (LL1) = (p1 - p2) - z1 * SE = -0.163125 - 1.96 * 0.03415 = -0.230243 Upper Limit (UL1) = (p1 - p2) + z1 * SE = -0.163125 + 1.96 * 0.03415 = -0.096007 99% Confidence Interval: Lower Limit (LL2) = (p1 - p2) - z2 * SE = -0.163125 - 2.576 * 0.03415 = -0.251511 Upper Limit (UL2) = (p1 - p2) + z2 * SE = -0.163125 + 2.576 * 0.03415 = -0.074739

Interpret the confidence intervals

The phrase "95% confident" means that the confidence interval we've calculated has a 95% probability of containing the true difference in population proportions. Similarly, "99% confident" means there is a 99% probability that the true difference lies within the computed interval. 95% Confidence Interval: (-0.230243, -0.096007) This means that we are 95% confident that the true difference in population proportions lies between -0.230243 and -0.096007. 99% Confidence Interval: (-0.251511, -0.074739) This means that we are 99% confident that the true difference in population proportions lies between -0.251511 and -0.074739.

Key concepts

Population Proportions

The concept of population proportion is central when dealing with binomial populations. A population proportion (p ext{Population}) represents the fraction of individuals in a population who have a particular characteristic or attribute. In the given exercise, the populations are the two different groups, and we're interested in the proportion of "successes" within each respective group. Let's explore how this concept applies with the given data.

For Population 1, we have a sample size of 800, with 337 successes. This allows us to estimate the proportion of successes as \( p_1 = \frac{x_1}{n_1} = \frac{337}{800} = 0.42125 \). For Population 2, using a similar computation, we find \( p_2 = \frac{x_2}{n_2} = \frac{374}{640} = 0.584375 \). Notice that these proportions are point estimates of the true population proportions. In statistics, we often want to estimate how these proportions compare between different populations, which leads us to the difference in proportions.

The difference in sample proportions, calculated as \( p_1 - p_2 \), is simply \( 0.42125 - 0.584375 = -0.163125 \). This negative value indicates that the proportion of successes in Population 1 is smaller than in Population 2.

Standard Error

The standard error (SE) helps us understand the variability or spread of the sample proportion estimates that we calculate from a population. When we want to find how precise the estimated difference between two population proportions is, SE plays a crucial role.

The formula for calculating the standard error of the difference between two proportions is given by:

• \[ \text{SE} = \sqrt{ \left(\frac{p_1 (1 - p_1)}{n_1}\right) + \left(\frac{p_2 (1 - p_2)}{n_2}\right) } \]

In the exercise, we substitute our calculated proportions and sample sizes. Calculating this, we obtain:

• \[ \text{SE} = \sqrt{ \left(\frac{0.42125 \times 0.57875}{800}\right) + \left(\frac{0.584375 \times 0.415625}{640}\right) } = 0.03415 \]

The smaller the standard error, the more confidently we can say that our sample proportion difference is a good estimate of the true population difference. This is because a small SE indicates low variability between sample statistics and thus tighter confidence bounds when constructing confidence intervals.

Critical Values

Critical values are a statistical concept that helps determine how far a sample statistic can deviate, before being too unlikely, from the expected value within a specified confidence level. They rely heavily on the standard normal distribution (also known as the Z-distribution).

When constructing a confidence interval, you choose a confidence level which reflects how certain you are that your interval contains the true parameter. This exercise uses 95% and 99% confidence levels. For these intervals, we apply critical values (z-scores) based on standard normal distribution rules.

- **For a 95% confidence interval**, the critical value, \( z_{1} \), is 1.96. This means that 95% of the data falls within 1.96 standard deviations of the mean in a normal distribution.- **For a 99% confidence interval**, the critical value, \( z_{2} \), is 2.576. As the interval is larger, it reflects a higher level of confidence that the interval captures the true population parameter.

These values allow us to calculate the range of the confidence interval as:

• **95% Confidence Interval:** Lower Limit = \( (p_1 - p_2) - z_{1} \times \text{SE} \); Upper Limit = \( (p_1 - p_2) + z_{1} \times \text{SE} \)
• **99% Confidence Interval:** Lower Limit = \( (p_1 - p_2) - z_{2} \times \text{SE} \); Upper Limit = \( (p_1 - p_2) + z_{2} \times \text{SE} \)

The resulting intervals inform us about the likely range of the difference between the population proportions, taking into account the chosen level of confidence.