Question

In Exercises 6.139 to \(6.142,\) use the normal distribution to find a confidence interval for a difference in proportions \(p_{1}-p_{2}\) given the relevant sample results. Give the best estimate for \(p_{1}-p_{2},\) the margin of error, and the confidence interval. Assume the results come from random samples. A \(90 \%\) confidence interval for \(p_{1}-p_{2}\) given that \(\hat{p}_{1}=0.20\) with \(n_{1}=50\) and \(\hat{p}_{2}=0.32\) with \(n_{2}=100\)

Short answer

The confidence interval for \( p_1 - p_2 \) is given by \( \hat{p}_1 - \hat{p}_2 \pm ME \), where \( \hat{p}_1 - \hat{p}_2 \) is the point estimate for the difference in proportions, and ME is the margin of error. This interval contains the range of plausible values for the true difference in population proportions based on the sample data.

Step-by-step solution

Calculate the point estimate

The point estimate for the difference in proportions is the difference of the sample proportions, or \( \hat{p}_1 - \hat{p}_2 \). Given that \( \hat{p}_1 = 0.20 \) and \( \hat{p}_2 = 0.32 \), we calculate: \( \hat{p}_1 - \hat{p}_2 = 0.20 - 0.32 = -0.12 \)

Compute the standard error

The standard error (SE) can be calculated using the formula: \( \sqrt{ \frac{{\hat{p}_1 * (1 - \hat{p}_1)}}{{n_1}} + \frac{{\hat{p}_2 * (1 - \hat{p}_2)}}{{n_2}} }\). Substituting in the given values, we get \(SE = \sqrt{ \frac{0.20 * 0.80}{50} + \frac{0.32 * 0.68}{100} }\)

Compute the margin of error

The margin of error (ME) is typically calculated as the product of the z-score representing the desired level of confidence and the standard error. For a 90% level of confidence, the z-score (from standard normal distribution tables) is approximately 1.645. Hence, \( ME = 1.645 * SE \).

Formulate the confidence interval

Finally, the confidence interval for the difference in population proportions can be found by adding and subtracting the margin of error from the point estimate: \( CI = \hat{p}_1 - \hat{p}_2 \pm ME \). Applying the found values gives the final result.

Key concepts

Point Estimate

Understanding the point estimate is key when analyzing differences between groups. It serves as the best singular value that represents the parameter of interest. In the context of proportions, as in our problem, the point estimate for the difference in proportions, noted as \( p_1 - p_2 \), is simply the difference between the estimated proportions from two independent samples. The calculation involved taking the proportions of successes from each group and subtracting them. For example, if Sample 1 has an estimated proportion of success of 20% \( (\hat{p}_1 = 0.20) \) and Sample 2 has 32% \( (\hat{p}_2 = 0.32) \) then the point estimate of the difference is \( -0.12 \) (0.20 - 0.32). This point estimate becomes the center value around which a confidence interval is built.

The point estimate is a vital part of hypothesis testing and confidence interval estimation. It provides a specific value to work with, which helps in making inference about the population parameters based on sample data.

Standard Error

The term standard error (SE) refers to a measure of the variability or precision of our point estimate. In simpler terms, it indicates how much our estimate from the sample is expected to vary from the true population parameter if we were to take many samples. It integrates both the sample size and the variability within the sample data.

For the difference between two proportions, the standard error helps to quantify the sampling variability by taking into account the size of each sample \( (n_1 \text{ and } n_2) \) and the estimated proportion of successes in each. Its calculation involves a bit of variability from both sample proportions, combined in a specific way. The formula used \( SE = \sqrt{ \frac{\hat{p}_1 * (1 - \hat{p}_1)}{n_1} + \frac{\hat{p}_2 * (1 - \hat{p}_2)}{n_2} } \) allows us to assess the precision of the point estimate \( \hat{p}_1 - \hat{p}_2 \). A larger standard error indicates less precision, while a smaller standard error suggests the point estimate is more likely to be close to the actual difference in proportions in the entire population.

Margin of Error

The margin of error (ME) represents a range above and below the point estimate wherein we expect the true population parameter to fall with a certain level of confidence. It's essentially an allowance for error in our estimate, taking into account the natural variability present in the sample data.

To construct a margin of error, we require two main inputs: the standard error, which we calculated previously, and a z-score associated with our desired confidence level. The z-score corresponds to the cutoff points on the normal distribution for the specified confidence interval. For a 90% confidence level, we typically refer to z-tables and identify the z-score (approximately 1.645 in our case). Multiplying this z-score by the standard error \( (ME = z * SE) \) produces the margin of error, which defines how

Z-Score

The z-score is a statistical critical value that corresponds to the desired confidence level in a standard normal distribution. It's a measure of how many standard deviations away from the mean a particular point is.

In practical terms, different z-scores are used for different confidence levels. For instance, a 90% confidence level typically uses a z-score of approximately 1.645. This signifies that we are looking at the point that is 1.645 standard deviations away from the mean of a standard normal distribution, leaving 5% in the tails (since it's a two-sided interval). These z-scores are the key to determining the margin of error for confidence intervals. By using the appropriate z-score, we adjust the width of the interval to correspond with our desired level of confidence, directly affecting our certainty about where the true parameter lies relative to our point estimate.

Normal Distribution

The normal distribution, often referred to as the bell curve, is a continuous probability distribution that is symmetrical and has its peak at the mean value. This distribution is crucial because many natural phenomena and statistical data tend to follow it, and it plays a central role in the central limit theorem.

For constructing confidence intervals for the difference in proportions, the normal distribution is used as an approximation provided certain criteria are met. These include having large enough sample sizes to ensure that the sampling distribution of the estimate \( \hat{p}_1 - \hat{p}_2 \) is nearly normal, which is the case when the original populations follow a binomial distribution.

Under these circumstances, we can use the properties of the normal distribution to make inferences about our population parameter with a certain level of confidence. The z-scores from this distribution help us to construct margins of error, which are then used to find the upper and lower bounds of our confidence intervals.