Question

Independent random samples were selected from two binomial populations, with sample sizes and the number of successes given in Exercises \(5-6 .\) Find the best point estimate for the difference in the population proportion of successes and calculate the margin of error: $$n_{1}=1250, n_{2}=1100, x_{1}=565, x_{2}=621$$

Short answer

The best point estimate for the difference in population proportions of successes is -0.113, and the margin of error at a 95% confidence level is approximately 0.0584.

Step-by-step solution

Calculate the sample proportions

To calculate the sample proportions, we will use the formula \(\widehat{p} = \dfrac{x}{n}\): For the first sample: $$\widehat{p_{1}} = \dfrac{x_{1}}{n_1} = \dfrac{565}{1250} = 0.452$$ For the second sample: $$\widehat{p_{2}} = \dfrac{x_{2}}{n_2} = \dfrac{621}{1100} = 0.565$$

Find the best point estimate for the difference in population proportions

Using the formula \(\widehat{p_{1}} - \widehat{p_{2}}\), we get: $$\widehat{p_{1}} - \widehat{p_{2}} = 0.452 - 0.565 = -0.113$$ The best point estimate for the difference in population proportions is -0.113.

Choose a confidence level and find the corresponding z-score

Suppose we choose a 95% confidence level, so the corresponding z-score can be found from the standard normal distribution table or using a calculator: $$z_{0.975} = 1.96$$

Calculate the margin of error

Now we will use the margin of error formula: $$MOE = z_{\alpha/2} \times \sqrt{\dfrac{\widehat{p_{1}}(1 - \widehat{p_{1}})}{n_1} + \dfrac{\widehat{p_{2}}(1 - \widehat{p_{2}})}{n_2}}$$ $$MOE = 1.96 \times \sqrt{\dfrac{0.452\times(1 - 0.452)}{1250} + \dfrac{0.565\times(1 - 0.565)}{1100}} = 1.96 \times \sqrt{0.00018411 + 0.00023694}$$ $$MOE = 1.96\times0.02983535 = 0.05841828$$ The margin of error is approximately 0.0584. So, the best point estimate for the difference in population proportions is -0.113, and the margin of error is 0.0584 at a 95% confidence level.

Key concepts

Binomial Distribution

The binomial distribution is a probability distribution that summarizes the likelihood that a value will take one of two independent states and is based on a series of identical and independent trials. For instance, suppose we are flipping a fair coin. Each flip represents a trial, and the coin can either land on heads (success) or tails (failure). If we flip the coin a fixed number of times, knowing the probability of getting a head, we can predict the distribution of the number of heads using the binomial distribution.

In the context of the textbook exercise, the binomial distribution applies to the number of successes in each sample (having or not having a particular attribute), with \( x_1 \) and \( x_2 \) representing the number of successes in the first and second sample respectively. The size of the sample \( n \) is fixed, and the probability of success—while unknown—is assumed to be the same across each trial within a sample.

Confidence Interval

A confidence interval is a range of values that is likely to contain a population parameter with a certain level of confidence. It is a tool used in statistics to express uncertainty in estimation. The confidence level represents the proportion of intervals that will enclose the true parameter if a large number of different samples is taken and interval calculations are performed on each one.

In simpler terms, if we were to collect 100 different samples and compute a 95% confidence interval for each sample, we would expect about 95 of the confidence intervals to contain the true population parameter. However, it's important to remember that a confidence interval does not guarantee the true value falls within it; instead, it reflects how confident we can be in our estimate, given the data collected and the method used to calculate the interval.

Point Estimate

A point estimate is a single value given as the estimate of a population parameter. It is the best guess based on sample data for what the true value of the parameter might be. For instance, if we're estimating the average height of a population, the average height from our sample is a point estimate of the population's average height.

In the example provided from the textbook, the point estimate is concerned with the difference in population proportions of successes (\( \widehat{p_{1}} - \widehat{p_{2}} \)). The calculated figure, -0.113, is the point estimate that suggests there is a 11.3% lower proportion of successes in the first population compared to the second one, based on the samples.

Margin of Error

The margin of error is a statistic that expresses the amount of random sampling error in a survey's results. It represents the radius of a confidence interval for a particular statistic. The margin of error is what allows us to quantify the notion of 'approximate': how close the point estimate is likely to be to the population parameter, taking into account the variability within our sample.

Calculating a margin error involves deciding on a confidence level (often 95%) and using a corresponding z-score—which represents the number of standard deviations a data-point is from the mean. The margin of error for the exercise \( MOE \) is calculated to be approximately 0.0584, which indicates that the true difference in population proportions would likely fall within ±5.84 percentage points of our point estimate if the same procedure were repeated on numerous samples.