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: $$n_{1}=60, n_{2}=60, x_{1}=43, x_{2}=36$$

Short answer

Question: Calculate the best point estimate for the difference in the population proportion of successes and the margin of error. Given two independent random samples with sample sizes (n₁, n₂) and the number of successes (x₁, x₂): - Sample 1: n₁ = 60, x₁ = 43 - Sample 2: n₂ = 60, x₂ = 36 Answer: The best point estimate for the difference in the population proportion of successes is: $$\hat{p}_1 - \hat{p}_2 = \frac{43}{60} - \frac{36}{60} = \frac{7}{60}$$ The margin of error, assuming a 95% confidence level, is: $$ME = 1.96 * SE$$ where SE is the standard error calculated in Step 3.

Step-by-step solution

Calculate sample proportions

To find the sample proportions of successes for both samples, we need to divide the number of successes by the sample size for each sample. The sample proportions are given by the formula: $$\hat{p}_1 = \frac{x_1}{n_1}, \hat{p}_2 = \frac{x_2}{n_2}$$ Using the given values, we can calculate the sample proportions: $$\hat{p}_1 = \frac{43}{60}, \hat{p}_2 = \frac{36}{60}$$

Calculate the point estimate

Now we have the sample proportions; we can find the best point estimate for the difference in population proportions using the following formula: $$\hat{p}_1 - \hat{p}_2$$ Substituting the values we found in Step 1, we get: $$\frac{43}{60} - \frac{36}{60}$$

Calculate the standard error

To find the margin of error, we first need to calculate the standard error of the difference in proportions. The formula for the standard error is: $$SE = \sqrt{\frac{\hat{p}_1 ( 1 - \hat{p}_1)}{n_1} + \frac{\hat{p}_2 ( 1 - \hat{p}_2 )}{n_2}}$$ Substituting the values we found in Step 1, we have: $$SE = \sqrt{\frac{\frac{43}{60}( 1 - \frac{43}{60})}{60} + \frac{\frac{36}{60}( 1 - \frac{36}{60})}{60}}$$

Calculate the margin of error

Now that we have the standard error, we can find the margin of error using the following formula: $$ME = Z * SE$$ In this problem, we are not given a specific confidence level, so we will assume a 95% confidence level (which is the most common). For a 95% confidence level, the corresponding \(Z\) score is approximately 1.96. Using the standard error we calculated in Step 3, we have: $$ME = 1.96 * SE$$

Finalize the result

Now we have everything we need to provide the final answer for this problem. The best point estimate for the difference in the population proportion of successes is: $$\hat{p}_1 - \hat{p}_2 = \frac{43}{60} - \frac{36}{60}$$ And the margin of error is: $$ME = 1.96 * SE$$ With these two values, we can report the best point estimate for the difference in population proportions and the margin of error.

Key concepts

Point Estimate

Understanding the concept of a point estimate is foundational when handling statistical analysis, particularly when dealing with population proportions. A point estimate represents the single best guess for a population parameter, like the mean or proportion, based on sample data. In the context of our exercise, the point estimate concerns the difference in population proportions. Here's how it plays out practically:

To arrive at a point estimate for the difference in population proportions, one simply subtracts the sample proportion of one group from the other as shown in the equation \( \hat{p}_1 - \hat{p}_2 \). This calculation provides an approximation for the true difference in population proportions, based solely on our sample data.

• In our problem, the calculation yields a point estimate of \( \frac{43}{60} - \frac{36}{60} \).
• This result is essentially the most probable estimate of the actual difference between the two populations, as per our sample.

Margin of Error

Whenever we make estimations based on sample data, there's always a level of uncertainty. This is where the margin of error (ME) becomes critical. The margin of error quantifies the range within which the true population parameter is expected to fall, with a certain degree of confidence. Greater margins of error suggest less precision, while smaller margins indicate more precise estimates.

To calculate ME, we multiply the standard error (SE) by a Z-score that corresponds to our desired confidence level. Typically, for a 95% confidence level, a Z-score of approximately 1.96 is used. If we're aiming for a 99% confidence level, the Z-score would be approximately 2.58.

• The formula is \( ME = Z * SE \), where Z is the Z-score for the chosen confidence level.
• In our example, with a given Z-score of 1.96, the ME will be calculated, reflecting our confidence that the true population parameter will be within this range from our point estimate.

Sample Proportions

Sample proportions play a vital role in estimating features of an entire population from a subset of that population. For each group in a study, the sample proportion is the total number of 'successes' divided by the total number of observations in the sample (\( \hat{p} = \frac{x}{n} \)).

'Success' usually means the characteristic, outcome, or behavior of interest in the study. In polling, for example, a 'success' might mean a respondent favoring a particular candidate or policy.

• In our textbook problem, the sample proportions are calculated separately for each group and are crucial for finding both the point estimate and the standard error.
• Their exact values (\( \hat{p}_1 = \frac{43}{60} \) and \( \hat{p}_2 = \frac{36}{60} \)) are used as part of the formula to compute the point estimate for the difference in population proportions.

Standard Error

The standard error (SE) of a statistical estimate denotes the degree to which it is expected to vary due to random sampling. In simpler terms, it measures the precision of the sample mean as an estimate of the population mean. For differences in proportions, as seen in our problem, the standard error helps gauge the variability expected when estimating the difference between two population proportions.

To calculate the standard error of the difference in sample proportions, we use the formula: \( SE = \sqrt{\frac{\hat{p}_1 ( 1 - \hat{p}_1)}{n_1} + \frac{\hat{p}_2 ( 1 - \hat{p}_2 )}{n_2}} \).

• This formula incorporates not only the sample proportions but also the sizes of the samples taken from each population.
• The SE is a key component when determining the margin of error and, thus, influences the confidence we have in our point estimate.
• In the provided problem, once the SE is calculated, it can be used together with the Z-score to give the margin of error around our estimate.