Problem 48 - Basic Statistics for Business and Economics with Student

Question

. In a poll to estimate presidential popularity, each person in a random sample of 1,000 voters was asked to agree with one of the following statements: 1\. The president is doing a good job. 2\. The president is doing a poor job. 3\. I have no opinion. A total of 560 respondents selected the first statement, indicating they thought the president was doing a good job. a. Construct a 95 percent confidence interval for the proportion of respondents who feel the president is doing a good job. b. Based on your interval in part \((a),\) is it reasonable to conclude that a majority (more than half) of the population believes the president is doing a qood iob?

Step-by-Step Solution

Answer

The 95% confidence interval is (0.529, 0.591), and it suggests that a majority (more than half) supports the president.

Step 1: Identify the Sample Proportion

First, identify the sample proportion of voters who think the president is doing a good job. The sample proportion \( p \) is given by the formula \( \hat{p} = \frac{x}{n} \), where \( x \) is the number of people who agree the president is doing a good job, and \( n \) is the total number of respondents. Here, \( x = 560 \) and \( n = 1000 \), so \( \hat{p} = \frac{560}{1000} = 0.56 \).

Step 2: Calculate the Standard Error

Next, compute the standard error (SE) for the proportion. The standard error for the sample proportion is calculated as \( SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \). Substituting the values, we get \( SE = \sqrt{\frac{0.56 \times 0.44}{1000}} = \sqrt{\frac{0.2464}{1000}} \approx 0.0157 \).

Step 3: Determine the Z-Score for 95% Confidence

For a 95% confidence interval, use a standard normal distribution z-score of 1.96.

Step 4: Compute the Confidence Interval

Now calculate the confidence interval using the formula: \( \hat{p} \pm z \cdot SE \). Substituting the known values, the lower bound of the interval is \( 0.56 - 1.96 \times 0.0157 \approx 0.529 \), and the upper bound is \( 0.56 + 1.96 \times 0.0157 \approx 0.591 \). Thus, the 95% confidence interval is approximately \( (0.529, 0.591) \).

Step 5: Evaluate If a Majority Exists

A majority means more than 50% or \( 0.5 \). The entire confidence interval (0.529, 0.591) is above 0.5, indicating more than half of the population believes the president is doing a good job. Therefore, it is reasonable to conclude that a majority of the population believes the president is doing a good job.

Key Concepts

Sample Proportion

The sample proportion is a crucial part of statistical analysis, especially when trying to draw conclusions about a larger population based on a sample. In the context of the poll about the president's popularity, the sample proportion represents the fraction of the sample that thinks the president is doing a good job. To calculate the sample proportion, we use the formula: \[ \hat{p} = \frac{x}{n} \]- Here, \( x \) is the number of respondents who agree that the president is doing a good job, and \( n \) is the total number of respondents.In this exercise, \( x = 560 \) and \( n = 1000 \), so the sample proportion \( \hat{p} \) is \( 0.56 \). This means 56% of the sample believes the president is doing a good job. Understanding your sample proportion is vital because it serves as the foundation for constructing confidence intervals, which help you understand what percentage of the entire population might share this opinion.

Standard Error

The standard error (SE) is a measure that indicates the accuracy of the sample proportion as an estimate of the true population proportion. It provides insight into the variability you might expect if you were to take many samples from the same population. To calculate the standard error for the sample proportion, we use this formula:\[ SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \]- In this equation, \( \hat{p} \) is the sample proportion, \( 1-\hat{p} \) is the proportion that does not hold the opinion, and \( n \) is the number of respondents.For this problem, substituting the values \( \hat{p} = 0.56 \) and \( n = 1000 \) into the formula gives us \( SE \approx 0.0157 \).A lower SE suggests that the sample proportion is a more precise estimate of the true population proportion. This is critical for making reliable conclusions about the population.

Z-Score

The Z-score is a critical value that corresponds to the desired confidence level when calculating a confidence interval. In this exercise, the confidence level is set to 95%, which is a common choice in statistics to ensure a balance between precision and reliability.For a 95% confidence interval, a Z-score of 1.96 is used. This Z-score implies that assuming a normal distribution, 95% of the sample means will fall within 1.96 standard deviations of the population mean.The Z-score helps you to determine the margin of error for the confidence interval:\[ MOE = z \times SE \]- This allows you to add and subtract this margin of error from the sample proportion to find the confidence interval.The role of the Z-score is essential because it reflects the number of standard deviations required to achieve a specified confidence level.

Majority Population

Determining if a majority of the population shares a particular opinion helps guide decisions and policies. In the context of this poll, "majority" means that more than 50% of the population agrees the president is doing a good job. A confidence interval that lies entirely above 0.5 supports the conclusion that a majority of the population holds this belief.- Given the computed confidence interval of approximately \( (0.529, 0.591) \), both bounds of the interval are above 0.5.This indicates that you can be 95% confident that the true proportion of the entire population who thinks the president is doing a good job is between 52.9% and 59.1%.Concluding with confidence about the majority of a population is what makes the confidence interval a powerful tool in statistical analysis, providing a range rather than a single estimate, which enhances reliability.