Problem 47 - Basic Statistics for Business and Economics with Student

Question

There are 20,000 eligible voters in York County, South Carolina. A random sample of 500 York County voters revealed 350 plan to vote to return Louella Miller to the state senate. Construct a 99 percent confidence interval for the proportion of voters in the county who plan to vote for Ms. Miller. From this sample information, can you confirm she will be re-elected?

Step-by-Step Solution

Answer

99% confidence interval is (0.6472, 0.7528); likely to be re-elected.

Step 1: Identify the Sample Proportion

First, identify the sample proportion, which is the number of voters who plan to vote for Louella Miller divided by the total sample size. In this case, it is \( \frac{350}{500} = 0.7 \).

Step 2: Determine the Z-Score for 99% Confidence

For a 99% confidence level, the Z-score (critical value) is typically 2.576. This is found using a Z-table or standard normal distribution.

Step 3: Calculate the Standard Error

The standard error (SE) measures the variability around the sample proportion and is calculated as \( SE = \sqrt{ \frac{p(1-p)}{n} } \), where \( p \) is the sample proportion and \( n \) is the sample size. Here, \( SE = \sqrt{ \frac{0.7 \times 0.3}{500} } = 0.0205 \).

Step 4: Compute the Margin of Error

The margin of error (MOE) gives the range above and below the sample proportion for the confidence interval and is calculated as \( MOE = Z \times SE \). Therefore, \( MOE = 2.576 \times 0.0205 = 0.0528 \).

Step 5: Construct the Confidence Interval

The confidence interval is found by adding and subtracting the margin of error from the sample proportion. The interval is \( 0.7 \pm 0.0528 \), which translates to \( (0.6472, 0.7528) \).

Step 6: Interpret the Confidence Interval

With 99% confidence, between 64.72% and 75.28% of the voters plan to vote for Ms. Miller. Since both ends of the interval are above 50%, it suggests she has a strong chance of being re-elected.

Key Concepts

Sample Proportion

The sample proportion is an essential concept when examining the voting intentions within a sample of a broader population. It's a simple calculation but powerful in statistical analysis. In this specific case, the sample proportion is the ratio of the number of people who plan to vote for a particular candidate to the total number of participants surveyed.
To find the sample proportion, divide the number of favorable responses by the total sample size. For instance, in the problem given, 350 out of 500 voters plan to vote for Louella Miller.
Hence, the sample proportion is calculated as follows:

• Total number of voters in favor: 350
• Total sample size: 500
• Sample proportion: \( \frac{350}{500} = 0.7 \)

This means 70% of the sampled voters expressed their intention to vote for the candidate. Understanding this proportion helps in predicting trends about the entire population of voters.

Standard Error

The standard error (SE) plays a vital role in statistics, especially when trying to understand the precision of a sample proportion. It measures how much the sample proportion would vary if different samples were taken from the population.
To compute the standard error for a proportion, use the formula:\[ SE = \sqrt{ \frac{p(1-p)}{n} } \]where \( p \) is the sample proportion and \( n \) is the sample size.
In our example, with a sample proportion of 0.7 and a sample size of 500, the standard error is:

• \( SE = \sqrt{ \frac{0.7 \times 0.3}{500} } \)
• Resulting SE is approximately 0.0205

This standard error indicates how much the sample proportion would vary if multiple samples were taken. Smaller SE indicates that the sample proportion is a more accurate reflection of the population's actual proportion.

Margin of Error

The margin of error (MOE) provides the range within which the true population proportion is likely to fall. It's the amount added and subtracted from the sample proportion to create the confidence interval.
The formula for calculating the margin of error is:\[ MOE = Z \times SE \]where \( Z \) is the Z-score corresponding to the desired confidence level, and \( SE \) is the standard error.
For our example, with a Z-score of 2.576 for a 99% confidence interval and SE of 0.0205, the MOE is calculated as:

• \( MOE = 2.576 \times 0.0205 \)
• This results in a MOE of approximately 0.0528

This margin means that the true proportion could reasonably be expected to fall 5.28% above or below the sample proportion.

Z-score

The Z-score is a statistical tool that helps determine the probability of a sample proportion within a certain confidence level. It shows how many standard deviations an element is from the mean of a standard normal distribution.
The Z-score is crucial when constructing confidence intervals, as it factors in the level of certainty desired in the result. For a confidence interval to cover 99% of the possible outcomes, a Z-score of 2.576 is typically utilized.
So, within our context, when calculating confidence intervals, the Z-score helps ensure that 99% of the time, the interval will encompass the true population proportion. It acts as a multiplier for the standard error in the margin of error formula, adjusting its breadth based on the sample size and desired confidence level. Understanding the Z-score and its application helps predict whether a candidate would likely win based on sampled data.