Problem 60 - Basic Statistics for Business & Economics

Question

Near the time of an election, a cable news service performs an opinion poll of 1,000 probable voters. It shows that the Republican contender has an advantage of \(52 \%\) to \(48 \% .\) a. Develop a \(95 \%\) confidence interval for the proportion favoring the Republican candidate. b. Estimate the probability that the Democratic candidate is actually leading. c. Repeat the above analysis based on a sample of 3,000 probable voters.

Step-by-Step Solution

Answer

a) Confidence interval for 1,000: (0.489, 0.551). For 3,000: (0.502, 0.538). b) Probability Democratic is leading is very low in both. c) Larger sample narrows interval, still supporting Republican lead.

Step 1: Determine the Sample Proportion

First, note that the Republican candidate received 52% support in a sample of 1,000 voters. This gives us the sample proportion \( \hat{p} = 0.52 \).

Step 2: Find the Standard Error

The standard error (SE) of the sample proportion is calculated using the formula:\[SE = \sqrt{ \frac{\hat{p}(1-\hat{p})}{n} }\]For \(\hat{p} = 0.52\) and \(n = 1000\), the SE is:\[SE = \sqrt{ \frac{0.52 \times 0.48}{1000} } \approx 0.0157\]

Step 3: Calculate the 95% Confidence Interval

A 95% confidence interval is calculated as \( \hat{p} \pm Z \times SE \), where \(Z\) is the z-score corresponding to 95% confidence (1.96).\[CI = 0.52 \pm 1.96 \times 0.0157\]This results in the interval:\[CI \approx (0.489, 0.551)\]

Step 4: Evaluate the Probability that Democratic Candidate is Leading

Since the confidence interval for the Republican candidate (0.489 to 0.551) is above 0.48, it indicates that there is very little probability that the Democratic candidate is actually leading.

Step 5: Repeat Steps 1-3 for a Sample of 3,000 Voters

Redo the calculations for a sample size of 3,000. The standard error becomes:\[SE = \sqrt{ \frac{0.52 \times 0.48}{3000} } \approx 0.009\]The confidence interval is:\[CI = 0.52 \pm 1.96 \times 0.009 \approx (0.502, 0.538)\]

Step 6: Evaluate with New Sample Size

With a sample of 3,000 probable voters, the Republican candidate's lower limit of the confidence interval (0.502) is still above 0.48, suggesting it's even less probable that the Democratic candidate is leading.

Key Concepts

Sample Proportion

The sample proportion is an essential concept in statistics, especially when dealing with surveys or polls. It represents the fraction of subjects in a sample that show a particular characteristic. In our election polling example, the sample proportion denotes the percentage of voters who support the Republican candidate out of the total sampled voters.
To calculate it, you take the number of favorable outcomes and divide it by the total number of samples. In this case, it is 52% or 0.52 for the Republican candidate.

• Consider a hypothetical poll of 1,000 likely voters.
• 52% of them support the Republican candidate.
• The sample proportion then is:
  \[\hat{p} = \frac{520}{1000} = 0.52\]

Understanding the sample proportion helps in estimating the true proportion of a population, which in this situation, is the actual voter support across all voters.

Standard Error

The standard error (SE) measures the variability or spread of the sample proportion. It’s vital when estimating the reliability of the sample proportion as a representation of the population proportion.
In simpler terms, a smaller SE suggests the sample proportion is a more accurate estimate of the true population proportion.
For our election poll, the SE is calculated using the formula:

• \[SE = \sqrt{ \frac{\hat{p}(1-\hat{p})}{n} }\]
• Where \( \hat{p} = 0.52 \) (sample proportion) and \( n = 1000 \) (sample size).
• In this exercise, \[ SE \approx 0.0157 \].

This indicates how much the sample proportion might vary from the actual population proportion. A larger sample, like 3,000 voters, reduces the SE to \(0.009\), thus increasing precision in estimating the population proportion.

Probability Estimation

Estimating probability with a confidence interval involves understanding the likelihood of certain population parameters based on sample data. When presenting polling results, a confidence interval provides a range in which the true proportion of support is expected to lie with a certain level of confidence, commonly 95%.
For example, from the 1,000 voter sample, a 95% confidence interval is calculated as:

• \[CI = \hat{p} \pm Z \times SE\]
• Using the z-score of 1.96 for 95% confidence, the interval is calculated as:
  \[ 0.52 \pm 1.96 \times 0.0157 = (0.489, 0.551) \]

The confidence interval shows we're 95% confident that the true support for the Republican candidate is between 48.9% and 55.1%. If this range is above the Democratic candidate's support of 48%, it indicates a low probability of that candidate leading.
When the sample size is increased to 3,000, this interval narrows to 0.502 to 0.538, providing a clearer picture of the Republican candidate's lead.