Problem 16 - Basic Statistics for Business & Economics

Question

Ms. Maria Wilson is considering running for mayor of Bear Gulch, Montana. Before completing the petitions, she decides to conduct a survey of voters in Bear Gulch. A sample of 400 voters reveals that 300 would support her in the November election. a. Estimate the value of the population proportion. b. Develop a \(99 \%\) confidence interval for the population proportion. c. Interpret your findings.

Step-by-Step Solution

Answer

Estimated population proportion is 0.75, with a 99% confidence interval of (0.69416, 0.80584).

Step 1: Find the Sample Proportion

First, determine the sample proportion \( \hat{p} \). This is calculated by dividing the number of voters who support Ms. Wilson by the total number of voters surveyed. Mathematically, it is given by:\[\hat{p} = \frac{300}{400} = 0.75\]So, the sample proportion \( \hat{p} \) is 0.75, or 75%.

Step 2: Estimate Population Proportion

The population proportion \( p \) is estimated using the sample proportion \( \hat{p} \). In this context, \( \hat{p} = 0.75 \) is taken as the point estimate for the true proportion \( p \) of all voters in Bear Gulch who would support Ms. Wilson.

Step 3: Calculate the Standard Error

The standard error \( SE \) of the sample proportion is computed as:\[SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} = \sqrt{\frac{0.75(1-0.75)}{400}} = \sqrt{\frac{0.1875}{400}} \approx 0.02165\]where \( n \) is the sample size, which is 400.

Step 4: Determine the Z-value for 99% Confidence

Using a standard normal distribution table, the Z-value for a 99% confidence interval is approximately 2.576. This Z-value represents the critical value for the two-tailed distribution.

Step 5: Calculate Confidence Interval

To calculate the confidence interval for the population proportion, use the formula:\[\hat{p} \pm Z \times SE = 0.75 \pm 2.576 \times 0.02165\]This gives:\[0.75 - 0.05584 \leq p \leq 0.75 + 0.05584\]Thus, the 99% confidence interval is \[(0.69416, 0.80584)\].

Step 6: Interpret the Confidence Interval

The 99% confidence interval of (0.69416, 0.80584) suggests that if we were to sample the population multiple times, 99% of such intervals would contain the true population proportion. Hence, we are 99% confident that between 69.42% and 80.58% of Bear Gulch voters support Ms. Wilson.

Key Concepts

Population Proportion

The population proportion, denoted by \( p \), is a parameter that represents the fraction of the entire population that possesses a certain attribute. In the case of Ms. Wilson's survey, the population proportion is the proportion of all Bear Gulch voters who would support her. Since we rarely have the resources to survey an entire population, we rely on sample data to estimate \( p \).

• The true population proportion cannot be known unless every individual is surveyed.
• We use sample data to provide a point estimate of the population proportion.
• This estimate is expressed through the sample proportion \( \hat{p} \).

In surveys, the sample proportion serves as the best estimate for the true population proportion. However, keep in mind that estimates can be subject to sampling errors, which is why we additionally calculate confidence intervals to better encompass the true proportion.

Sample Proportion

The sample proportion, \( \hat{p} \), is the estimate calculated from the sample data, representing the proportion of the sample that possesses the attribute of interest. In Ms. Wilson's survey, the sample proportion is obtained by dividing the number of supporting voters by the total number of respondents:\[\hat{p} = \frac{300}{400} = 0.75\]

• This value of 0.75, or 75%, indicates that 75% of the respondents support Ms. Wilson.
• It is used as the point estimate for the population proportion \( p \).

Being a direct representation from the sample, \( \hat{p} \) is influenced by the sample size and variability. Therefore, larger samples tend to give a more accurate reflection of the population parameter, while smaller samples can lead to more variability in \( \hat{p} \). This is crucial in determining how trustworthy our sample estimate is.

Standard Error

The standard error (SE) is a statistical term that measures the accuracy with which a sample proportion estimates the population proportion. It indicates the variability or spread of the sample proportion in repeated sampling: \[SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\]For Ms. Wilson’s survey:\[SE = \sqrt{\frac{0.75(1-0.75)}{400}} = 0.02165\]

• The SE helps in understanding the potential error margin in the point estimate.
• It reflects how much the sample proportion is expected to fluctuate if different samples are taken.

A smaller SE suggests that the sample proportion is more precise as it indicates less variability between possible sample proportions derived from different samples. SE is also key in constructing confidence intervals, providing us with a range rather than a fixed estimate.

Z-value

The Z-value is a critical value from the standard normal distribution, used in statistics to determine the boundaries of a confidence interval. For Ms. Wilson's survey, a Z-value is applied because we want to build a confidence interval around our sample estimate that would contain the true population proportion with a specified level of confidence.

• For a 99% confidence level, the corresponding Z-value is 2.576.
• This Z-value separates the middle 99% of the distribution from the extreme 1%, split between the two tails (0.5% in each tail).

The Z-value ensures that the interval calculated is wide enough to account for expected variability, but not too wide to be uninformative. By multiplying this Z-value with the standard error (SE), you determine the margin of error that frames the confidence interval around the sample proportion. This results in a comprehensive range that is highly likely to include the true population proportion.