Problem 16 - Basic Statistics for Business and Economics with Student

Question

16\. Ms. Maria Wilson is considering running for mayor of the town 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 percent confidence interval for the population proportion. c. Interpret your findings.

Step-by-Step Solution

Answer

a. Estimated proportion = 75%. b. 99% Confidence interval = (0.6942, 0.8058). c. 99% confident the true support is between 69.42% and 80.58%.

Step 1: Estimate the Population Proportion

To estimate the population proportion, divide the number of survey respondents who support Ms. Maria Wilson by the total number of respondents. Therefore, \( \hat{p} = \frac{300}{400} = 0.75 \). So, the estimated population proportion is 0.75 or 75%.

Step 2: Identify the Significance Level and Critical Value

For a 99% confidence interval, the significance level \( \alpha \) is 0.01. Divide this by two to find the critical value for each tail: \( \alpha/2 = 0.005 \). From a standard normal distribution table, the critical value \( Z \) for 0.005 in each tail is approximately 2.576.

Step 3: Calculate the Standard Error

Use the formula for standard error: \( SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \). With our values, \( SE = \sqrt{\frac{0.75 \times 0.25}{400}} = \sqrt{0.00046875} \approx 0.02165 \).

Step 4: Determine the Confidence Interval

Use the formula for the confidence interval: \( \hat{p} \pm Z \times SE \). Compute each term: \( Z \times SE \approx 2.576 \times 0.02165 \approx 0.0558 \). Therefore, the confidence interval is \( 0.75 \pm 0.0558 \), which results in the interval \( (0.6942, 0.8058) \).

Step 5: Interpret the Confidence Interval

The 99% confidence interval for the population proportion is \((0.6942, 0.8058)\). This means we can be 99% confident that the true proportion of voters in Bear Gulch who would support Ms. Maria Wilson falls between 69.42% and 80.58%.

Key Concepts

Population Proportion

The term "population proportion" refers to the fraction of the entire population that has a particular attribute, which in this case is supporting Ms. Maria Wilson. To estimate this proportion, we use data from a sample. In our exercise, the sample data shows that 300 out of 400 voters support her. Thus, we calculate the population proportion as:

• Number of supporters = 300
• Total respondents = 400
• Estimated population proportion, \( \hat{p} = \frac{300}{400} = 0.75 \)

This calculation suggests that about 75% of the population could potentially support Ms. Wilson based on the sample. This proportion allows for analysis and predictions about the larger population.

Standard Error

Standard error measures the estimated variability or dispersion of a sample statistic, in this case, the population proportion. It helps in understanding how much our sample proportion might differ from the true population proportion. The formula for standard error of the population proportion \( \hat{p} \) is:\[SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\]Using our values:

• \( \hat{p} = 0.75 \)
• Sample size \( n = 400 \)
• Calculate: \( SE = \sqrt{\frac{0.75 \times 0.25}{400}} = \sqrt{0.00046875} \approx 0.02165 \)

The standard error of approximately 0.02165 indicates the extent to which the sample proportion may vary from the actual population proportion. Smaller standard errors represent more reliable estimates of the true population proportion.

Significance Level

In statistical hypothesis testing, the significance level, often denoted by \( \alpha \), represents the probability of rejecting the null hypothesis when it is actually true. It indicates how sure we want to be about our confidence interval containing the true population proportion. For a 99% confidence interval, the significance level is:

• \( \alpha = 1 - 0.99 = 0.01 \)
• This means there is a 1% risk of concluding that the population proportion lies outside the interval when it actually doesn't.

The significance level divides into two tails when determining the critical value for a two-tailed confidence interval as we see in this exercise. By dividing \( \alpha \) by two, we ensure both tails are considered for a complete confidence range.

Critical Value

A critical value in statistics is a point on a test's distribution that is compared to the test statistic to determine whether to reject the null hypothesis. For a given significance level, it separates likely from unlikely values. It helps build a confidence interval by dictating how many standard errors to move away from the sample statistic. For a 99% confidence interval, we find the critical value using the standard normal distribution table:

• With \( \alpha = 0.01 \), split for each tail, \( \alpha/2 = 0.005 \)
• The corresponding \( Z \) value is approximately 2.576

This critical value of 2.576, when multiplied by the standard error and added and subtracted from the sample proportion, provides the boundaries of the confidence interval. This helps ensure our interval has the necessary coverage probability of the true population proportion.