Confidence Interval Examples in Inferential Statistics

Explore practical examples of confidence intervals in inferential statistics to understand their applications.
By Jamie

Understanding Confidence Intervals in Inferential Statistics

Confidence intervals are a fundamental concept in inferential statistics, providing a range of values that likely contains the true population parameter. They help researchers understand the precision of their estimates and make informed decisions based on sample data. Below are three diverse, practical examples of confidence intervals in various contexts.

Example 1: Estimating Average Height of Adults

In a health study, researchers want to estimate the average height of adults in a city. They randomly sample 100 adults and find the sample mean height to be 170 cm with a standard deviation of 10 cm. Using a 95% confidence level, they calculate the confidence interval for the average height of the entire adult population.

To calculate the confidence interval, the formula used is:
CI = sample mean ± (z * (standard deviation / √n))

Here, the z-value for a 95% confidence level is approximately 1.96. Plugging in the values:

  • Standard Error (SE) = 10 / √100 = 1
  • Margin of Error (ME) = 1.96 * 1 = 1.96
  • Confidence Interval = 170 ± 1.96, which gives (168.04, 171.96)

Thus, the researchers can say with 95% confidence that the true average height of adults in the city lies between 168.04 cm and 171.96 cm.

Notes:

  • If a higher confidence level is desired (e.g., 99%), the z-value increases, resulting in a wider confidence interval.
  • This method assumes a normal distribution of heights in the population.

Example 2: Average Time Taken to Complete a Task

A company conducts a study to understand how long it takes employees to complete a specific task. They sample 50 employees and find that the average time taken is 45 minutes, with a standard deviation of 5 minutes. The company wants to establish a 90% confidence interval for the average time taken to complete the task.

For a 90% confidence level, the z-value is approximately 1.645. Applying the confidence interval formula:

  • Standard Error (SE) = 5 / √50 ≈ 0.707
  • Margin of Error (ME) = 1.645 * 0.707 ≈ 1.165
  • Confidence Interval = 45 ± 1.165, which provides (43.835, 46.165)

The company can be 90% confident that the average time to complete the task for all employees lies between 43.835 minutes and 46.165 minutes.

Notes:

  • The sample size affects the width of the confidence interval; larger samples yield more precise estimates.
  • The assumption of normality is valid due to the Central Limit Theorem as the sample size is adequate.

Example 3: Survey on Customer Satisfaction

A retail store conducts a survey to measure customer satisfaction regarding their services. From a sample of 200 customers, they find that 80% are satisfied with their service. The store wishes to calculate a 95% confidence interval for the proportion of all customers who are satisfied.

For proportions, the formula for the confidence interval is:
CI = p̂ ± (z * √(p̂(1 - p̂) / n))

Where p̂ is the sample proportion (0.8 in this case). The z-value for a 95% confidence level is 1.96. Thus:

  • Standard Error (SE) = √(0.8 * (1 - 0.8) / 200) = √(0.8 * 0.2 / 200) = √(0.0008) ≈ 0.0283
  • Margin of Error (ME) = 1.96 * 0.0283 ≈ 0.0556
  • Confidence Interval = 0.8 ± 0.0556, resulting in (0.7444, 0.8556)

The store can conclude with 95% confidence that between 74.44% and 85.56% of all customers are satisfied with their service.

Notes:

  • This example is particularly useful for businesses in assessing customer feedback.
  • A larger sample size would lead to a narrower confidence interval, providing more precise estimates.