Practical Confidence Interval for a Median Examples

Understanding Confidence Intervals for a Median

Confidence intervals are a statistical tool used to estimate the range within which a population parameter lies. When we focus specifically on the median, we can assess the center of a dataset, which is particularly useful in skewed distributions. Below, we present three practical examples of confidence intervals for a median, illustrating their application in real-world scenarios.

Example 1: Household Income Distribution

In a study of household incomes within a metropolitan area, researchers aim to understand the financial landscape of the population. They collect a random sample of 250 households and calculate the median income.

• Context: Understanding socio-economic status in urban planning.
• Calculated Median: $55,000.
• Confidence Interval: Using a non-parametric method, they determine a 95% confidence interval for the median income to be between \(52,000 and \)58,000.

This means that researchers can be 95% confident that the true median income of the entire population lies within this range. Such information is crucial for policymakers when allocating resources or planning community services.

Notes:

• Variations in sampling size can affect the width of the confidence interval. A larger sample could yield a narrower interval, reflecting more precision.

Example 2: Testing Plant Growth under Different Conditions

A botany researcher is experimenting with the growth of a specific plant species under various lighting conditions. To quantify the median height of the plants after six weeks, they take measurements from three different groups exposed to different light intensities.

• Context: Analyzing the impact of light on plant growth.
• Group Data: 30 plants under low light, 30 under medium light, and 30 under high light.
• Calculated Medians: 12 cm (low), 18 cm (medium), and 25 cm (high).
• Confidence Intervals: The researcher determines confidence intervals for each group at 90% confidence as follows:
  • Low Light: (10 cm, 14 cm)
  • Medium Light: (16 cm, 20 cm)
  • High Light: (23 cm, 27 cm)

This data indicates that as light intensity increases, the height of the plants also tends to rise, with high confidence in the median heights.

Notes:

• The confidence intervals reflect the variability in plant growth under different conditions.
• Non-parametric methods are often preferred here due to potential skewness in growth data.

Example 3: Surveying Customer Satisfaction Scores

A company conducts a survey to measure customer satisfaction regarding a new product. They collect satisfaction scores on a scale from 1 to 10 from 500 customers and want to analyze the median score.

• Context: Understanding customer feedback for product improvement.
• Calculated Median: 7.5.
• Confidence Interval: Using bootstrapping techniques, they establish a 95% confidence interval for customer satisfaction scores of (7.2, 7.8).

This indicates that the company can be 95% confident that the true median customer satisfaction score lies between these two values, guiding them in making informed decisions about product enhancements.

Notes:

• Bootstrapping is a powerful method for constructing confidence intervals, especially when the data does not meet normality assumptions.
• The larger the sample size, the more reliable the confidence interval becomes.

These examples illustrate how confidence intervals for a median can be applied across different fields, providing valuable insights into the data being analyzed.