Understanding Confidence Intervals

Confidence intervals are a fundamental statistical concept that provides a range of values within which a population parameter is likely to fall. They are widely used in various fields, including science and economics, to convey the uncertainty associated with sample estimates. Here are three practical examples that illustrate the application of confidence intervals in lab reports.

Example 1: Estimating Average Height of Students

In a study conducted at a local university, researchers aimed to estimate the average height of students enrolled in a specific program. They collected a random sample of 50 students and measured their heights.

To analyze the data, the researchers calculated the sample mean height as 170 cm with a standard deviation of 10 cm. They chose a confidence level of 95% to construct the confidence interval.

Using the formula for the confidence interval:

• Confidence Interval = Sample Mean ± (Z * (Standard Deviation / √n))

Where Z for a 95% confidence level is approximately 1.96, n is the sample size, and the standard deviation is known.

• Confidence Interval = 170 ± (1.96 * (10 / √50))
• Confidence Interval = 170 ± (1.96 * 1.414)
• Confidence Interval = 170 ± 2.77
• Confidence Interval = (167.23, 172.77)

This means the researchers are 95% confident that the average height of all students in the program falls between 167.23 cm and 172.77 cm.

Notes

• For different confidence levels (e.g., 90% or 99%), the Z value will change, affecting the interval width.

Example 2: Surveying Customer Satisfaction

A retail company conducted a survey to assess customer satisfaction with their services. They randomly surveyed 200 customers and found that 150 expressed satisfaction.

The proportion of satisfied customers is calculated as:

• Sample Proportion (p) = 150 / 200 = 0.75

To calculate the 95% confidence interval for the proportion of satisfied customers, the researchers used the formula:

• Confidence Interval = p ± Z * √((p(1 - p)) / n)

Substituting the values:

• Confidence Interval = 0.75 ± 1.96 * √((0.75 * (1 - 0.75)) / 200)
• Confidence Interval = 0.75 ± 1.96 * √(0.75 * 0.25 / 200)
• Confidence Interval = 0.75 ± 1.96 * √(0.09375 / 200)
• Confidence Interval = 0.75 ± 1.96 * 0.2165
• Confidence Interval = 0.75 ± 0.425
• Confidence Interval = (0.325, 1.175)

However, since proportions cannot exceed 1, the adjusted confidence interval is (0.325, 1.0).

Notes

• The margin of error reflects the uncertainty in customer satisfaction estimates, and larger samples yield narrower confidence intervals.

Example 3: Analyzing Test Scores

A high school wanted to analyze the performance of students in a standardized mathematics test. A random sample of 30 students was selected, and their average score was found to be 78 with a standard deviation of 12.

To construct a 90% confidence interval for the mean score, the following formula was used:

• Confidence Interval = Sample Mean ± t * (Standard Deviation / √n)

For a 90% confidence level and 29 degrees of freedom, the t-value is approximately 1.699.

Substituting the values into the formula:

• Confidence Interval = 78 ± 1.699 * (12 / √30)
• Confidence Interval = 78 ± 1.699 * (2.19)
• Confidence Interval = 78 ± 3.72
• Confidence Interval = (74.28, 81.72)

This interval suggests that the researchers are 90% confident that the true average score of all students in the school is between 74.28 and 81.72.

Notes

• The choice of t-distribution is appropriate for smaller sample sizes, especially when the population standard deviation is unknown.