Confidence Interval Examples for Small Samples

Explore practical examples of confidence intervals for small sample sizes in various contexts.
By Jamie

Understanding Confidence Intervals for Small Sample Sizes

Confidence intervals (CIs) are a fundamental concept in statistics, providing a range of values that likely contain the population parameter (such as the mean or proportion). When working with small sample sizes, the calculation of confidence intervals can be challenging due to increased variability. This article presents three practical examples of confidence intervals for small sample sizes in different contexts.

Example 1: Estimating Average Height of a Class

In a small class of 10 students, a teacher wants to estimate the average height. To do this, she measures the heights of all students and calculates the confidence interval.

The heights (in cm) are: [150, 160, 155, 165, 158, 162, 155, 157, 159, 161].

  1. Calculate the Sample Mean (X̄):

    • X̄ = (150 + 160 + 155 + 165 + 158 + 162 + 155 + 157 + 159 + 161) / 10 = 157.7 cm

  2. Calculate the Sample Standard Deviation (s):

    • s = 3.87 cm (using the formula for standard deviation).

  3. Determine the t-score for a 95% confidence level with 9 degrees of freedom (df = n - 1):

    • t ≈ 2.262 (from t-distribution table).

  4. Calculate the Standard Error (SE):

    • SE = s / √n = 3.87 / √10 = 1.22 cm

  5. Construct the Confidence Interval:

    • CI = X̄ ± (t * SE) = 157.7 ± (2.262 * 1.22)
    • CI = 157.7 ± 2.76
    • CI = (154.94, 160.46)

This means the teacher can be 95% confident that the true average height of all students in her class falls between 154.94 cm and 160.46 cm.

Notes:

  • This example highlights the impact of small sample sizes on the width of the confidence interval. The smaller the sample, the wider the interval tends to be, reflecting greater uncertainty.

Example 2: Surveying Satisfaction Levels in a Small Business

A small café owner wants to determine customer satisfaction levels. She surveys 15 customers and records their satisfaction on a scale from 1 to 10.

The satisfaction scores are: [8, 9, 7, 6, 8, 10, 5, 7, 9, 8, 9, 6, 7, 8, 9].

  1. Calculate the Sample Mean (X̄):

    • X̄ = (8 + 9 + 7 + 6 + 8 + 10 + 5 + 7 + 9 + 8 + 9 + 6 + 7 + 8 + 9) / 15 = 8.13

  2. Calculate the Sample Standard Deviation (s):

    • s = 1.33.

  3. Determine the t-score for a 95% confidence level with 14 degrees of freedom:

    • t ≈ 2.145.

  4. Calculate the Standard Error (SE):

    • SE = s / √n = 1.33 / √15 = 0.34.

  5. Construct the Confidence Interval:

    • CI = X̄ ± (t * SE) = 8.13 ± (2.145 * 0.34)
    • CI = 8.13 ± 0.73
    • CI = (7.40, 8.86)

This indicates the café owner can be 95% confident that the average satisfaction level of all customers lies between 7.40 and 8.86.

Notes:

  • The small sample size means the owner should consider conducting larger surveys over time to achieve more reliable results.

Example 3: Analyzing Test Scores in a Small Study Group

A university professor conducts a study on the effectiveness of a new teaching method. She assesses 8 students and records their test scores.

The test scores are: [78, 85, 82, 90, 88, 76, 85, 80].

  1. Calculate the Sample Mean (X̄):

    • X̄ = (78 + 85 + 82 + 90 + 88 + 76 + 85 + 80) / 8 = 82.50.

  2. Calculate the Sample Standard Deviation (s):

    • s = 4.69.

  3. Determine the t-score for a 95% confidence level with 7 degrees of freedom:

    • t ≈ 2.365.

  4. Calculate the Standard Error (SE):

    • SE = s / √n = 4.69 / √8 = 1.66.

  5. Construct the Confidence Interval:

    • CI = X̄ ± (t * SE) = 82.50 ± (2.365 * 1.66)
    • CI = 82.50 ± 3.93
    • CI = (78.57, 86.43)

This suggests that the professor can be 95% confident that the true average test score of students taught with the new method is between 78.57 and 86.43.

Notes:

  • The results indicate the potential variability in test scores due to the small sample size, reinforcing the importance of interpreting findings with caution.