Exploring Confidence Intervals Using the T-Distribution

In this article, we will delve into confidence intervals using the t-distribution. You'll learn how to calculate and interpret confidence intervals, especially when dealing with small sample sizes or unknown population variances.
By Jamie

What is a Confidence Interval?

A confidence interval (CI) is a range of values that is used to estimate the true population parameter. It provides an interval estimate that, with a certain level of confidence, contains the parameter. The t-distribution is particularly useful when the sample size is small (n < 30) or when the population standard deviation is unknown.

Example 1: Calculating a Confidence Interval for a Small Sample

Scenario:
Suppose a researcher wants to estimate the average height of students in a small class of 10 students. The heights (in cm) are as follows:

  • 160, 165, 170, 155, 162, 168, 172, 158, 166, 164

Step 1: Calculate the Sample Mean and Standard Deviation

  • Sample Mean ():
    = (160 + 165 + 170 + 155 + 162 + 168 + 172 + 158 + 166 + 164) / 10 = 164.6 cm

  • Sample Standard Deviation (s):

    s = sqrt(Σ(x_i - )² / (n - 1))
    = sqrt(((-4.6)² + (-0.6)² + (5.4)² + (-9.6)² + (-2.6)² + (3.4)² + (7.4)² + (-6.6)² + (1.4)² + (-0.6)²) / 9)
    = sqrt(86.4 / 9) ≈ 3.13 cm

Step 2: Determine the Critical Value
For a 95% confidence level with 9 degrees of freedom (n-1), we can look up the t-distribution table or use statistical software. The critical value (t*) is approximately 2.262.

Step 3: Calculate the Margin of Error (ME)

ME = t* × (s / sqrt(n))
= 2.262 × (3.13 / sqrt(10))
≈ 2.262 × 0.991 ≈ 2.24 cm

Step 4: Construct the Confidence Interval

CI = ( - ME,  + ME)
= (164.6 - 2.24, 164.6 + 2.24)
= (162.36, 166.84)

Interpretation:
We are 95% confident that the true average height of students in this class lies between 162.36 cm and 166.84 cm.

Example 2: Using a Confidence Interval to Make Decisions

Scenario:
A company wants to estimate the average time (in hours) employees spend on training during a month. A sample of 15 employees yields the following training times:

  • 2, 2.5, 3, 4, 1.5, 3.5, 2, 3.2, 2.8, 3.1, 4.5, 2.7, 3.8, 2.9, 3.3

Step 1: Calculate the Sample Mean and Standard Deviation

  • Sample Mean ():
    = (2 + 2.5 + 3 + 4 + 1.5 + 3.5 + 2 + 3.2 + 2.8 + 3.1 + 4.5 + 2.7 + 3.8 + 2.9 + 3.3) / 15 ≈ 3.04 hours

  • Sample Standard Deviation (s):

    s ≈ 0.78 hours (calculation omitted for brevity)

Step 2: Determine the Critical Value
For a 90% confidence level with 14 degrees of freedom, the critical value (t*) is approximately 1.761.

Step 3: Calculate the Margin of Error (ME)

ME = t* × (s / sqrt(n))
≈ 1.761 × (0.78 / sqrt(15)) ≈ 0.35 hours

Step 4: Construct the Confidence Interval

CI = ( - ME,  + ME)
= (3.04 - 0.35, 3.04 + 0.35)
= (2.69, 3.39)

Interpretation:
With 90% confidence, the company can estimate that the average training time for employees lies between 2.69 hours and 3.39 hours. This information may help in planning future training sessions and resource allocation.

Conclusion

Understanding how to calculate and interpret confidence intervals using the t-distribution is crucial for making informed decisions based on sample data, especially when dealing with small sample sizes or unknown population variances. By following the examples outlined above, you can apply these concepts to your own data analysis tasks.