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How to Compute t-statistic?

Published in Statistical Test Calculation 4 mins read

The t-statistic is calculated to determine if there's a significant difference between the means of two groups or if a sample mean differs significantly from a known or hypothesized population mean. Here's a step-by-step guide on how to compute the t-statistic:

Understanding the t-statistic

The t-statistic is a crucial tool in hypothesis testing. It uses a t-distribution to assess the likelihood of getting your observed results by chance if the null hypothesis were true. This is particularly useful when dealing with small sample sizes or when the population standard deviation is unknown.

Steps to Calculate the t-statistic

Here's how you compute the t-statistic according to the provided reference:

  1. Calculate the Sample Mean (x̄):

    • The sample mean (x̄) is the average of your data set. You find it by adding up all the values in your sample and then dividing by the number of values in the sample.

  2. Determine the Population Mean (μ):

    • The population mean (μ) is the average value of the entire population you are interested in. This value may be known, hypothesized, or assumed.

  3. Calculate the Sample Standard Deviation (s):

    • The sample standard deviation (s) measures the dispersion or spread of the data in your sample. It’s calculated by taking the square root of the variance.

    • Variance: Find the variance by doing the following:

    1. Subtract the mean from each data point.
    2. Square each result.
    3. Sum these squared results.
    4. Divide this sum by n-1 (where n is the number of data points).
    • Standard deviation: Calculate the square root of the variance.

  4. Compute the t-statistic:

    • The t-statistic is calculated using the formula:

      t = (x̄ - μ) / (s / √n)

      Where:

      • t is the t-statistic
      • x̄ is the sample mean
      • μ is the population mean
      • s is the sample standard deviation
      • n is the sample size

Example Calculation

Let’s walk through an example.

Suppose you are testing a new teaching method.

  • Sample Data: 80, 85, 90, 75, 95
  • Hypothesized Population Mean (μ): The established method results in an average score of 82 (μ = 82)
  • Sample Size (n): 5 (number of students tested)
  1. Calculate the Sample Mean (x̄): (80+85+90+75+95) / 5 = 425 / 5 = 85
  2. Population mean (μ): 82
  3. Calculate the Sample Standard Deviation (s):
  • Variance:
 1. Subtract the mean from each data point: 80-85 = -5, 85-85 = 0, 90-85=5, 75-85=-10, 95-85 =10
 2. Square each result: (-5)² = 25, 0²=0, 5²=25, (-10)²=100, 10²=100
 3. Sum these squared results: 25+0+25+100+100 = 250
 4. Divide the sum by *n-1*: 250 / (5-1) = 250 / 4 = 62.5
  • Standard deviation: Square root of 62.5 = 7.91
  1. Calculate the t-statistic: t = (85-82) / (7.91 / √5) = 3 / (7.91 / 2.236) = 3/ 3.54 = 0.847

So, the calculated t-statistic is 0.847. This value would then be used in conjunction with a t-distribution table and your determined alpha to determine if the null hypothesis is rejected.

Practical Insights

  • Degrees of Freedom: Remember to consider the degrees of freedom when using a t-table to find the p-value. Degrees of freedom for a one sample t-test are calculated as n-1, where n is the sample size.
  • Interpreting the t-statistic: A larger absolute value of the t-statistic indicates that the sample mean is more likely to be significantly different from the population mean.
  • Tails: t-tests can be one-tailed or two-tailed depending on the hypothesis.

Conclusion

Computing the t-statistic involves understanding key parameters like the sample mean, population mean, sample standard deviation, and sample size. Following these steps provides a quantitative measure for statistical inference regarding differences between means.

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