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Last updated on October 6, 2025
In statistics, the t-distribution is a probability distribution that is used to estimate population parameters when the sample size is small and/or the population variance is unknown. In this topic, we will learn the formula for the t-distribution.
The t-distribution is a key concept in statistics, especially useful in hypothesis testing and constructing confidence intervals. Let’s learn the formula for the t-distribution.
The t-distribution is used when you have a small sample size or when the population standard deviation is unknown.
The formula for calculating the t-score is: \([ t = \frac{\bar{x} - \mu}{s/\sqrt{n}} ] \) where \((\bar{x})\) is the sample mean,\( (\mu)\) is the population mean, \((s)\) is the sample standard deviation, and n is the sample size.
In mathematics and real-life applications, the t-distribution formula is crucial for analyzing data and making inferences about populations. Here are some important aspects of the t-distribution:
Students may find statistical formulas tricky. Here are some tips to help memorize the t-distribution formula:
The t-distribution plays a significant role in various real-life scenarios, especially in the field of statistics and research. Here are some applications:
Students often make errors when calculating t-scores. Here are some mistakes and how to avoid them.
A sample of 10 students has a mean score of 85 with a standard deviation of 5. If the population mean is 80, what is the t-score?
The t-score is 3.16
Using the formula\( ( t = \frac{\bar{x} - \mu}{s/\sqrt{n}} )\), we have: \((\bar{x} = 85), (\mu = 80), (s = 5), (n = 10).\)
\(( t = \frac{85 - 80}{5/\sqrt{10}} = \frac{5}{1.58} = 3.16 ).\)
A quality control manager tests 15 samples with a mean weight of 2.2 kg and a standard deviation of 0.3 kg. The expected population mean is 2 kg. What is the t-score?
The t-score is 2.58
Using the formula\(( t = \frac{85 - 80}{5/\sqrt{10}} = \frac{5}{1.58} = 3.16 ).\), we have: \((\bar{x} = 2.2), (\mu = 2), (s = 0.3), (n = 15).\)
\(( t = \frac{2.2 - 2}{0.3/\sqrt{15}} = \frac{0.2}{0.077} = 2.58 ).\)
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