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Step 1: Understand the t-distribution
The t -distribution, also known as Student's t -distribution, is a probability distribution that arises in the problem of estimating the mean of a normally distributed population when the sample size is small and the population standard deviation is unknown. The formula for the t -statistic involves the sample mean xˉ, population mean μ, sample standard deviation s, and sample size n.
Step 2: Determine the range of the t-statistic
The t -statistic is a ratio of a normally distributed variable (the difference between sample and population means) to a scaled chi-squared distributed variable (related to the sample standard deviation). Both the numerator and denominator can take on values that allow the t -statistic to range from negative infinity to positive infinity. This is similar to the standard normal distribution (z -distribution), which also has a range from −∞ to ∞.