The range of variable 't' of the t-distribution is :

Answer: The range of variable ' $t$ ' of the $t$ -distribution is $(-\infty, \infty)$ .

Step-by-step solution

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 $\bar{x}$ , population mean $\mu$ , 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 $-\infty$ to $\infty$ .

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