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The Student's t-distribution is a probability distribution used in hypothesis testing and confidence intervals.

The t-distribution is similar to the standard normal distribution, but is used when the sample size is small or the population standard deviation is unknown. It is a continuous probability distribution that is symmetric and bell-shaped, with a mean of 0 and a variance of 1.

The t-distribution is characterized by its degrees of freedom (df), which is the sample size minus one. As the sample size increases, the t-distribution approaches the standard normal distribution.

The t-distribution is used in hypothesis testing to determine whether a sample mean is significantly different from a population mean. The test statistic is calculated as:

t = (x̄ - μ) / (s / √n)

where x̄ is the sample mean, μ is the population mean, s is the sample standard deviation, and n is the sample size. The test statistic is compared to a critical value from the t-distribution with df = n-1 and a chosen level of significance.

The t-distribution is also used to construct confidence intervals for the population mean. The confidence interval is calculated as:

x̄ ± tα/2 (s / √n)

where x̄ is the sample mean, s is the sample standard deviation, n is the sample size, and tα/2 is the critical value from the t-distribution with df = n-1 and a chosen level of confidence.

In summary, the Student's t-distribution is a probability distribution used in hypothesis testing and confidence intervals when the sample size is small or the population standard deviation is unknown. It is characterized by its degrees of freedom and is used to determine whether a sample mean is significantly different from a population mean.

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