# T Test Formula

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Definition: - t test, which can be used to analyzed hypotheses about a single population mean for small sample sizes when σ is unknown if the population is normally distributed for the measurement being studied.
sometimes a researcher is testing hypotheses about a single population mean and, for reasons such as time, money, convenience, or availability, is able to gather only a small random sample (n<30) of data. In such cases, if the data are normally distributed in the population and σ is known, the z test can be used. However, in reality the sample standard deviation is often used as an estimate for the population standard deviation in hypothesis testing about the population mean because the population standard deviation is unknown. Thus, the z test has limited usage for small- sample analysis of single population means.

The formula for testing such hypotheses follows.

t= (x? - µ) / (s/√n)
Degree of freedom for t test:
df = n-1                                  Where, n= sample size.

Example: - A random sample of size 20 is taken, resulting in a sample mean of 16.45 and a sample standard deviation of 3.59. Assume x is normally distributed with population mean 16. Find the t statistic.

Solution:-
t= (x? - µ) / (s/√n)
= (16.45- 16)/ (3.59/√20)
= 0.56

Example 2: - With the same above example if the sample size is 25 then what is the value of t statistic now.
t= (x? - µ) /(s/√n)
= (16.45- 16)/ (3.59/√25)
= 0.63