Explain student‘t’ test and its applications.

Student's t-test is a test of significance for means when the sample is small (usually \(n < 30\)) and the population standard deviation is unknown; it uses the t-distribution with the appropriate degrees of freedom.

It was introduced by W. S. Gosset, who published under the pen name "Student". The t-distribution is bell-shaped and symmetric like the normal curve but has thicker tails; as the degrees of freedom increase it approaches the normal distribution.

Assumptions: data come from a normal population, samples are random and independent, and (for two-sample tests) the population variances are equal.

1. One-sample t-test (compare a sample mean with a known value \(\mu\)):

\[t = \frac{\bar x - \mu}{s/\sqrt n},\quad df = n-1\]

where \(s = \sqrt{\sum(x-\bar x)^2/(n-1)}\).

2. Independent two-sample t-test (compare two group means):

\[t = \frac{\bar x_1 - \bar x_2}{s_p\sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}\]

with pooled SD \(s_p\) and \(df = n_1 + n_2 - 2\).

3. Paired t-test (same subjects measured twice): \(t = \dfrac{\bar d}{s_d/\sqrt n}\), \(df = n-1\), where \(d\) = difference in each pair.

Decision. If \(|t_{\text{cal}}|\) > \(t_{\text{table}}\) at the chosen level and df, reject \(H_0\).

Applications

• Testing whether a machine fills packets with the stated mean weight (one-sample).
• Comparing mean marks of two sections or yields of two crop varieties (independent samples).
• Checking whether a training programme, diet or drug changed results by measuring before and after (paired).
• Testing the significance of a correlation or regression coefficient.
• A/B testing of two designs with small samples.