What is "when using the student's t distribution to test 𝜇?

When using the <a href="https://www.wikiwhat.page/kavramlar/Student's%20t-distribution">Student's t-distribution</a> to test a hypothesis about the population mean (𝜇), here's some crucial information:

• When to Use: You use the t-distribution when the population standard deviation (σ) is unknown and you are estimating it using the <a href="https://www.wikiwhat.page/kavramlar/sample%20standard%20deviation">sample standard deviation</a> (s). This is most common in real-world scenarios. If the population standard deviation is known, you'd use the <a href="https://www.wikiwhat.page/kavramlar/z-distribution">z-distribution</a> instead.

• Assumptions:

  • The data should be a random sample from the population.
  • The population should be approximately normally distributed. While the t-test is fairly robust to violations of normality, especially with larger sample sizes, significant departures from normality can affect the validity of the test, particularly for small samples.

• Hypothesis Testing: The goal is to determine if there's enough evidence to reject the null hypothesis (H₀) about the population mean. Common null hypotheses are:

  • H₀: 𝜇 = some specified value (e.g., 𝜇 = 0)
  • H₀: 𝜇 ≥ some specified value (right-tailed test)
  • H₀: 𝜇 ≤ some specified value (left-tailed test)

• Test Statistic: The t-statistic is calculated as:

  t = (sample mean - hypothesized mean) / (sample standard deviation / square root of sample size) or t = (x̄ - 𝜇₀) / (s / √n)

  Where: * x̄ is the sample mean * 𝜇₀ is the hypothesized population mean under the null hypothesis * s is the sample standard deviation * n is the sample size

• Degrees of Freedom: The t-distribution's shape depends on the <a href="https://www.wikiwhat.page/kavramlar/degrees%20of%20freedom">degrees of freedom</a> (df), which is calculated as n - 1 (sample size minus one). A smaller sample size results in a lower degree of freedom which flattens and widens the distribution.

• P-value: The p-value represents the probability of observing a test statistic as extreme as, or more extreme than, the one calculated from your sample data, assuming the null hypothesis is true. You compare the p-value to your chosen significance level (α).

• Decision Rule:

  • If p-value ≤ α: Reject the null hypothesis. There is sufficient evidence to conclude that the population mean is significantly different from (or greater/less than, depending on the test type) the hypothesized value.
  • If p-value > α: Fail to reject the null hypothesis. There is not enough evidence to conclude that the population mean is significantly different from the hypothesized value.

• Types of T-tests:

  • One-sample t-test: Compares the mean of a single sample to a hypothesized population mean.
  • Independent samples t-test (two-sample t-test): Compares the means of two independent groups.
  • Paired samples t-test: Compares the means of two related groups (e.g., before and after measurements on the same individuals).

• Interpreting Results: Beyond simply rejecting or failing to reject the null hypothesis, consider the practical significance of your findings. A statistically significant result might not be practically meaningful if the difference between the sample mean and hypothesized mean is very small. Also, consider the <a href="https://www.wikiwhat.page/kavramlar/confidence%20interval">confidence interval</a> for the population mean.