Lecture 11: Inference from a Normal Population

What is Student’s t-distribution used for?

Student’s t-distribution is used to make inferences about a normally distributed population when the sample size is small and the population standard deviation (σ) is unknown.

What are the key properties of Student’s t-distribution?

• Symmetric and bell-shaped, centred at zero.

• Has fatter tails than the standard normal distribution (z-distribution). This accounts for the extra uncertainty of estimating the population standard deviation (σ) from a small sample.

• Its shape depends on the degrees of freedom (df = n − 1).

• As sample size (n), and thus the degrees of freedom increase, the t-distribution becomes taller, narrower, and increasingly resembles the standard normal distribution.

How can we estimate the true population mean (μ) using Student’s t-distribution?

Different random samples produce different estimates of the population mean (μ). Therefore, instead of relying on a single estimate, we calculate a confidence interval to provide a range of plausible values.

When σ is unknown, we use:

What is a one-sample t-test used for?

A one-sample t-test is used to determine whether a sample mean is significantly different from a known or hypothesized population mean (μ₀).

What are the steps involved in conducting a one-sample t-test?

1) State your hypotheses:

• Null hypothesis (H₀): μ = μ₀

• Alternative hypothesis (H_A): μ ≠ μ₀

2) Check the assumptions:

• Random, independent observations.

• Normality: The population should be approximately normally distributed.

3) Calculate the test statistic:

4) Find the critical value:

Use the t-table with df = n − 1 and the chosen significance level (α).

5) Apply the decision rule:

For a two-tailed test:

• If |t| ≥ t* → Reject H₀

• If |t| ≥ t* → Fail to reject H₀

How does sample size affect standard error and statistical power?

The standard error of the sample mean is:

• Larger sample size (n): Smaller standard error, greater precision, and greater statistical power to detect true differences.

• Smaller sample size (n): Larger standard error, less precision, and lower statistical power, increasing the likelihood of a Type II error (failing to reject a false null hypothesis).

Why do we use the chi-squared distribution instead of the t-distribution when estimating population variance (σ²)?

We use the chi-squared distribution because variance cannot be negative, and the chi-squared distribution starts at zero and extends toward positive values.

Its critical values are asymmetric, so both tails must be calculated separately.

We calculate a confidence interval to provide a range of plausible values likely to contain the true population variance (σ²) using the formula below: