The t-Test (AQA A-Level Further Maths): Revision Notes

The t-Test

Introduction to the t-test

When conducting hypothesis tests about the mean of a Normal distribution, the method depends on whether the population variance is known. If the population variance is known, you can use a standard Normal distribution test (z-test). However, when the population variance is unknown, which is often the case in real-world situations, you must use a different approach called the t-test.

The t-test uses the sample variance as an estimate for the unknown population variance. This estimation introduces additional uncertainty, particularly when working with small sample sizes. The t-test accounts for this extra uncertainty by using a different probability distribution called the t-distribution.

Understanding the sample mean distribution

Consider a situation where you believe a process follows a Normal distribution with an expected mean of 12.0. You collect a sample of 10 observations:

11.9	8.37	14.8	14.6	11.2
11.1	14.9	12.2	10.7	15.7

The sample mean of these values is $\bar{X} = 12.547$. This value is relatively close to the hypothesised population mean of 12.0, but it could also be far from it depending on the variability in the data. To make a proper decision, you need to understand the probability distribution of the sample mean.

The sample mean $\bar{X}$ is itself a random variable with its own probability distribution. An important statistical principle states that the mean of the sample mean distribution (also called the expected value of $\bar{X}$) equals the population mean $\mu$. In other words, on average, the sample mean provides an unbiased estimate of the population mean. However, any individual sample mean may differ from the population mean due to random variation.

The t-distribution

Just as there is a standard Normal distribution that other Normal distributions can be related to, there are standard t-distributions that describe the distribution of sample means for different sample sizes.

When you know the population variance $\sigma^2$, the test statistic:

$\frac{\bar{X} - \mu_0}{\sigma/\sqrt{n}}$

follows a standard Normal distribution with mean 0 and variance 1.

However, when the population variance is unknown, you must estimate it using the sample variance. This estimate becomes less reliable as the sample size decreases. The sample variance for a sample of size $n$ is calculated as:

$S^2 = \frac{1}{n-1} \sum_{i=1}^{n} (X_i - \bar{X})^2 = \frac{1}{n-1}\left(\sum_{i=1}^{n} X_i^2 - n\bar{X}^2\right)$

For the sample data shown earlier with $n = 10$:

$S^2 = \frac{1}{10-1}\left((11.9)^2 + (8.37)^2 + \ldots + (15.7)^2 - 10 \times (12.547)^2\right) = 5.56$

The t-distribution is similar in shape to the Normal distribution (bell-shaped and symmetric) but has heavier tails, meaning it assigns greater probability to extreme values. This reflects the additional uncertainty from estimating the population variance. The exact shape of the t-distribution depends on the degrees of freedom, which for a single-sample t-test equals $n - 1$.

The t-test statistic

The t-test statistic is calculated using the formula:

$t = \frac{\bar{X} - \mu_0}{S/\sqrt{n}}$

where:

• $\bar{X}$ is the sample mean
• $\mu_0$ is the hypothesised population mean (from the null hypothesis)
• $S$ is the sample standard deviation (the square root of the sample variance)
• $n$ is the sample size

The test statistic follows a t-distribution with $n - 1$ degrees of freedom.

Degrees of freedom

The degrees of freedom represent the number of independent pieces of information available to estimate the population variance. For a single-sample t-test, the degrees of freedom equal:

$\text{df} = n - 1$

Conducting a hypothesis test using the t-test

The procedure for conducting a t-test follows the standard hypothesis testing framework:

Step 1: State the hypotheses

• Null hypothesis $H_0$: $\mu = \mu_0$ (population mean equals hypothesised value)
• Alternative hypothesis $H_1$: This depends on the context (could be $\mu \neq \mu_0$, $\mu > \mu_0$, or $\mu < \mu_0$)

Step 2: Choose a significance level (e.g., 5%, 10%, 1%)

Step 3: Calculate the test statistic using $t = \frac{\bar{X} - \mu_0}{S/\sqrt{n}}$

Step 4: Find the degrees of freedom: df = $n - 1$

Step 5: Determine the p-value or critical region using the t-distribution tables or calculator

Step 6: Make a decision by comparing the p-value to the significance level, or by checking if the test statistic falls in the critical region

Step 7: State your conclusion in context

Confidence intervals using the t-distribution

The t-distribution can also be used to construct confidence intervals for the population mean when the population variance is unknown and the sample size is small.

For a sample of size $n$ taken from a Normal distribution $N(\mu, \sigma^2)$, the $p\%$-confidence interval is:

$\bar{X} - t \times \frac{s}{\sqrt{n}} < \mu < \bar{X} + t \times \frac{s}{\sqrt{n}}$

where:

• $t$ is the critical value from the t-distribution with $n-1$ degrees of freedom
• $\bar{X}$ is the sample mean
• $s$ is the sample standard deviation (or $\sqrt{S^2}$)

For a $p\%$-confidence interval, you need to find $t = t_{n-1}^{-1}\left(\frac{1+p}{2}\right)$ using your calculator's inverse t-distribution function.

When to use the t-test vs Normal test

Understanding when to use each test is crucial for exams:

Conditions for validity

For a hypothesis test using the t-distribution to give valid results, two conditions must be met:

Key formulas summary

Exam tips and common traps

Tip 1: Always check whether the variance is known or unknown. If it's unknown, use the t-test; if it's known, use the Normal distribution.

Tip 2: Don't forget to subtract 1 when calculating degrees of freedom. It's n - 1, not $n$.

Tip 3: For two-tailed tests, remember to consider both tails of the distribution when finding p-values or critical regions.

Tip 4: When stating conclusions, always write them in context. Don't just say "reject $H_0$" – explain what this means for the original problem.

Tip 5: Check validity conditions before using the t-test. If the sample isn't random or the population isn't Normal, the test results may not be valid.

Remember!