Small samples Test for single mean

Definition

A small samples test for a single mean is a statistical hypothesis test used to determine whether the population mean $\mu$ equals a specified value when the sample size is small, the population standard deviation is unknown, and the data are assumed to come from a normally distributed population or approximately normal population.

The test statistic used is:

$$t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}$$

where:

• $\bar{x}$ = sample mean
• $\mu_0$ = hypothesized population mean
• $s$ = sample standard deviation
• $n$ = sample size

This statistic follows a t-distribution with $n-1$ degrees of freedom under the null hypothesis.

Main Content

1. First Concept: Hypothesis Testing for a Single Mean

• The first step in a small sample test is to clearly state the null hypothesis and alternative hypothesis.

• The null hypothesis usually states that the population mean is equal to a claimed value: $$H_0: \mu = \mu_0$$ The alternative hypothesis may be:

• Two-tailed: $H_1: \mu \ne \mu_0$

• Right-tailed: $H_1: \mu > \mu_0$
• Left-tailed: $H_1: \mu < \mu_0$

Hypothesis testing is necessary because a sample mean alone may not be enough to conclude that the population mean differs from the claimed value. Random variation can make small samples misleading, so the test measures whether the difference between the sample mean and hypothesized mean is statistically significant.

Example:
Suppose a manufacturer claims that the average weight of a chocolate bar is 100 g. A sample of 8 bars shows an average of 97 g. We need to test whether this difference is due to chance or whether the true mean weight is actually lower than 100 g.

2. Second Concept: t-Distribution and Degrees of Freedom

• For small samples, the population standard deviation is usually unknown, so the sample standard deviation is used instead.
• Because of this extra uncertainty, the test statistic does not follow the normal distribution but follows the Student’s t-distribution.

The t-distribution:

• is bell-shaped and symmetric like the normal distribution,
• has heavier tails than the normal distribution,
• depends on degrees of freedom, which for a single mean test is: $$df = n - 1$$

As the sample size increases, the t-distribution approaches the normal distribution. For very small samples, the tails are heavier to reflect the greater uncertainty in estimating the population standard deviation.

Illustration of t-distribution shape:

Density
  ^
  |                 __
  |               _/  \_
  |             _/      \_
  |           _/          \_
  |__________/______________\__________> t
             -2   0   2

Heavier tails than normal distribution

This concept is crucial because using the normal distribution instead of the t-distribution for small samples may lead to incorrect conclusions.

3. Third Concept: Assumptions and Decision Rule

• The test is valid only if certain assumptions are satisfied:
• the sample is random,
• observations are independent,
• the population is approximately normal, especially for very small samples,
• the population standard deviation is unknown.
• Once the test statistic is calculated, it is compared with the critical t-value from the t-table or evaluated using the p-value approach.

The decision rule is:

• If the test statistic falls in the rejection region, reject $H_0$.
• If the p-value is less than or equal to the significance level $\alpha$, reject $H_0$.
• Otherwise, fail to reject $H_0$.

This decision helps determine whether there is enough evidence to support the alternative hypothesis.

Example:
If a researcher tests whether the mean time taken to complete a task is 30 minutes using a sample of 6 people, the data must be reasonably normal. If the computed t-value is more extreme than the critical value, the null hypothesis is rejected.

Working / Process

1. Formulate the hypotheses

• State the null hypothesis $H_0: \mu = \mu_0$.
• State the alternative hypothesis based on the claim:
  • $H_1: \mu \ne \mu_0$
  • $H_1: \mu > \mu_0$
  • $H_1: \mu < \mu_0$

2. Choose the significance level and compute the test statistic

• Select a significance level such as $\alpha = 0.05$.
• Calculate the sample mean $\bar{x}$, sample standard deviation $s$, and sample size $n$.

• Use the formula: $$t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}$$

• Determine the degrees of freedom: $$df = n - 1$$

3. Make the decision and interpret the result

• Compare the calculated t-value with the critical t-value or use the p-value method.
• If the calculated value lies in the rejection region, reject $H_0$.
• Interpret the result in the context of the problem.

Worked example:
A teacher believes the average score of a small group of students is 75. A sample of 9 students has:

• $\bar{x} = 80$
• $s = 12$
• $n = 9$

Test whether the population mean is 75 at 5% significance.

$$t = \frac{80 - 75}{12/\sqrt{9}} = \frac{5}{4} = 1.25$$

Degrees of freedom:

$$df = 9 - 1 = 8$$

If this is a two-tailed test at $\alpha = 0.05$, compare $t = 1.25$ with the critical value from the t-table. Since 1.25 is usually smaller than the critical value, we fail to reject $H_0$. This means there is not enough evidence to conclude that the true mean differs from 75.

Advantages / Applications

• Useful when only a small amount of data is available and a reliable inference about the mean is needed.
• Commonly used in quality control, medicine, psychology, agriculture, education, and engineering where collecting large samples may be difficult or expensive.
• Helps test practical claims about averages, such as average life of a battery, average marks of students, average production weight, or average reaction time.

Summary

• The small samples test for a single mean is usually a t-test.
• It is used when the sample size is small and the population standard deviation is unknown.
• The test compares the sample mean with a hypothesized population mean using the t-distribution.

Important terms to remember

• sample mean, population mean, t-distribution, degrees of freedom, significance level, null hypothesis, alternative hypothesis.