Question

What assumptions are made when Student's \(t\) -test is used to test a hypothesis concerning a population mean?

Short answer

Answer: The key assumptions made when using the Student's t-test are: (1) normal distribution of the population, (2) independent observations, (3) random sampling, (4) equal variances for two-sample t-test, and (5) ratio or interval scale data.

Step-by-step solution

Assumption 1: Normal Distribution

The first assumption is that the population from which the sample is taken follows a normal distribution. In other words, the variable being tested should be normally distributed in the population.

Assumption 2: Independent Observations

The second assumption is that the observations in the sample are independent of each other. This means that the outcome of one observation does not affect the outcome of any other observation.

Assumption 3: Random Sampling

The third assumption is that the sample is randomly selected from the population. This ensures that every member of the population has an equal chance of being included in the sample, minimizing any selection bias.

Assumption 4: Equal Variances (for Two-sample t-test)

When conducting a two-sample t-test, an additional assumption must be made. The assumption is that the variances of the two populations being compared are equal. If this assumption is violated, alternative versions of the t-test, such as the Welch's t-test, can be used instead.

Assumption 5: Ratio Scale or Interval Scale Data

The last assumption is that the data being tested in the t-test must be measured on a ratio scale or an interval scale. This means that the data must have a clear, numerical ordering and the differences between the data points can be meaningfully measured. In summary, there are several key assumptions made when using the Student's t-test to test a hypothesis concerning a population mean: normal distribution of the population, independent observations, random sampling, equal variances for two-sample t-test, and ratio or interval scale data.

Key concepts

Normal Distribution

One of the fundamental assumptions of the Student's t-test is that the data come from a population that follows a normal distribution. This assumption is crucial because the t-test relies on the normality of data to accurately calculate probabilities and critical values.

Why is normal distribution important? Statistical tests assume the bell-shaped curve, which represents how data naturally distributes around the mean. It helps in determining the probability of a sample mean by referencing this shape.

In practice, small deviations from normality are often acceptable, especially with larger sample sizes due to the central limit theorem, which states that the mean of a sufficiently large sample will approximate a normal distribution. However, when data is heavily skewed, it may be necessary to transform them or consider non-parametric alternatives to the t-test.

Independent Observations

The assumption of independent observations means each data point in a sample doesn't influence or relate to another. Outcomes should not affect one another, ensuring that each provides unique information about the population.

Why is this crucial? If observations aren't independent, it may inflate or deflate the variability, leading to unreliable t-test results. For instance, measurements taken from the same person over time would violate independence.

Ensuring independence often involves careful experimental design, such as randomizing the order of treatments or the selection of subjects, to prevent systemic bias.

Random Sampling

Random sampling is a key assumption to ensure that every individual in the population has an equal chance of being chosen. This creates a sample that is representative of the population, reducing bias.

Why does it matter? A non-random sample may not reflect the true characteristics of the population, thereby compromising the validity of the test results.

Various methods, like simple random sampling, stratified sampling, or systematic sampling, can be used to achieve random sampling. Proper attention to this step increases the generalizability of the findings.

Equal Variances

When conducting a two-sample t-test, it is assumed that both samples come from populations with equal variances. This means that the spread of data should be similar in both groups.

Implication of this assumption: Unequal variances might lead to incorrect test results, as the t-distribution used in calculations assumes equal spread. In cases of unequal variances, the Welch's t-test, which doesn’t assume equal population variances, can be used as an alternative.

Checking and ensuring this assumption typically involves preliminary tests like Levene's test or analyzing box plots.

Measurement Scales

The Student's t-test requires data measured on either the ratio scale or interval scale. These scales imply that data is not only ordered but also that the intervals between data points are meaningful, allowing for precise calculations.

Why are these scales needed? They facilitate the arithmetic operations necessary in statistical analysis, such as subtraction and division, which underpin hypothesis testing.

For example, temperature (in Celsius or Fahrenheit) and weight are typical examples of interval and ratio scales, respectively. Without these scales, the mathematical integrity of differences between values may not be preserved, making the test unsuitable.