Question

Explain what one-sample hypothesis testing is used for and why it is used

Provide three specific examples of situations in which a person within a business setting would use hypothesis tests and why they should be used in those situations

Explain what two-sample hypothesis testing is used for and why it is used

Provide three specific examples of situations in which a person within a business setting would use hypothesis tests and why they should be used in those situations

Answer 1

Home | Academic Solutions | Directory of Statistical Analyses | (M)ANOVA Analysis | One Sample T-Test

One Sample T-Test

The one sample t-test is a statistical procedure used to determine whether a sample of observations could have been generated by a process with a specific mean. Suppose you are interested in determining whether an assembly line produces laptop computers that weigh five pounds. To test this hypothesis, you could collect a sample of laptop computers from the assembly line, measure their weights, and compare the sample with a value of five using a one-sample t-test.

Hypotheses

There are two kinds of hypotheses for a one sample t-test, the null hypothesis and the alternative hypothesis. The alternative hypothesis assumes that some difference exists between the true mean (μ) and the comparison value (m0), whereas the null hypothesis assumes that no difference exists. The purpose of the one sample t-test is to determine if the null hypothesis should be rejected, given the sample data. The alternative hypothesis can assume one of three forms depending on the question being asked. If the goal is to measure any difference, regardless of direction, a two-tailed hypothesis is used. If the direction of the difference between the sample mean and the comparison value matters, either an upper-tailed or lower-tailed hypothesis is used. The null hypothesis remains the same for each type of one sample t-test. The hypotheses are formally defined below:

• • The null hypothesis (H0) assumes that the difference between the true mean (μ) and the comparison value (m0) is equal to zero.
• • The two-tailed alternative hypothesis (H1) assumes that the difference between the true mean (μ) and the comparison value (m0) is not equal to zero.
• • The upper-tailed alternative hypothesis (H1) assumes that the true mean (μ) of the sample is greater than the comparison value (m0).
• • The lower-tailed alternative hypothesis (H1) assumes that the true mean (μ) of the sample is less than the comparison value (m0).

The mathematical representations of the null and alternative hypotheses are defined below:

• H0: μ = m0
• H1: μ ≠ m0    (two-tailed)
• H1: μ > m0    (upper-tailed)
• H1: μ < m0    (lower-tailed)

Note. It is important to remember that hypotheses are never about data, they are about the processes which produce the data. If you are interested in knowing whether the mean weight of a sample of laptops is equal to five pounds, the real question being asked is whether the process that produced those laptops has a mean of five.

Assumptions

As a parametric procedure (a procedure which estimates unknown parameters), the one sample t-test makes several assumptions. Although t-tests are quite robust, it is good practice to evaluate the degree of deviation from these assumptions in order to assess the quality of the results. The one sample t-test has four main assumptions:

• • The dependent variable must be continuous (interval/ratio).
• • The observations are independent of one another.
• • The dependent variable should be approximately normally distributed.
• • The dependent variable should not contain any outliers.

Level of Measurement

The one sample t-test requires the sample data to be numeric and continuous, as it is based on the normal distribution. Continuous data can take on any value within a range (income, height, weight, etc.). The opposite of continuous data is discrete data, which can only take on a few values (Low, Medium, High, etc.). Occasionally, discrete data can be used to approximate a continuous scale, such as with Likert-type scales.

Independence

Independence of observations is usually not testable, but can be reasonably assumed if the data collection process was random without replacement. In our example, we would want to select laptop computers at random, compared to using any systematic pattern. This ensures minimal risk of collecting a biased sample that would yield inaccurate results.

Normality

To test the assumption of normality, a variety of methods are available, but the simplest is to inspect the data visually using a histogram or a Q-Q scatterplot. Real-world data are almost never perfectly normal, so this assumption can be considered reasonably met if the shape looks approximately symmetric and bell-shaped. The data in the example figure below is approximately normally distributed.