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Chapter 10: Problem 10

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

Short Answer

Expert verified
Answer: The assumptions made when using the Student's t-test include: 1. Normality: The data follows a normal distribution. 2. Independence: The observations in the sample are independent of each other. 3. Homoscedasticity: The population variances of the two groups being compared are equal. 4. Measurement scale: The data is measured on an interval or ratio scale. 5. Random sampling: The samples have been drawn using random sampling techniques.

Step by step solution

01

Assumption 1: Normality

The first assumption of the Student's t-test is that the data follows a normal distribution. This means that the underlying population from which the sample is drawn should be normally distributed. The t-test is robust to violations of this assumption, especially when the sample size is large, but it's a good idea to use alternative tests, such as non-parametric tests, if the data is significantly non-normal.
02

Assumption 2: Independence

The second assumption of the Student's t-test is that the observations in the sample are independent of each other. In other words, the value of one observation does not influence or is not influenced by the value of another observation. This can be ensured by using a simple random sampling method when selecting the sample.
03

Assumption 3: Homoscedasticity

The third assumption of the Student's t-test is that the population variances of the two groups being compared are equal, also known as the assumption of homoscedasticity. When this assumption is not met, it's recommended to use alternative tests, like Welch's t-test, which do not assume equal variances.
04

Assumption 4: Measurement scale

The fourth assumption of the Student's t-test is that the data is measured on an interval or ratio scale. This means that the data must be quantitative or continuous, rather than qualitative or categorical. The t-test is not suitable for ordinal or nominal scale data.
05

Assumption 5: Random sampling

The last assumption of the Student's t-test is that the samples have been drawn using random sampling techniques. This ensures that any observed differences between the groups are not due to sampling bias or uncontrolled variables. Using random sampling helps to ensure the generalizability of the results.

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Most popular questions from this chapter

Industrial wastes and sewage dumped into our rivers and streams absorb oxygen and thereby reduce the amount of dissolved oxygen available for fish and other forms of aquatic life. One state agency requires a minimum of 5 parts per million (ppm) of dissolved oxygen in order for the oxygen content to be sufficient to support aquatic life. Six water specimens taken from a river at a specific location during the low-water season (July) gave readings of \(4.9,5.1,4.9,5.0,5.0,\) and \(4.7 \mathrm{ppm}\) of dissolved oxygen. Do the data provide sufficient evidence to indicate that the dissolved oxygen content is less than 5 ppm? Test using \(\alpha=.05 .\)

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Use the Student's \(t\) Probabilities applet to find the following critical values: a. an upper one-tailed rejection region with \(\alpha=.05\) and 11 df b. a two-tailed rejection region with \(\alpha=.05\) and 7 df c. a lower one-tailed rejection region with \(\alpha=.01\) and \(15 d f\)
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