Question

A) Suppose we were not sure if the distribution of a population was normal. In which of the following circumstances would we NOT be safe using a t procedure?

A. A histogram of the data shows moderate skewness.
B. The mean and median of the data are nearly equal.
C. A stemplot of the data has a large outlier.
D. The sample standard deviation is large.

(b) Which of the following is an example of a matched pairs design?

A. A teacher compares the pre-test and post-test scores of students.
B. A teacher calculates the average scores of students on a pair of tests and wishes to see if this average is larger than 80%.
C. A teacher compares the scores of students using a computer based method of instruction with scores of other students using a traditional method of instruction.
D. A teacher compares the scores of students in her class on a standardized test with the national average score.

(c) You are thinking of using a t procedure to test hypotheses about the mean of a population using a significance level of 0.05. You suspect the distribution of the population is not normal and may be moderately skewed. Which of the following statements is correct?

A. You may use the t procedure, but you should probably only claim the significance level is 0.10.
B. You may not use the t procedure. t procedures are robust to nonnormality for confidence intervals but not for tests of hypotheses.
C. You may use the t procedure, provided your sample size is large, say at least fifty.
D. You should not use the t procedure because the population does not have a normal distribution.

Answer 1

1. Answer: C) stemplot with a large outlier

The large outlier will skew the data to make it nonnormal. A moderate skew isn't as bad as a large one, so choice A is a fairly safer choice to go with in terms of using the T procedure. The size of the standard deviation does not affect the normality of the distribution.

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2. Answer: A) Teacher comparing pre-test and post-test results

A matched pairs design is basically a before and after sort of thing, most of the time anyway. Each person's pre-test score is paired up with their post-test score to see if any significant improvement has been made. This is also known as a paired T test

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3. Answer: C) You may use the t procedure, provided your sample size is large, say at least fifty.

Some textbooks say n > 30 is sufficient. It will depend on the textbook and your teacher of course. The idea is that if n is large enough, then by the CLT (central limit theorem), the distribution of sample means will be approximately normal, which allows you to use the T procedure and T distribution even if the underlying distribution may not be normal and/or skewed.