The *t-*distribution is a relative of the normal distribution. It has a bell shape with values more spread out around the middle. That is, it's not as sharply curved as the normal distribution, which reflects its ability to work with problems that may not be exactly normal but are close.

Solve the following problems about the *t-*distribution, its traits, and how it compares to the *Z-*distribution.

## Sample questions

Which of the following is true of the

*t-*distribution, as compared to the*Z-*distribution? (Assume a low number of degrees of freedom.)(A) The

*t-*distribution has thicker tails than the*Z-*distribution.(B) The

*t-*distribution has a proportionately larger standard deviation than the*Z-*distribution.(C) The

*t-*distribution is bell-shaped but has a lower peak than the*Z-*distribution.(D) Choices (A) and (C)

(E) Choices (A), (B), and (C)

**Answer: E.**Choices (A), (B), and (C) (The*t-*distribution has thicker tails than the*Z-*distribution; the*t-*distribution has a proportionately larger standard deviation than the*Z-*distribution; the*t-*distribution is bell-shaped but has a lower peak than the*Z-*distribution.)Compared to the

*Z-*distribution, the*t-*distribution has thicker tails and a proportionately larger standard deviation. It's still bell-shaped, but it has a lower peak than the*Z-*distribution.Which

*t-*distribution do you use for a study involving one population with a sample size of 30?**Answer:***t*_{29}A

*t-*distribution for a study with one population with a sample size of 30 has*n –*1 = 30 – 1 = 29 degrees of freedom, so the correct distribution is*t*_{29}.If you graphed a standard normal distribution (

*Z-*distribution) on the same number line as a*t-*distribution with 15 degrees of freedom, how would you expect them to differ?**Answer:**The peak of the*Z-*distribution would be higher, and the t-distribution would have thicker tails.In general, the

*t-*distribution is bell-shaped but is flatter and has a lower peak than the standard normal (*Z-*) distribution, particularly with smaller degrees of freedom for the*t-*distribution.Given different

*t-*distributions with the following degrees of freedom, which one would you expect to most closely resemble the*Z-*distribution: 5, 10, 20, 30, or 100?**Answer:**100As the degrees of freedom increase, the

*t-*distribution tends to look more like the*Z-*distribution. So the*t-*distribution with the highest degrees of freedom most resembles the*Z-*distribution.

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