# 8.2 - One Sample Mean

One sample mean tests are covered in Section 6.2 of the Lock5 textbook.

Concerning one sample mean, the Central Limit Theorem states that if the sample size is large, then the distribution of sample means will be approximately normally distributed with a standard deviation (i.e., standard error) equal to $$\frac{\sigma}{\sqrt n}$$. In this course, a "large" sample size will be defined as one where $$n \ge 30$$.

When constructing confidence interval and conducting hypothesis tests we often do not know the value of $$\sigma$$. In those cases, $$\sigma$$ may be estimated using the sample standard deviation ($$s$$). When we are using $$s$$ to estimate $$\sigma$$ our sampling distribution will not follow a $$z$$ distribution exactly.  Instead, we use what is known as the $$t$$ distribution.  Like the $$z$$ distribution, the $$t$$ distribution is symmetrical. The difference is that its height varies depending on the sample size. By doing so, the distribution becomes more conservative for smaller sample sizes to account for some error that may occur from estimating $$\sigma$$ with $$s$$ from a small sample. As $$n$$ approaches infinity ($$\infty$$) the $$t$$ distribution approaches the standard normal distribution. The next page compares the $$z$$ and $$t$$ distributions.

When constructing confidence intervals and conducting hypothesis tests we will usually be using the $$t$$ distribution when working with one mean. The only exception would be in cases where $$\sigma$$ is known. This scenario is most common in the fields of education and psychology where some tests are normed to have a certain $$\mu$$ and $$\sigma$$. In those cases, the $$z$$ distribution can be used.

In terms of language, all of these tests could be called "single sample mean tests" or "one sample mean tests."  We could also specify the sampling distribution by using the term "single sample mean $$t$$ test" or "single sample mean $$z$$ test."

The flow chart below may help you in determining which method should be used when constructing a sampling distribution for one sample mean.

#### One Sample Mean

Identify when z and t distributions should be used.

• Is the population known to be normally distributed?
• Is the population standard deviation known?
• Is the sample size at least 30?