8.2 - One Sample Mean

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One sample mean tests are covered in section 6.2 of the Lock^5 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 \(\frac{\sigma}{\sqrt n}\)

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 height of the t distribution is determined by the number of degrees of freedom. For a one sample mean test the degrees of freedom (df) is \(n-1\).

Below are a few plots comparing the standard normal distribution to t distributions with various sample sizes. In each plot, the solid black line is the standard normal distribution; this does not change regardless of the sample size. The dashed red line is the t distribution which does change depending on the sample size.

In the first graph, if \(n=3\) then \(df=2\). Here, the tails of the t distribution are higher than the tails of the normal distribution.

Plot comparing z and t for df=2

If you think about the area under the curve, the higher tails mean that more area will fall in the tails. For example, as seen in the following two plots, for \(n=3\), \(P(z>2.00)=.02275\) while \(P(t_{df=2}>2.00)=.09175\).

Plot of area above z=2.00

Plot of area above t=2.00

The next plot compares the t distribution and standard normal distribution for \(n=11\). Notice that the two distributions are becoming more similar as the sample size increases.

Plot comparing z and t for df=10

The next plot compares the t distribution and standard normal distribution for \(n=30\). 

Plot comparing z and t for df=29

In the final graph, we see \(df=500\). Here, the t distribution and standard normal distribution are nearly identical. This is because as n approaches infinity, the t distribution approaches the standard normal distribution.

Plot comparing z and t for df=500

 

When constructing confidence intervals and conducting hypothesis tests we will usually be using the t distribution. 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 use.

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. 

Identify when z and t distributions should be used, and randomization methods