27.4 The test statistic and $t$-scores: Observation

The sampling distributions describes what to expect from the sample mean, assuming $\mu ={37.0}^{\circ }\text{C}$. The value of $\overline{x}$ that is observed, however, is $\overline{x}={36.8051}^{\circ }$ How likely is it that such a value could occur by chance?

The value of the observed sample mean can be located the picture of the sampling distribution (Fig. 27.3). The value $\overline{x}={36.8051}^{\circ }\text{C}$ is unusually small. About how many standard deviations is $\overline{x}$ away from $\mu =37$? A lot…

FIGURE 27.3: The sample mean of $\overline{x}={36.8041}^{\circ }$C is very unlikely to have been observed if the poulation mean really was ${37}^{\circ }$C, and $n=130$

Relatively speaking, the distance that the observed sample mean (of $\overline{x}=36.8051$) is from the mean of the sampling distribution (Fig. 27.3). is found by computing how many standard deviations the value of $\overline{x}$ is from the mean of the distribution; that is, computing something like a $z$-score. (Remember that the standard deviation in Fig. 27.3 is the the standard error: the amount of variation in the sample means.)

Since the mean and standard deviation (i.e. the standard error) of this normal distribution are known, the number of standard deviations that $\overline{x}=36.8051$ is from the mean is

$$\frac{36.8051-37.0}{0.035724}=-\mathrm{5.453.}$$ This value is like a $z$-score. However, this is actually called a $t$-score because it has been computed when the population standard deviation is unknown, and the best estimate (the sample standard deviation) is used when $\text{s.e.}\left(\overline{x}\right)$ was computed.

Both $t$ and $z$ scores measure the number of standard deviations that an observation is from the mean: $z$-scores use $\sigma$ and $t$-scores use $s$. Here, the distribution of the sample statistic is relevant, so the appropriate standard deviation is the standard deviation of the sampling distribution: the standard error.

Like $z$-scores, $t$-scores measure the number of standard deviations that an observation is from the mean. $z$-scores are calculated using the population standard deviation, and $t$-scores are calculated using the sample standard deviation.

In hypothesis testing, $t$-scores are more commonly used than $z$-scores, because almost always the population standard deviation is unknown, and the sample standard deviation is used instead.

In this course, it is sufficient to think of $z$-scores and $t$-scores as approximately the same. Unless sample sizes are small, this is a reasonable approximation.

So the calculation is:

$$t=\frac{36.8051-37.0}{0.035724}=-5.453;$$ the observed sample mean is more than five standard deviation below the population mean. This is highly unusual based on the 68–95–99.7 rule, as seen in Fig. 27.3.

In general, a $t$-score in hypothesis testing is

$$\begin{array}{}t=\frac{\text{sample statistic}-\text{assumed population parameter}}{\text{standard error of the sample statistic}}.& \text{(27.1)}\end{array}$$