### Chapter 7. Hypothesis Testing: Introduction to Hypothesis Testing (p-value Approach)

### Computing the Test Statistic

Once the hypotheses of the test have been formulated and the significance level of the test has been set, it is time to collect the sample data and compute the *test statistic*.

Test Statistic

A **test statistic **is a single numerical value that quantifies the difference between the *observed *sample data and what you would *expect to observe *if the null hypothesis of the test is true.

In general, larger test statistics are indicative of stronger evidence *against *the null hypothesis being tested.

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The test statistic used in a #Z#-test is the #Z#*-statistic*.

Z-Statistic

The #\boldsymbol{Z}#-**statistic **for a #Z#*-test for a population mean *#\mu# is obtained by transforming the sample mean #\bar{X}# into a #Z#-score:

\[Z=\cfrac{\bar{X}-\mu_0}{\sigma_{\bar{X}}} =\cfrac{\bar{X}-\mu_0}{\sigma/\sqrt{n}}\]

If the population from which the sample is drawn is normally distributed, then the sampling distribution of the #Z#-statistic is the *Standard Normal Distribution*. That is #Z\sim N(0,1)#.

If the population is not normally distributed, but the sample size is large (#n>30#), the *Central Limit Theorem *allows us to proceed as if #Z\sim N(0,1)#.

A lowercase #z# will be used to denote the measured value of #Z# after the data has been collected. The value of #z# is used to assess the strength of the evidence *against *the null hypothesis.

As the difference between the *observed* sample mean #\bar{X}# and the *hypothesized* population mean #\mu_0# increases, the value of #z# becomes larger and the evidence against the null hypothesis becomes stronger.

Consider a normally-distributed population with unknown mean #\mu# and standard deviation #\sigma = 3.1#.

Suppose a researcher has formulated the following research hypotheses about the population mean #\mu#:

- #H_0:\mu=15#
- #H_a:\mu \neq 15#

Calculate the #Z#-statistic for this sample. Round your answer to #3# decimal places.

The #Z#-statistic is calculated with the following formula:

\[z = \cfrac{\bar{X}-\mu_0}{\sigma/\sqrt{n}} =\cfrac{13.5-15}{0.313} = -4.790\]

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