### Chapter 7. Hypothesis Testing: Introduction to Hypothesis Testing (Critical Region Approach

### Computing the Test Statistic and Making a Decision

Once the critical region has been determined, the next step is to collect the data and calculate the *test statistic*.

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Test Statistic

**Definition**

A **test statistic** is generally composed of a ratio. The numerator of the ratio is the obtained difference between the sample statistic and the hypothesized population parameter.

The denominator of the ratio is the standard error which measures how much difference is expected by chance.

**Formula**

\[\text{test statistic}=\cfrac{\text{obtained difference}}{\text{expected difference}}\]

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If the value of the test statistic falls inside the critical region, the null hypothesis is rejected. If the test statistic does not fall inside the critical region, the null hypothesis is not rejected.

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

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z-Statistic

**Definition**

A #\boldsymbol{z}#-**statistic **is obtained by transforming the mean of a sample into a #z#-score using the parameters of the sampling distribution of the sample mean.

**Formulas**

\[z=\cfrac{\bar{X}-\mu_0}{\sigma_{\bar{X}}}\]

\[\sigma_{\bar{X}}=\cfrac{\sigma}{\sqrt{n}}\]

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4. Test Statistic Computation and Decision

After the conclusion of the Summer Course, the students are tested on their statistical knowledge. It turns out that the mean grade of the #100# students who attended the Summer Course is #\bar{X} = 7.0#.

This sample mean is then converted into a #z#-score, which will serve as the test statistic:

\[z=\dfrac{\bar{X} - \mu_0}{\sigma_{\bar{X}}} = \dfrac{7.0 - 6.5}{0.1} = \dfrac{0.5}{0.1} = 5.00\]

Since #z=5.00 \gt 1.96#, the sample mean falls inside the critical region and the null hypothesis #H_0:\mu_0=6.5# should be rejected.

The university, therefore, concludes that participating in the Summer Course has had a significant impact on the mean grade of the students.

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