4 ### Understanding Alpha Level and Critical Regions in Hypothesis TestingIn hypothesis testing, the alpha level \(\alpha\) is set to determine the probability threshold for rejecting the null hypothesis. Using the Distributions tool, we can identify which samples lie in the extreme ends and are less consistent with the null hypothesis. It is assumed to be nondirectional, meaning the critical region is split between both tails of the distribution.#### Z-Score BoundariesFor an alpha level \(\alpha = 0.05\), the z-score boundaries are:- \(Z = 3.29\) and \(Z = -3.29\)- \(Z = 2.58\) and \(Z = -2.58\)- **\(Z = 1.96\) and \(Z = -1.96\)** (This is typically the correct z-score for \(\alpha = 0.05\))#### Using the Distribution Tool- To find z-score boundaries: 1. Click the icon featuring two orange lines. 2. Adjust the orange lines so that the area within the critical region matches the alpha level. The probability will be split evenly between the tails. - To evaluate the hypothesis: 1. Use the icon with the purple line. 2. Position the orange lines on the critical values. 3. Align the purple line with the z statistic.#### Diagram ExplanationThe diagram depicts a **Standard Normal Distribution** with:- Mean = 0.0- Standard Deviation = 1.0The shaded areas on both tails (colored orange) illustrate the critical regions at the specified z-scores, highlighting extreme values in the distribution. The critical values determine where the distribution splits based on selected alpha levels. #### Alpha Level \(\alpha = 0.001\)- For stricter levels like \(\alpha = 0.001\), similar steps apply to identify tighter z-score boundaries. Through this process, hypothesis testing becomes a methodical procedure to determine if data supports or contradicts a given claim based on statistical evidence. # Hypothesis Testing Using Z-Scores## Understanding Z-Scores and Critical RegionsTo use the tool to identify the z-score boundaries, click on the icon with two orange lines, and slide the orange lines until the area in the critical region equals the alpha level. Remember that the probability will need to be split between the two tails.To use the tool to help you evaluate the hypothesis, click on the icon with the purple line, place the two orange lines on the critical values, and then place the purple line on the z statistic.### Visual Representation#### Graph: Standard Normal Distribution- **Description**: The graph shows a standard normal distribution with a mean of 0 and a standard deviation of 1. - **Elements**: - The distribution is symmetrical around the mean of 0. - The area under the curve is highlighted, with: - Two orange sections representing the tails of the distribution. - A central blue section denoting the non-critical region. - The distribution is marked with z-scores: - 0.67 and -0.67 delineate the boundaries for the central 0.5000 probability, with 0.2500 in each tail.### Identifying Critical Regions**The critical region is**:- [ ] the area between the z-scores - [ ] the area in the tails beyond each z-score**The z-score boundaries**:- [ ] \( z = 1.96 \) and \( z = -1.96 \)- [ ] \( z = 2.58 \) and \( z = -2.58 \)- [ ] \( z = 3.29 \) and \( z = -3.29 \)### Hypothesis Testing ExampleSuppose that the calculated z statistic for a particular hypothesis test is 4.12 and the alpha is 0.001. This z statistic is \( \text{______} \) the critical region. Therefore, the researcher \( \text{______} \) reject the null hypothesis and \( \text{______} \) conclude the alternative hypothesis is probably correct.

a) Given : Significance level = 0.05 The Critical region is the area in the tails beyond each Z…

the 2-score boundaries, click on the icon with two orange lines, and slide the orange lines until the area in the critical region equals the alpha level. Remember that the probability will need to be split between the two tails. To use the tool to help you evaluate the hypothesis, click on the icon with the purple line, place the two orange lines on the critical values, and then place the purple line on the z statistic. Standard Normal Distribution Mean 00 Standard Devistion = 1.0 .5000 2500 2500 -3 -1 -0.67 0.67 The critical region is the area between the z-scores The z-score boundar the area in the tails beyond each z-score O z = 1.96 al O z = 2.58 and z = -2.58 O z = 3.29 and z = -3.29 Suppose that the calculated z statistic for a particular hypothesis test is 4.12 and the alpha is 0.001. This z statistic is the critical region Therefore, the researcher v reject the null hypothesis, and he conclude the alternative hypothesis is probably correct.