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![### Hypothesis Testing for Population Means
A hypothesis test is to be performed for a population mean with the following null hypothesis:
\[ H_0: \mu = \mu_0 \]
The test statistic used will be:
\[ z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}} \]
Find the critical value(s) for a two-tailed test with \( \alpha = 0.02 \) and draw a graph that illustrates your answer.
#### Options:
1. \(-2.05\)
This graph displays a standard normal distribution (bell curve) with a critical value at \(-2.05\). The regions to the left of \(-2.05\) are labeled "Reject \( H_0 \)". The region between \( -2.05 \) and \( 2 \) is labeled "Do not reject \( H_0 \)".
2. \(-2.05, 2.05\)
This graph displays a standard normal distribution (bell curve) with critical values at \(-2.05\) and \(2.05\). The regions to the left of \(-2.05\) and to the right of \(2.05\) are labeled "Reject \( H_0 \)". The region between \(-2.05\) and \(2.05\) is labeled "Do not reject \( H_0 \)".
In hypothesis testing, the critical values divide the distribution into regions where we decide whether to reject the null hypothesis \( H_0 \) or not. For a two-tailed test with an alpha level of \( \alpha = 0.02 \), the critical values are typically determined based on the Z-distribution. This results in critical values that establish the reject regions for the hypothesis test.
Note: Make sure to double-check the critical values using standard Z-tables or statistical software as they might vary slightly depending on the exact significance level cut-offs used.](https://content.bartleby.com/qna-images/question/cfef55d1-e74e-4acc-a65e-b6e34bf1c85d/b0446876-c570-44af-ac98-769f9f90b436/7b3hfqs_processed.jpeg)
Transcribed Image Text:### Hypothesis Testing for Population Means
A hypothesis test is to be performed for a population mean with the following null hypothesis:
\[ H_0: \mu = \mu_0 \]
The test statistic used will be:
\[ z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}} \]
Find the critical value(s) for a two-tailed test with \( \alpha = 0.02 \) and draw a graph that illustrates your answer.
#### Options:
1. \(-2.05\)
This graph displays a standard normal distribution (bell curve) with a critical value at \(-2.05\). The regions to the left of \(-2.05\) are labeled "Reject \( H_0 \)". The region between \( -2.05 \) and \( 2 \) is labeled "Do not reject \( H_0 \)".
2. \(-2.05, 2.05\)
This graph displays a standard normal distribution (bell curve) with critical values at \(-2.05\) and \(2.05\). The regions to the left of \(-2.05\) and to the right of \(2.05\) are labeled "Reject \( H_0 \)". The region between \(-2.05\) and \(2.05\) is labeled "Do not reject \( H_0 \)".
In hypothesis testing, the critical values divide the distribution into regions where we decide whether to reject the null hypothesis \( H_0 \) or not. For a two-tailed test with an alpha level of \( \alpha = 0.02 \), the critical values are typically determined based on the Z-distribution. This results in critical values that establish the reject regions for the hypothesis test.
Note: Make sure to double-check the critical values using standard Z-tables or statistical software as they might vary slightly depending on the exact significance level cut-offs used.
![## Understanding Hypothesis Testing through Critical Regions
In the context of statistical hypothesis testing, it's crucial to understand the concept of critical regions. These regions are defined by certain boundaries on the standard normal distribution curve. Let's explore two scenarios depicted in the graphs.
### Graph 1: Critical Regions Defined by \[-2.33, 2.33\]
The first graph illustrates a standard normal distribution (bell curve) divided by critical values at -2.33 and 2.33.
- **Regions outside these boundaries (i.e., less than -2.33 or greater than 2.33):** Fall into the "Reject \( H_0 \)" zone. This implies that if a test statistic falls within these regions, you will reject the null hypothesis \( H_0 \). These regions correspond to the tails of the distribution.
- **Region between these boundaries (i.e., between -2.33 and 2.33):** Labeled as the "Do not reject \( H_0 \)" zone. If a test statistic falls within this middle region, you do not reject the null hypothesis.
This graph shows a two-tailed test with a critical value at 0.01 on each tail, indicating a 1% significance level per tail (or 2% overall).
### Graph 2: Critical Regions Defined by \[-1.75, 1.75\]
The second graph shows another standard normal distribution with critical values at -1.75 and 1.75.
- **Regions outside these boundaries (i.e., less than -1.75 or greater than 1.75):** Designated as the "Reject \( H_0 \)" zone. A test statistic in these areas leads to the rejection of the null hypothesis \( H_0 \).
- **Region between these boundaries (i.e., between -1.75 and 1.75):** Labeled as the "Do not reject \( H_0 \)" zone. A test statistic here would mean you do not reject the null hypothesis.
This graph indicates a critical region at 2.5% on each tail, totaling a 5% significance level.
### Key Takeaways
- **Critical Values:** Define the boundaries of rejection regions. The specific values depend on the chosen significance level of the test.
- **Rejection Regions:** Areas where we would reject the null hypothesis. Anything outside the critical values falls into this category.
- **Non-Re](https://content.bartleby.com/qna-images/question/cfef55d1-e74e-4acc-a65e-b6e34bf1c85d/b0446876-c570-44af-ac98-769f9f90b436/6k8flub_processed.jpeg)
Transcribed Image Text:## Understanding Hypothesis Testing through Critical Regions
In the context of statistical hypothesis testing, it's crucial to understand the concept of critical regions. These regions are defined by certain boundaries on the standard normal distribution curve. Let's explore two scenarios depicted in the graphs.
### Graph 1: Critical Regions Defined by \[-2.33, 2.33\]
The first graph illustrates a standard normal distribution (bell curve) divided by critical values at -2.33 and 2.33.
- **Regions outside these boundaries (i.e., less than -2.33 or greater than 2.33):** Fall into the "Reject \( H_0 \)" zone. This implies that if a test statistic falls within these regions, you will reject the null hypothesis \( H_0 \). These regions correspond to the tails of the distribution.
- **Region between these boundaries (i.e., between -2.33 and 2.33):** Labeled as the "Do not reject \( H_0 \)" zone. If a test statistic falls within this middle region, you do not reject the null hypothesis.
This graph shows a two-tailed test with a critical value at 0.01 on each tail, indicating a 1% significance level per tail (or 2% overall).
### Graph 2: Critical Regions Defined by \[-1.75, 1.75\]
The second graph shows another standard normal distribution with critical values at -1.75 and 1.75.
- **Regions outside these boundaries (i.e., less than -1.75 or greater than 1.75):** Designated as the "Reject \( H_0 \)" zone. A test statistic in these areas leads to the rejection of the null hypothesis \( H_0 \).
- **Region between these boundaries (i.e., between -1.75 and 1.75):** Labeled as the "Do not reject \( H_0 \)" zone. A test statistic here would mean you do not reject the null hypothesis.
This graph indicates a critical region at 2.5% on each tail, totaling a 5% significance level.
### Key Takeaways
- **Critical Values:** Define the boundaries of rejection regions. The specific values depend on the chosen significance level of the test.
- **Rejection Regions:** Areas where we would reject the null hypothesis. Anything outside the critical values falls into this category.
- **Non-Re
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