### Sampling Distributions and Type I Error#### b. Which sketch below shows the sampling distribution for the rejection region $ z < -1.645 $?- **Option A:** - Description: This graph displays a standard normal distribution with two tails, both extending beyond $ |z| = 1.645 $ on either side (left: $ z = -1.645 $ and right: $ z = 1.645 $). Both tails have shaded regions, each labeled $\frac{\alpha}{2}$, with rejection regions indicating $ H_0 $ rejection.- **Option B (Correct):** - Description: This sketch illustrates a standard normal distribution with a single tail extending beyond $ z = -1.645 $. The left tail is shaded and labeled $\alpha$, representing the rejection region where $ H_0 $ is rejected. Only one side of the distribution (left) is considered for the rejection.#### What is the probability that a Type I error will be made for $ z < -1.645 $?- **Question:** - Calculate the significance level ($\alpha$) that corresponds to $ z < -1.645 $.- **Answer Input Box:** - A blank input box is provided for entering the value of $\alpha$, to be rounded to three decimal places as needed.### Detailed Explanation#### Standard Normal Distribution Graphs- **Graph A:** Represents a two-tailed test scenario. - **Sections:** - **Left Tail** ($ z < -1.645 $): Shaded region marking $\frac{\alpha}{2}$. - **Right Tail** ($ z > 1.645 $): Shaded region marking $\frac{\alpha}{2}$. - This structure implies that the null hypothesis ($ H_0 $) can be rejected if the test statistic falls in either tail region.- **Graph B:** Represents the correct one-tailed test scenario for $ z < -1.645 $. - **Section:** - **Left Tail** ($ z < -1.645 $): Shaded region marking $\alpha$. The null hypothesis ($ H_0 $) is rejected if the test statistic is less than $-1.645$.### Conclusion- **Selection**: Sketch B correctly illustrates the sampling distribution for the rejection region

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Description: ### Sampling Distributions and Type I Error#### b. Which sketch below shows the sampling distribution for the rejection region $ z < -1.645 $?- **Option A:** - Description: This graph displays a standard normal distribution with two tails, both e

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1.645? B. 2. -1.645 0. 1.645 -1.645 Reject Ho Reject Ho Reject Ho What is the probability that a Type I error will be made for z< - 1.645? (Round to three decimal places as needed.)

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Concept explainers

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.

Topic Video

Question

### Sampling Distributions and Type I Error

#### b. Which sketch below shows the sampling distribution for the rejection region $ z < -1.645 $?

- **Option A:**
- Description: This graph displays a standard normal distribution with two tails, both extending beyond $ |z| = 1.645 $ on either side (left: $ z = -1.645 $ and right: $ z = 1.645 $). Both tails have shaded regions, each labeled $\frac{\alpha}{2}$, with rejection regions indicating $ H_0 $ rejection.

- **Option B (Correct):**
- Description: This sketch illustrates a standard normal distribution with a single tail extending beyond $ z = -1.645 $. The left tail is shaded and labeled $\alpha$, representing the rejection region where $ H_0 $ is rejected. Only one side of the distribution (left) is considered for the rejection.

#### What is the probability that a Type I error will be made for $ z < -1.645 $?

- **Question:**
- Calculate the significance level ($\alpha$) that corresponds to $ z < -1.645 $.

- **Answer Input Box:**
- A blank input box is provided for entering the value of $\alpha$, to be rounded to three decimal places as needed.

### Detailed Explanation

#### Standard Normal Distribution Graphs

- **Graph A:** Represents a two-tailed test scenario.
- **Sections:**
- **Left Tail** ($ z < -1.645 $): Shaded region marking $\frac{\alpha}{2}$.
- **Right Tail** ($ z > 1.645 $): Shaded region marking $\frac{\alpha}{2}$.
- This structure implies that the null hypothesis ($ H_0 $) can be rejected if the test statistic falls in either tail region.

- **Graph B:** Represents the correct one-tailed test scenario for $ z < -1.645 $.
- **Section:**
- **Left Tail** ($ z < -1.645 $): Shaded region marking $\alpha$. The null hypothesis ($ H_0 $) is rejected if the test statistic is less than $-1.645$.

### Conclusion

- **Selection**: Sketch B correctly illustrates the sampling distribution for the rejection region

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