You want O determine whether the reasons given by workers for continuing their education s related to job type. In the study, you randomly collect the data shown in the contingency table. At x=0.05, can you conclude that the reason and type of worker are dependent? Complete parts (a) through (d).
You want O determine whether the reasons given by workers for continuing their education s related to job type. In the study, you randomly collect the data shown in the contingency table. At x=0.05, can you conclude that the reason and type of worker are dependent? Complete parts (a) through (d).
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter10: Statistics
Section10.5: Comparing Sets Of Data
Problem 3BGP
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Question
![**Hypothesis Testing: Independence of Reasons for Continuing Education and Job Type**
---
**Problem Explanation:**
You want to determine whether the reasons given by workers for continuing their education are related to their job type. In the study, you randomly collect the data shown in the contingency table. At \( \alpha = 0.05 \), can you conclude that the reason and type of worker are dependent? Complete parts (a) through (d).
*Contingency Table:*
| | **Reason** |
|-----------------|--------------------|
| **Type of Worker** | *Professional* | *Personal* | *Both* |
| Technical | 36 | 34 | 33 |
| Other | 40 | 23 | 37 |
---
**Part (a) Identify the Claim and State the Null and Alternative Hypotheses**
\( H_0 \): Reasons are \( \boxed{\text{independent of}} \) the type of worker.
\( H_a \): Reasons are \( \boxed{\text{dependent on}} \) the type of worker.
The \( \boxed{\text{alternative hypothesis}} \) is the claim.
---
**Part (b) Determine the Degrees of Freedom, Find the Critical Value, and Identify the Rejection Region**
*Degrees of Freedom:*
\[ \text{d.f.} = \boxed{2} \]
*Find the Critical Value:*
\[ \chi^2_0 = \boxed{5.991} \]
(Round to three decimal places as needed)
By completing these steps, you will be able to test if the reasons for continuing education are independent of the job type at the 0.05 significance level.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fac4255ba-2787-4b8b-b63d-664d949df058%2F01b9d939-416d-4759-a46f-ab2691baa038%2Fweb7ls_processed.png&w=3840&q=75)
Transcribed Image Text:**Hypothesis Testing: Independence of Reasons for Continuing Education and Job Type**
---
**Problem Explanation:**
You want to determine whether the reasons given by workers for continuing their education are related to their job type. In the study, you randomly collect the data shown in the contingency table. At \( \alpha = 0.05 \), can you conclude that the reason and type of worker are dependent? Complete parts (a) through (d).
*Contingency Table:*
| | **Reason** |
|-----------------|--------------------|
| **Type of Worker** | *Professional* | *Personal* | *Both* |
| Technical | 36 | 34 | 33 |
| Other | 40 | 23 | 37 |
---
**Part (a) Identify the Claim and State the Null and Alternative Hypotheses**
\( H_0 \): Reasons are \( \boxed{\text{independent of}} \) the type of worker.
\( H_a \): Reasons are \( \boxed{\text{dependent on}} \) the type of worker.
The \( \boxed{\text{alternative hypothesis}} \) is the claim.
---
**Part (b) Determine the Degrees of Freedom, Find the Critical Value, and Identify the Rejection Region**
*Degrees of Freedom:*
\[ \text{d.f.} = \boxed{2} \]
*Find the Critical Value:*
\[ \chi^2_0 = \boxed{5.991} \]
(Round to three decimal places as needed)
By completing these steps, you will be able to test if the reasons for continuing education are independent of the job type at the 0.05 significance level.
![**Hypothesis Testing for Independence**
Choose the correct rejection region below:
**A.** \( \chi^2 < \chi^2_0 \)
**B.** \( \chi^2 > \chi^2_0 \)
**C.** \( \chi^2 \geq \chi^2_0 \)
**D.** \( \chi^2 \leq \chi^2_0 \)
**(c)** Calculate the test statistic. If convenient, use technology.
\[ \chi^2 = \boxed{\phantom{000}} \]
*(Round to three decimal places as needed.)*
**(d)** Decide to reject or fail to reject the null hypothesis. Can you conclude that the reason and type of worker are dependent? Choose the correct conclusion below.
**A.** Fail to reject the null hypothesis. There is not enough evidence to conclude that reasons are dependent on the type of worker.
**B.** Reject the null hypothesis. There is not enough evidence to conclude that reasons are dependent on the type of worker.
**C.** Fail to reject the null hypothesis. There is enough evidence to conclude that reasons are dependent on the type of worker.
**D.** Reject the null hypothesis. There is enough evidence to conclude that reasons are dependent on the type of worker.
---
**Explanation:**
In this exercise, we are conducting a chi-square test to determine if there is a dependency between two categorical variables: the reason and type of worker. The steps for this analysis include:
1. Choosing the correct rejection region based on the chi-square distribution.
2. Calculating the chi-square test statistic using the provided data. This value will be essential in determining whether to reject or fail to reject the null hypothesis.
3. Interpreting the test result to draw a conclusion about the independence or dependence of the two categorical variables under study.
These steps are fundamental to understanding the relationships in categorical data and are frequently applied in various fields such as social sciences, business, and healthcare for data-driven decision-making.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fac4255ba-2787-4b8b-b63d-664d949df058%2F01b9d939-416d-4759-a46f-ab2691baa038%2Fqlijsa_processed.png&w=3840&q=75)
Transcribed Image Text:**Hypothesis Testing for Independence**
Choose the correct rejection region below:
**A.** \( \chi^2 < \chi^2_0 \)
**B.** \( \chi^2 > \chi^2_0 \)
**C.** \( \chi^2 \geq \chi^2_0 \)
**D.** \( \chi^2 \leq \chi^2_0 \)
**(c)** Calculate the test statistic. If convenient, use technology.
\[ \chi^2 = \boxed{\phantom{000}} \]
*(Round to three decimal places as needed.)*
**(d)** Decide to reject or fail to reject the null hypothesis. Can you conclude that the reason and type of worker are dependent? Choose the correct conclusion below.
**A.** Fail to reject the null hypothesis. There is not enough evidence to conclude that reasons are dependent on the type of worker.
**B.** Reject the null hypothesis. There is not enough evidence to conclude that reasons are dependent on the type of worker.
**C.** Fail to reject the null hypothesis. There is enough evidence to conclude that reasons are dependent on the type of worker.
**D.** Reject the null hypothesis. There is enough evidence to conclude that reasons are dependent on the type of worker.
---
**Explanation:**
In this exercise, we are conducting a chi-square test to determine if there is a dependency between two categorical variables: the reason and type of worker. The steps for this analysis include:
1. Choosing the correct rejection region based on the chi-square distribution.
2. Calculating the chi-square test statistic using the provided data. This value will be essential in determining whether to reject or fail to reject the null hypothesis.
3. Interpreting the test result to draw a conclusion about the independence or dependence of the two categorical variables under study.
These steps are fundamental to understanding the relationships in categorical data and are frequently applied in various fields such as social sciences, business, and healthcare for data-driven decision-making.
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