In a trial of a new allergy medicine, 75 people were given the medicine and 75 were given a placebo. Of those given the medicine, 50 showed improvement in their allergy symptoms. Of those given the placebo, 55 did not show improvement. Complete parts (a) through (d) below.**a. Make a two-way table summarizing the reviews:**| | Improvement | No Improvement ||----------------|-------------|----------------|| Medicine | 50 | 25 || Placebo | 20 | 55 |**b. Make a Venn diagram from the table in part (a). Choose the correct answer below.**[The correct choice is Image C.]**Explanation of Venn Diagram (Correct Image C):**- The Venn diagram has two circles, labeled "Medicine" and "Improve".- "Medicine" circle contains the numbers 25 and 50. - 25 represents the participants who did not improve with medicine. - 50 represents the participants who improved with medicine (overlapping with "Improve" circle).- "Improve" circle contains the number 20 within it and in the overlap: - 20 represents the participants who improved with placebo (overlapping with "Improve" circle).**c. How many people who received the placebo did not improve?**Answer: 55**d. How many people who received medicine improved?**Answer: 50 ### Understanding Venn Diagrams: A Study GuideIn this section, we explore how Venn diagrams visually represent the relationships between different sets.#### Example Question:**What can be concluded based on the Venn diagram to the right?**Below the question, you will be asked to choose the correct answer from the available options. The provided Venn diagram consists of two circles:- Circle G is contained entirely within Circle E.#### Answer Choices:1. **Option A:** - G is disjoint from E. Venn diagrams use circles to represent sets and the relations between them. When one circle lies entirely inside another circle, the circles are called disjointed.2. **Option B:** - E is a subset of G. Venn diagrams use circles to represent sets and the relations between them. Because the circle for E is in the circle for G, E is a subset of G.3. **Option C:** - G is a subset of E. Venn diagrams use circles to represent sets and the relations between them. Because the circle for G is in the circle for E, G is a subset of E.4. **Option D:** - G is a subset of E, because G is first in the list of the letters, alphabetically.5. **Option E:** - E is a subset of G, because G is first in the list of the letters, alphabetically.6. **Option F:** - G is disjoint from E. Venn diagrams use circles to represent sets and the relations between them. Because the circle for G does not overlap the circle for E, G is disjoint from E.#### Detailed Diagram Explanation:In the Venn diagram:- Circle G represents set G.- Circle E represents set E.- Circle G lies entirely within Circle E, meaning that every element of G is also an element of E.#### Conclusion:Based on the definition and the specific positioning of the circles in the Venn diagram, the correct answer is **Option C: G is a subset of E**. This interpretation follows the rule that if one circle (representing set G) is completely within another circle (representing set E), then set G is a subset of set E.

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a. Make a two-way table summarizing the reviews. Improvement No Improvement Medicine 25 Placebo 20 (Type whole numbers.) b. Make a Venn diagram from the table in part (a). Choose the correct answer below. OA. OB. OC. OD. Aledicine A Improve Medicine Xlmprove Aledicine Aimprove Stedicine AImprove 25 55 50 20 25 20 55 50 c. How many people who received the placebo did not improve? Type a whole number.) d. How many people who received medicine improved? (Type a whole number.)

a. Make a two-way table summarizing the reviews. Improvement No Improvement Medicine 25 Placebo 20 (Type whole numbers.) b. Make a Venn diagram from the table in part (a). Choose the correct answer below. OA. OB. OC. OD. Aledicine A Improve Medicine Xlmprove Aledicine Aimprove Stedicine AImprove 25 55 50 20 25 20 55 50 c. How many people who received the placebo did not improve? Type a whole number.) d. How many people who received medicine improved? (Type a whole number.)

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Question

In a trial of a new allergy medicine, 75 people were given the medicine and 75 were given a placebo. Of those given the medicine, 50 showed improvement in their allergy symptoms. Of those given the placebo, 55 did not show improvement. Complete parts (a) through (d) below.

**a. Make a two-way table summarizing the reviews:**

| | Improvement | No Improvement |
|----------------|-------------|----------------|
| Medicine | 50 | 25 |
| Placebo | 20 | 55 |

**b. Make a Venn diagram from the table in part (a). Choose the correct answer below.**

[The correct choice is Image C.]

**Explanation of Venn Diagram (Correct Image C):**
- The Venn diagram has two circles, labeled "Medicine" and "Improve".
- "Medicine" circle contains the numbers 25 and 50.
- 25 represents the participants who did not improve with medicine.
- 50 represents the participants who improved with medicine (overlapping with "Improve" circle).
- "Improve" circle contains the number 20 within it and in the overlap:
- 20 represents the participants who improved with placebo (overlapping with "Improve" circle).

**c. How many people who received the placebo did not improve?**

Answer: 55

**d. How many people who received medicine improved?**

Answer: 50

### Understanding Venn Diagrams: A Study Guide

In this section, we explore how Venn diagrams visually represent the relationships between different sets.

#### Example Question:
**What can be concluded based on the Venn diagram to the right?**

Below the question, you will be asked to choose the correct answer from the available options.

The provided Venn diagram consists of two circles:
- Circle G is contained entirely within Circle E.

#### Answer Choices:
1. **Option A:**
- G is disjoint from E. Venn diagrams use circles to represent sets and the relations between them. When one circle lies entirely inside another circle, the circles are called disjointed.

2. **Option B:**
- E is a subset of G. Venn diagrams use circles to represent sets and the relations between them. Because the circle for E is in the circle for G, E is a subset of G.

3. **Option C:**
- G is a subset of E. Venn diagrams use circles to represent sets and the relations between them. Because the circle for G is in the circle for E, G is a subset of E.

4. **Option D:**
- G is a subset of E, because G is first in the list of the letters, alphabetically.

5. **Option E:**
- E is a subset of G, because G is first in the list of the letters, alphabetically.

6. **Option F:**
- G is disjoint from E. Venn diagrams use circles to represent sets and the relations between them. Because the circle for G does not overlap the circle for E, G is disjoint from E.

#### Detailed Diagram Explanation:

In the Venn diagram:
- Circle G represents set G.
- Circle E represents set E.
- Circle G lies entirely within Circle E, meaning that every element of G is also an element of E.

#### Conclusion:
Based on the definition and the specific positioning of the circles in the Venn diagram, the correct answer is **Option C: G is a subset of E**. This interpretation follows the rule that if one circle (representing set G) is completely within another circle (representing set E), then set G is a subset of set E.

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