MATLAB: An Introduction with Applications
6th Edition
ISBN: 9781119256830
Author: Amos Gilat
Publisher: John Wiley & Sons Inc
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![### Hypothesis Testing in Medical Research: Vaccine Effectiveness against Influenza Strain X
#### Problem Statement
A medical researcher is studying the effectiveness of a newly-developed influenza vaccine compared to last year's vaccine. The researcher’s claim is that the proportion of patients receiving 200 mg of the new vaccine who contract influenza strain X is less than the proportion of patients receiving 200 mg of last year's vaccine who contract the same strain.
#### Data Collection
- **Last Year's Vaccine:** Out of 320 patients, 114 contract influenza strain X.
- **New Vaccine:** Out of 350 patients, 112 contract influenza strain X.
The task is to define the proportion of patients contracting influenza X when using each vaccine. Let:
- \( p_N \) be the proportion of patients receiving the new vaccine who contract influenza strain X.
- \( p_L \) be the proportion of patients receiving last year's vaccine who contract influenza strain X.
#### Hypothesis Formulation
We need to state the null and alternative hypotheses and determine the value of the test statistic.
1. **Null Hypothesis (H₀)**: The proportions of patients contracting influenza strain X are the same for both vaccines.
\[
\text{H₀: } p_N = p_L
\]
2. **Alternative Hypothesis (Hₐ)**: The proportion of patients contracting influenza strain X is less for the new vaccine compared to last year's vaccine.
\[
\text{Hₐ: } p_N < p_L
\]
3. **Test Statistic Calculation and Analysis**
Using the collected data and appropriate statistical tests:
\[
\text{Test Statistic (Z) = 0.99}
\]
#### Possible Outcomes from Multiple-Choice Answers
Students are presented with multiple choices to state the null and alternative hypotheses with corresponding test statistics:
1. **Choice 1:**
\[
\text{H₀: } p_N = p_L \text{ versus Hₐ: } p_N > p_L; \text{ Test statistic: } Z = 0.99
\]
2. **Choice 2:**
\[
\text{H₀: } p_N = p_L \text{ versus Hₐ: } p_N = p_L; \text{ Test statistic: } Z = 1.55
\]
3. **Choice 3](https://content.bartleby.com/qna-images/question/a7e444e2-a441-4a28-bca8-26e04764874a/db8c8839-7355-403d-86fe-5f92b68e289d/3jqkjk_processed.jpeg)
Transcribed Image Text:### Hypothesis Testing in Medical Research: Vaccine Effectiveness against Influenza Strain X
#### Problem Statement
A medical researcher is studying the effectiveness of a newly-developed influenza vaccine compared to last year's vaccine. The researcher’s claim is that the proportion of patients receiving 200 mg of the new vaccine who contract influenza strain X is less than the proportion of patients receiving 200 mg of last year's vaccine who contract the same strain.
#### Data Collection
- **Last Year's Vaccine:** Out of 320 patients, 114 contract influenza strain X.
- **New Vaccine:** Out of 350 patients, 112 contract influenza strain X.
The task is to define the proportion of patients contracting influenza X when using each vaccine. Let:
- \( p_N \) be the proportion of patients receiving the new vaccine who contract influenza strain X.
- \( p_L \) be the proportion of patients receiving last year's vaccine who contract influenza strain X.
#### Hypothesis Formulation
We need to state the null and alternative hypotheses and determine the value of the test statistic.
1. **Null Hypothesis (H₀)**: The proportions of patients contracting influenza strain X are the same for both vaccines.
\[
\text{H₀: } p_N = p_L
\]
2. **Alternative Hypothesis (Hₐ)**: The proportion of patients contracting influenza strain X is less for the new vaccine compared to last year's vaccine.
\[
\text{Hₐ: } p_N < p_L
\]
3. **Test Statistic Calculation and Analysis**
Using the collected data and appropriate statistical tests:
\[
\text{Test Statistic (Z) = 0.99}
\]
#### Possible Outcomes from Multiple-Choice Answers
Students are presented with multiple choices to state the null and alternative hypotheses with corresponding test statistics:
1. **Choice 1:**
\[
\text{H₀: } p_N = p_L \text{ versus Hₐ: } p_N > p_L; \text{ Test statistic: } Z = 0.99
\]
2. **Choice 2:**
\[
\text{H₀: } p_N = p_L \text{ versus Hₐ: } p_N = p_L; \text{ Test statistic: } Z = 1.55
\]
3. **Choice 3
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