In a study of treatments for very painful "cluster" headaches, 145 patients were treated with oxygen and 144 other patients were given a placebo consisting of ordinary air. Among the 145 patients in the oxygen treatment aroup. 124 were free from headaches 15 minutes after treatment. Amona the 144 patients aiven the placebo. The 90% confidence interval isO< (P1 - P2)
In a study of treatments for very painful "cluster" headaches, 145 patients were treated with oxygen and 144 other patients were given a placebo consisting of ordinary air. Among the 145 patients in the oxygen treatment aroup. 124 were free from headaches 15 minutes after treatment. Amona the 144 patients aiven the placebo. The 90% confidence interval isO< (P1 - P2)
MATLAB: An Introduction with Applications
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ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
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In a study of treatments for very painful "cluster" headaches,...
Find the 90% confidence interval using technology.
![### Study on Treatments for Cluster Headaches
In a study investigating treatments for very painful "cluster" headaches, 145 patients received oxygen treatment while another 144 patients were given a placebo consisting of ordinary air. The outcomes were as follows:
- **Oxygen Treatment Group:** Among the 145 patients treated with oxygen, 124 were free from headaches 15 minutes after treatment.
- **Placebo Group:** Among the 144 patients given the placebo, [data not provided in the image].
To assess the effectiveness of the treatments, the 90% confidence interval for the difference in proportions (P1 - P2) between the two groups is used. A confidence interval provides a range of values within which the true difference in treatment effects is likely to lie.
### Confidence Interval Calculation
The notation used for the confidence interval is:
\[ \boxed{} < (P1 - P2) < \boxed{} \]
For accurate results, the confidence interval needs to be calculated and rounded to three decimal places as necessary.
---
Remember, confidence intervals are useful in indicating the reliability and precision of your estimates. A narrower interval suggests higher precision.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F242a0dac-d7d5-4e5c-b656-00cdaf0b077c%2Fe90dd835-2068-4714-90c8-d5e9ea2930ff%2Fyt00496.png&w=3840&q=75)
Transcribed Image Text:### Study on Treatments for Cluster Headaches
In a study investigating treatments for very painful "cluster" headaches, 145 patients received oxygen treatment while another 144 patients were given a placebo consisting of ordinary air. The outcomes were as follows:
- **Oxygen Treatment Group:** Among the 145 patients treated with oxygen, 124 were free from headaches 15 minutes after treatment.
- **Placebo Group:** Among the 144 patients given the placebo, [data not provided in the image].
To assess the effectiveness of the treatments, the 90% confidence interval for the difference in proportions (P1 - P2) between the two groups is used. A confidence interval provides a range of values within which the true difference in treatment effects is likely to lie.
### Confidence Interval Calculation
The notation used for the confidence interval is:
\[ \boxed{} < (P1 - P2) < \boxed{} \]
For accurate results, the confidence interval needs to be calculated and rounded to three decimal places as necessary.
---
Remember, confidence intervals are useful in indicating the reliability and precision of your estimates. A narrower interval suggests higher precision.
![**Statistical Analysis: Testing the Effectiveness of Oxygen Treatment on Headaches**
In this example, we are evaluating whether an oxygen treatment is effective for relieving headaches. We will use a significance level of 0.05 to test this claim. Here are the details of the study and our analysis step-by-step:
### Study Details:
- **Experimental Group**: There were 124 patients in this group who received the oxygen treatment. After 15 minutes of treatment, 124 were free from headaches.
- **Control Group**: There were 144 patients in this group who received a placebo (no actual treatment). After 15 minutes, 32 were free from headaches.
### Hypothesis Testing:
We will conduct hypothesis testing to evaluate the effectiveness of the treatment.
### Confidence Interval Calculation:
We are required to calculate the 90% confidence interval for the difference in proportions of patients relieved from headaches between the experimental and control groups.
### Steps:
1. **Determine the Sample Proportions**:
- \( p_1 \): Proportion of patients in the experimental group who were free from headaches.
- \( p_2 \): Proportion of patients in the control group who were free from headaches.
2. **Confidence Interval Calculation**:
Use the appropriate statistical formula to determine the 90% confidence interval for the difference \( p_1 - p_2 \).
The formula for the confidence interval for the difference between two proportions is:
\[
(p_1 - p_2) \pm Z \sqrt{\left( \frac{p_1(1 - p_1)}{n_1} \right) + \left( \frac{p_2(1 - p_2)}{n_2} \right)}
\]
Here \(Z\) represents the Z-score corresponding to the confidence level.
### Result:
The 90% confidence interval calculated is:
\[
\boxed{} < (p_1 - p_2) < \boxed{}
\]
### Additional Note:
Please ensure the values are rounded to three decimal places as needed.
**Conclusion:**
By constructing and analyzing the confidence interval, we can determine whether the oxygen treatment significantly reduces headaches compared to the placebo. If the interval does not include zero, we can reject the null hypothesis and conclude that there is a significant difference in effectiveness between the oxygen treatment and the placebo.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F242a0dac-d7d5-4e5c-b656-00cdaf0b077c%2Fe90dd835-2068-4714-90c8-d5e9ea2930ff%2Feo0osft.png&w=3840&q=75)
Transcribed Image Text:**Statistical Analysis: Testing the Effectiveness of Oxygen Treatment on Headaches**
In this example, we are evaluating whether an oxygen treatment is effective for relieving headaches. We will use a significance level of 0.05 to test this claim. Here are the details of the study and our analysis step-by-step:
### Study Details:
- **Experimental Group**: There were 124 patients in this group who received the oxygen treatment. After 15 minutes of treatment, 124 were free from headaches.
- **Control Group**: There were 144 patients in this group who received a placebo (no actual treatment). After 15 minutes, 32 were free from headaches.
### Hypothesis Testing:
We will conduct hypothesis testing to evaluate the effectiveness of the treatment.
### Confidence Interval Calculation:
We are required to calculate the 90% confidence interval for the difference in proportions of patients relieved from headaches between the experimental and control groups.
### Steps:
1. **Determine the Sample Proportions**:
- \( p_1 \): Proportion of patients in the experimental group who were free from headaches.
- \( p_2 \): Proportion of patients in the control group who were free from headaches.
2. **Confidence Interval Calculation**:
Use the appropriate statistical formula to determine the 90% confidence interval for the difference \( p_1 - p_2 \).
The formula for the confidence interval for the difference between two proportions is:
\[
(p_1 - p_2) \pm Z \sqrt{\left( \frac{p_1(1 - p_1)}{n_1} \right) + \left( \frac{p_2(1 - p_2)}{n_2} \right)}
\]
Here \(Z\) represents the Z-score corresponding to the confidence level.
### Result:
The 90% confidence interval calculated is:
\[
\boxed{} < (p_1 - p_2) < \boxed{}
\]
### Additional Note:
Please ensure the values are rounded to three decimal places as needed.
**Conclusion:**
By constructing and analyzing the confidence interval, we can determine whether the oxygen treatment significantly reduces headaches compared to the placebo. If the interval does not include zero, we can reject the null hypothesis and conclude that there is a significant difference in effectiveness between the oxygen treatment and the placebo.
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