MATLAB: An Introduction with Applications
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Author: Amos Gilat
Publisher: John Wiley & Sons Inc
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![**Clinical Trial Analysis on Drug Effectiveness for Treating Insomnia in Older Subjects**
**Overview:**
A clinical trial was conducted to evaluate the effectiveness of a drug designed to treat insomnia in older subjects. Here's a summary and analysis of the trial results:
**Trial Details:**
- **Before Treatment:**
- Number of subjects: 25
- Mean wake time: 105.0 minutes
- **After Treatment:**
- Number of subjects: 25
- Mean wake time: 100.3 minutes
- Standard deviation: 21.7 minutes
**Objective:**
To determine if the mean wake time after treatment significantly differs from the mean wake time before treatment by constructing a 90% confidence interval estimate.
**Analysis:**
- **Assumption:**
- The 25 sample values are assumed to be from a normally distributed population.
**Steps for Constructing the 90% Confidence Interval:**
1. **Identify sample mean ( \(\bar{x} \) ) and standard deviation (s):**
- Sample mean after treatment ( \(\bar{x} \) ): 100.3 minutes
- Standard deviation (s): 21.7 minutes
2. **Calculate the standard error of the mean (SEM):**
\[
SEM = \frac{s}{\sqrt{n}} = \frac{21.7}{\sqrt{25}} = \frac{21.7}{5} = 4.34
\]
3. **Determine the critical value (Z*) for a 90% confidence interval (from Z-tables):**
- For 90% confidence, Z* ≈ 1.645
4. **Calculate the margin of error (ME):**
\[
ME = Z* \times SEM = 1.645 \times 4.34 = 7.14
\]
5. **Construct the confidence interval:**
\[
CI = \bar{x} \pm ME = 100.3 \pm 7.14
\]
- Lower bound: 100.3 - 7.14 = 93.2
- Upper bound: 100.3 + 7.14 = 107.4
**Conclusion:**
- **90% Confidence Interval for the Mean Wake Time post-treatment:**
\[
93.2 \text](https://content.bartleby.com/qna-images/question/11fb6ed7-9250-48c6-8339-b5d076b64c81/5155267c-3a17-44ed-95f2-db830b89eec1/dms60l7.jpeg)
Transcribed Image Text:**Clinical Trial Analysis on Drug Effectiveness for Treating Insomnia in Older Subjects**
**Overview:**
A clinical trial was conducted to evaluate the effectiveness of a drug designed to treat insomnia in older subjects. Here's a summary and analysis of the trial results:
**Trial Details:**
- **Before Treatment:**
- Number of subjects: 25
- Mean wake time: 105.0 minutes
- **After Treatment:**
- Number of subjects: 25
- Mean wake time: 100.3 minutes
- Standard deviation: 21.7 minutes
**Objective:**
To determine if the mean wake time after treatment significantly differs from the mean wake time before treatment by constructing a 90% confidence interval estimate.
**Analysis:**
- **Assumption:**
- The 25 sample values are assumed to be from a normally distributed population.
**Steps for Constructing the 90% Confidence Interval:**
1. **Identify sample mean ( \(\bar{x} \) ) and standard deviation (s):**
- Sample mean after treatment ( \(\bar{x} \) ): 100.3 minutes
- Standard deviation (s): 21.7 minutes
2. **Calculate the standard error of the mean (SEM):**
\[
SEM = \frac{s}{\sqrt{n}} = \frac{21.7}{\sqrt{25}} = \frac{21.7}{5} = 4.34
\]
3. **Determine the critical value (Z*) for a 90% confidence interval (from Z-tables):**
- For 90% confidence, Z* ≈ 1.645
4. **Calculate the margin of error (ME):**
\[
ME = Z* \times SEM = 1.645 \times 4.34 = 7.14
\]
5. **Construct the confidence interval:**
\[
CI = \bar{x} \pm ME = 100.3 \pm 7.14
\]
- Lower bound: 100.3 - 7.14 = 93.2
- Upper bound: 100.3 + 7.14 = 107.4
**Conclusion:**
- **90% Confidence Interval for the Mean Wake Time post-treatment:**
\[
93.2 \text
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