### Confidence Interval AnalysisThe following data represent the length of time, in days, to recovery for patients randomly treated with one of two medications to clear up severe bladder infections. We need to find a 95% confidence interval for the difference $\mu_2 - \mu_1$ between the mean recovery times for the two medications, assuming normal populations with equal variances.#### Data Summary:| Medication | $ n $ (Sample Size) | $ \overline{x} $ (Sample Mean) | $ s^2 $ (Sample Variance) ||------------|-----------------------|---------------------------------|----------------------------|| Medication 1 | $ n_1 = 13 $ | $ \overline{x}_1 = 14 $ | $ s_1^2 = 1.7 $ || Medication 2 | $ n_2 = 17 $ | $ \overline{x}_2 = 18 $ | $ s_2^2 = 1.3 $ |#### Confidence Interval Formula:The formula for the confidence interval for the difference between two means ($\mu_2 - \mu_1$) with equal variances is given by:\[ \left( \overline{x}_2 - \overline{x}_1 \right) \pm t_{\alpha/2, \, df} \cdot \sqrt{ \left( \frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2} \right) \left( \frac{1}{n_1} + \frac{1}{n_2} \right) } \]#### Calculations:- Calculate the pooled variance: \[ s_p^2 = \frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2} \] - Insert the sample values: \[ s_p^2 = \frac{(13 - 1) \cdot 1.7 + (17 - 1) \cdot 1.3}{13 + 17 - 2} = \frac{12 \cdot 1.7 + 16 \cd

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The following data represent the length of time, in days, to recovery for patients randomly treated with one of two medications to clear up severe bladder infections Find a 95% confidence interval for the difference p,-H, between in the mean recovery times for the two medications, assuming normal populations with equal variances. Medication 1 s =17 s3-13 n = 13 X1 = 14 Medication 2 n2 = 17 X2 = 18 The confidence interval is <µ,-< (Round to two decimal places as needed.) Enter your answer in the edit fields and then click Check Answer All parts showing Clear All Fal Cherk

The following data represent the length of time, in days, to recovery for patients randomly treated with one of two medications to clear up severe bladder infections Find a 95% confidence interval for the difference p,-H, between in the mean recovery times for the two medications, assuming normal populations with equal variances. Medication 1 s =17 s3-13 n = 13 X1 = 14 Medication 2 n2 = 17 X2 = 18 The confidence interval is <µ,-< (Round to two decimal places as needed.) Enter your answer in the edit fields and then click Check Answer All parts showing Clear All Fal Cherk

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

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### Confidence Interval Analysis

The following data represent the length of time, in days, to recovery for patients randomly treated with one of two medications to clear up severe bladder infections. We need to find a 95% confidence interval for the difference $\mu_2 - \mu_1$ between the mean recovery times for the two medications, assuming normal populations with equal variances.

#### Data Summary:
| Medication | $ n $ (Sample Size) | $ \overline{x} $ (Sample Mean) | $ s^2 $ (Sample Variance) |
|------------|-----------------------|---------------------------------|----------------------------|
| Medication 1 | $ n_1 = 13 $ | $ \overline{x}_1 = 14 $ | $ s_1^2 = 1.7 $ |
| Medication 2 | $ n_2 = 17 $ | $ \overline{x}_2 = 18 $ | $ s_2^2 = 1.3 $ |

#### Confidence Interval Formula:
The formula for the confidence interval for the difference between two means ($\mu_2 - \mu_1$) with equal variances is given by:

\[ \left( \overline{x}_2 - \overline{x}_1 \right) \pm t_{\alpha/2, \, df} \cdot \sqrt{ \left( \frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2} \right) \left( \frac{1}{n_1} + \frac{1}{n_2} \right) } \]

#### Calculations:
- Calculate the pooled variance:
\[ s_p^2 = \frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2} \]

- Insert the sample values:
\[ s_p^2 = \frac{(13 - 1) \cdot 1.7 + (17 - 1) \cdot 1.3}{13 + 17 - 2} = \frac{12 \cdot 1.7 + 16 \cd

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