Answer:
The confidence interval is ![25.16 < \mu < 26.85](https://tex.z-dn.net/?f=25.16%20%20%3C%20%5Cmu%20%3C%2026.85)
Step-by-step explanation:
From the question we are given a data set
25.5, 26.1, 26.8, 23.2, 24.2, 28.4, 25.0, 27.8, 27.3, and 25.7.
The mean of the this sample data is
![\= x = \frac{\sum x_i}{n}](https://tex.z-dn.net/?f=%5C%3D%20x%20%20%3D%20%5Cfrac%7B%5Csum%20x_i%7D%7Bn%7D)
where is the sample size with values n = 10
![\= x = \frac{25.5+ 26.1+ 26.8+23.2+ 24.2+ 28.4+ 25.0+ 27.8+ 27.3+ 25.7}{10}](https://tex.z-dn.net/?f=%5C%3D%20x%20%3D%20%5Cfrac%7B25.5%2B%2026.1%2B%2026.8%2B23.2%2B%2024.2%2B%2028.4%2B%2025.0%2B%2027.8%2B%2027.3%2B%2025.7%7D%7B10%7D)
![\= x = 26](https://tex.z-dn.net/?f=%5C%3D%20x%20%3D%2026)
The standard deviation is evaluated as
![\sigma = \sqrt{\frac{\sum (x-\= x)}{n} }](https://tex.z-dn.net/?f=%5Csigma%20%20%3D%20%20%5Csqrt%7B%5Cfrac%7B%5Csum%20%28x-%5C%3D%20x%29%7D%7Bn%7D%20%7D)
substituting values
![= \sqrt{\frac{ ( 25.5-26)^2, (26.1-26)^2, (26.8-26)^2, (23.2-26)^2}{10} }](https://tex.z-dn.net/?f=%3D%20%5Csqrt%7B%5Cfrac%7B%20%28%2025.5-26%29%5E2%2C%20%2826.1-26%29%5E2%2C%20%2826.8-26%29%5E2%2C%20%2823.2-26%29%5E2%7D%7B10%7D%20%7D)
![\cdot \ \cdot \ \cdot \sqrt{\frac{ ( 24.2-26)^2, (28.4-26)^2+( 25.0-26)^2+ (27.8-26)^2+( 27.3-26)^2+( 25.7-26)^2}{10} }](https://tex.z-dn.net/?f=%5Ccdot%20%5C%20%5Ccdot%20%5C%20%5Ccdot%20%20%5Csqrt%7B%5Cfrac%7B%20%28%2024.2-26%29%5E2%2C%20%2828.4-26%29%5E2%2B%28%2025.0-26%29%5E2%2B%20%2827.8-26%29%5E2%2B%28%2027.3-26%29%5E2%2B%28%2025.7-26%29%5E2%7D%7B10%7D%20%7D)
![\sigma = 1.625](https://tex.z-dn.net/?f=%5Csigma%20%20%3D%20%201.625)
The confidence level is given as 90% hence the level of significance is calculated as
![\alpha = 100 -90](https://tex.z-dn.net/?f=%5Calpha%20%20%3D%20%20100%20-90)
%
![\alpha = 0.10](https://tex.z-dn.net/?f=%5Calpha%20%3D%200.10)
Now the critical values of
is obtained from the normal distribution table as
![Z_{\frac{\alpha }{2} } = 1.645](https://tex.z-dn.net/?f=Z_%7B%5Cfrac%7B%5Calpha%20%7D%7B2%7D%20%7D%20%3D%20%201.645)
The reason we are obtaining the critical values of
instead of
is because we are considering two tails of the area under the normal curve
The margin of error is evaluated as
![MOE = Z_{\frac{\alpha }{2} } * \frac{\sigma }{\sqrt{n} }](https://tex.z-dn.net/?f=MOE%20%3D%20%20Z_%7B%5Cfrac%7B%5Calpha%20%7D%7B2%7D%20%7D%20%2A%20%20%5Cfrac%7B%5Csigma%20%7D%7B%5Csqrt%7Bn%7D%20%7D)
substituting values
![MOE = 1.645 * \frac{1.625 }{\sqrt{10} }](https://tex.z-dn.net/?f=MOE%20%3D%20%201.645%20%2A%20%20%5Cfrac%7B1.625%20%7D%7B%5Csqrt%7B10%7D%20%7D)
![MOE = 0.845](https://tex.z-dn.net/?f=MOE%20%3D%200.845)
The 90%, two sided confidence interval is mathematically evaluated as
![\= x - MOE < \mu < \= x + MOE](https://tex.z-dn.net/?f=%5C%3D%20x%20%20-%20MOE%20%20%3C%20%5Cmu%20%3C%20%5C%3D%20x%20%20%2B%20MOE)
![26 - 0.845 < \mu < 26 + 0.845](https://tex.z-dn.net/?f=26%20%20-%200.845%20%20%3C%20%5Cmu%20%3C%2026%20%20%2B%200.845)
![25.16 < \mu < 26.85](https://tex.z-dn.net/?f=25.16%20%20%3C%20%5Cmu%20%3C%2026.85)
Given that the lower and the upper limit is greater than 25 then we can assure the manufactures that the battery life exceeds 25 hours