Answer:
The 99% confidence interval is (228.035, 234.233).
Step-by-step explanation:
The sample size selected to compute the 95% confidence interval for the true average natural frequency (Hz) of delaminated beams of a certain type is
<em>n</em> = 5
The sample size is very small and the population standard deviation is also not known.
So, we will use <em>t</em>-interval for the confidence interval.
The (1 - <em>α</em>)% confidence interval for the true mean is:
![CI=\bar x\pm t_{\alpha/2, (n-1)}\times \frac{s}{\sqrt{n}}](https://tex.z-dn.net/?f=CI%3D%5Cbar%20x%5Cpm%20t_%7B%5Calpha%2F2%2C%20%28n-1%29%7D%5Ctimes%20%5Cfrac%7Bs%7D%7B%5Csqrt%7Bn%7D%7D)
The 95% confidence interval for the true average natural frequency (Hz) of delaminated beams of a certain type is:
(Upper limit, Lower limit) = (229.266, 233.002).
Compute the value of sample mean as follows:
![\bar x=\frac{UL+LL}{2}=\frac{233.002+229.266}{2}=231.134](https://tex.z-dn.net/?f=%5Cbar%20x%3D%5Cfrac%7BUL%2BLL%7D%7B2%7D%3D%5Cfrac%7B233.002%2B229.266%7D%7B2%7D%3D231.134)
The critical value of <em>t</em> for <em>α</em> = 0.05 and <em>n</em> = 5 is:
![t_{\alpha/2, (n-1)}=t_{0.05/2, (5-1)}=t_{0.025,4}=2.776](https://tex.z-dn.net/?f=t_%7B%5Calpha%2F2%2C%20%28n-1%29%7D%3Dt_%7B0.05%2F2%2C%20%285-1%29%7D%3Dt_%7B0.025%2C4%7D%3D2.776)
*Use a <em>t</em>-table.
Compute the value of sample standard deviation as follows:
![\frac{UL-LL}{2}=t_{\alpha/2, (n-1)}\times \frac{s}{\sqrt{n}}\\\frac{233.002-229.266}{2}=2.776\times\frac{s}{\sqrt{5}}\\s=1.505](https://tex.z-dn.net/?f=%5Cfrac%7BUL-LL%7D%7B2%7D%3Dt_%7B%5Calpha%2F2%2C%20%28n-1%29%7D%5Ctimes%20%5Cfrac%7Bs%7D%7B%5Csqrt%7Bn%7D%7D%5C%5C%5Cfrac%7B233.002-229.266%7D%7B2%7D%3D2.776%5Ctimes%5Cfrac%7Bs%7D%7B%5Csqrt%7B5%7D%7D%5C%5Cs%3D1.505)
The critical value of <em>t</em> for <em>α</em> = 0.01 and <em>n</em> = 5 is:
![t_{\alpha/2, (n-1)}=t_{0.01/2, (5-1)}=t_{0.005,4}=4.604](https://tex.z-dn.net/?f=t_%7B%5Calpha%2F2%2C%20%28n-1%29%7D%3Dt_%7B0.01%2F2%2C%20%285-1%29%7D%3Dt_%7B0.005%2C4%7D%3D4.604)
*Use a <em>t</em>-table.
Construct the 99% confidence interval as follows:
![CI=\bar x\pm t_{\alpha/2, (n-1)}\times \frac{s}{\sqrt{n}}\\=231.134\pm 4.604\times \frac{1.505}{\sqrt{5}}\\=231.134\pm3.099\\=(228.035, 234.233)](https://tex.z-dn.net/?f=CI%3D%5Cbar%20x%5Cpm%20t_%7B%5Calpha%2F2%2C%20%28n-1%29%7D%5Ctimes%20%5Cfrac%7Bs%7D%7B%5Csqrt%7Bn%7D%7D%5C%5C%3D231.134%5Cpm%204.604%5Ctimes%20%5Cfrac%7B1.505%7D%7B%5Csqrt%7B5%7D%7D%5C%5C%3D231.134%5Cpm3.099%5C%5C%3D%28228.035%2C%20234.233%29)
Thus, the 99% confidence interval is (228.035, 234.233).