Answer:
a) ![\bar X =\frac{67.401+67.400+67.402+67.396+67.406+67.401+67.396+67.401+67.405+67.404}{10}=67.401](https://tex.z-dn.net/?f=%5Cbar%20X%20%3D%5Cfrac%7B67.401%2B67.400%2B67.402%2B67.396%2B67.406%2B67.401%2B67.396%2B67.401%2B67.405%2B67.404%7D%7B10%7D%3D67.401)
b) The sample deviation is calculated from the following formula:
![s=\sqrt{\frac{\sum_{i=1}^n (X_i -\bar X)^2}{n-1}}](https://tex.z-dn.net/?f=s%3D%5Csqrt%7B%5Cfrac%7B%5Csum_%7Bi%3D1%7D%5En%20%28X_i%20-%5Cbar%20X%29%5E2%7D%7Bn-1%7D%7D)
And for this case after replace the values and with the sample mean already calculated we got:
![s=0.0036](https://tex.z-dn.net/?f=%20s%3D0.0036)
If we assume that the data represent a population then the standard deviation would be given by:
![\sigma=\sqrt{\frac{\sum_{i=1}^n (X_i -\bar X)^2}{n}}](https://tex.z-dn.net/?f=%5Csigma%3D%5Csqrt%7B%5Cfrac%7B%5Csum_%7Bi%3D1%7D%5En%20%28X_i%20-%5Cbar%20X%29%5E2%7D%7Bn%7D%7D)
And then the deviation would be:
![\sigma =0.00319](https://tex.z-dn.net/?f=%5Csigma%20%3D0.00319)
Step-by-step explanation:
For this case we have the following dataset:
67.401, 67.400, 67.402, 67.396, 67.406, 67.401, 67.396, 67.401, 67.405, and 67.404
Part a: Determine the most probable value.
For this case the most probably value would be the sample mean given by this formula:
![\bar X =\frac{\sum_{i=1}^n X_i}{n}](https://tex.z-dn.net/?f=%20%5Cbar%20X%20%3D%5Cfrac%7B%5Csum_%7Bi%3D1%7D%5En%20X_i%7D%7Bn%7D)
And if we replace we got:
![\bar X =\frac{67.401+67.400+67.402+67.396+67.406+67.401+67.396+67.401+67.405+67.404}{10}=67.401](https://tex.z-dn.net/?f=%5Cbar%20X%20%3D%5Cfrac%7B67.401%2B67.400%2B67.402%2B67.396%2B67.406%2B67.401%2B67.396%2B67.401%2B67.405%2B67.404%7D%7B10%7D%3D67.401)
Part b: Determine the standard deviation
The sample deviation is calculated from the following formula:
![s=\sqrt{\frac{\sum_{i=1}^n (X_i -\bar X)^2}{n-1}}](https://tex.z-dn.net/?f=s%3D%5Csqrt%7B%5Cfrac%7B%5Csum_%7Bi%3D1%7D%5En%20%28X_i%20-%5Cbar%20X%29%5E2%7D%7Bn-1%7D%7D)
And for this case after replace the values and with the sample mean already calculated we got:
![s=0.0036](https://tex.z-dn.net/?f=%20s%3D0.0036)
If we assume that the data represent a population then the standard deviation would be given by:
![\sigma=\sqrt{\frac{\sum_{i=1}^n (X_i -\bar X)^2}{n}}](https://tex.z-dn.net/?f=%5Csigma%3D%5Csqrt%7B%5Cfrac%7B%5Csum_%7Bi%3D1%7D%5En%20%28X_i%20-%5Cbar%20X%29%5E2%7D%7Bn%7D%7D)
And then the deviation would be:
![\sigma =0.00319](https://tex.z-dn.net/?f=%5Csigma%20%3D0.00319)