Answer:
a) And if we replace we got: ![\bar X= 78](https://tex.z-dn.net/?f=%5Cbar%20X%3D%2078)
![s = 15.391](https://tex.z-dn.net/?f=%20s%20%3D%2015.391)
b)
So on this case the 99% confidence interval would be given by (62.182;93.818)
Step-by-step explanation:
Dataset given: 109 67 58 76 65 80 96 86 71 72
Part a
For this case we can calculate the sample mean with the following formula:
![\bar X = \frac{\sum_{i=1}^n X_i}{n}](https://tex.z-dn.net/?f=%5Cbar%20X%20%3D%20%5Cfrac%7B%5Csum_%7Bi%3D1%7D%5En%20X_i%7D%7Bn%7D)
And if we replace we got: ![\bar X= 78](https://tex.z-dn.net/?f=%5Cbar%20X%3D%2078)
And the deviation is given by:
![s =\sqrt{\frac{\sum_{i=1}^n (X_i -\bar X)^2}{n-1}}](https://tex.z-dn.net/?f=%20s%20%3D%5Csqrt%7B%5Cfrac%7B%5Csum_%7Bi%3D1%7D%5En%20%28X_i%20-%5Cbar%20X%29%5E2%7D%7Bn-1%7D%7D)
And if we replace we got ![s = 15.391](https://tex.z-dn.net/?f=%20s%20%3D%2015.391)
Part b
The confidence interval for the mean is given by the following formula:
(1)
In order to calculate the critical value
we need to find first the degrees of freedom, given by:
Since the Confidence is 0.99 or 99%, the value of
and
, and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.005,9)".And we see that
Now we have everything in order to replace into formula (1):
So on this case the 99% confidence interval would be given by (62.182;93.818)