Answer:
a)![\bar X =\frac{\sum_{i=1}^n x_i}{n}=\frac{1675}{20}=83.75](https://tex.z-dn.net/?f=%5Cbar%20X%20%3D%5Cfrac%7B%5Csum_%7Bi%3D1%7D%5En%20x_i%7D%7Bn%7D%3D%5Cfrac%7B1675%7D%7B20%7D%3D83.75)
![s=\sqrt{\frac{\sum_{i=1}^n (x_i -\bar X)^2}{n-1}}=28.97](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%3D28.97)
b) The 90% confidence interval would be given by (72.551;94.949)
c)We are 90% confident that the true mean temperature for the sleeping bags it's between 72.551 and 94.949
Step-by-step explanation:
Data set given
80,90,100,120,75,37,30,23,100,110 105,95,105,60,110,120,95,90,60,70
Part a
We can calculate the sample mean and the sample deviation with the following formulas:
![\bar X =\frac{\sum_{i=1}^n x_i}{n}=\frac{1675}{20}=83.75](https://tex.z-dn.net/?f=%5Cbar%20X%20%3D%5Cfrac%7B%5Csum_%7Bi%3D1%7D%5En%20x_i%7D%7Bn%7D%3D%5Cfrac%7B1675%7D%7B20%7D%3D83.75)
![s=\sqrt{\frac{\sum_{i=1}^n (x_i -\bar X)^2}{n-1}}=28.97](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%3D28.97)
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 0.90 or 90%, 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(0.05,19)".And we see that
Now we have everything in order to replace into formula (1):
So on this case the 90% confidence interval would be given by (72.551;94.949)
Part c
We are 90% confident that the true mean temperature for the sleeping bags it's between 72.551 and 94.949