Respuesta :
Answer:
a)[tex]\bar X =\frac{\sum_{i=1}^n x_i}{n}=\frac{1675}{20}=83.75[/tex]
[tex]s=\sqrt{\frac{\sum_{i=1}^n (x_i -\bar X)^2}{n-1}}=28.97[/tex]
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:
[tex]\bar X =\frac{\sum_{i=1}^n x_i}{n}=\frac{1675}{20}=83.75[/tex]
[tex]s=\sqrt{\frac{\sum_{i=1}^n (x_i -\bar X)^2}{n-1}}=28.97[/tex]
Part b
The confidence interval for the mean is given by the following formula:
[tex]\bar X \pm t_{\alpha/2}\frac{s}{\sqrt{n}}[/tex] (1)
In order to calculate the critical value [tex]t_{\alpha/2}[/tex] we need to find first the degrees of freedom, given by:
[tex]df=n-1=20-1=19[/tex]
Since the Confidence is 0.90 or 90%, the value of [tex]\alpha=0.1[/tex] and [tex]\alpha/2 =0.05[/tex], 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 [tex]t_{\alpha/2}=1.73[/tex]
Now we have everything in order to replace into formula (1):
[tex]83.75-1.73\frac{28.97}{\sqrt{20}}=72.551[/tex]
[tex]83.75+1.73\frac{28.97}{\sqrt{20}}=94.949[/tex]
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
