Respuesta :
Answer:
a) And if we replace we got: [tex]\bar X= 78[/tex]
[tex] s = 15.391[/tex]
b) [tex]78-3.25\frac{15.391}{\sqrt{10}}=62.182[/tex]
[tex]78-3.25\frac{15.391}{\sqrt{10}}=93.818[/tex]
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:
[tex]\bar X = \frac{\sum_{i=1}^n X_i}{n}[/tex]
And if we replace we got: [tex]\bar X= 78[/tex]
And the deviation is given by:
[tex] s =\sqrt{\frac{\sum_{i=1}^n (X_i -\bar X)^2}{n-1}}[/tex]
And if we replace we got [tex] s = 15.391[/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=10-1=9[/tex]
Since the Confidence is 0.99 or 99%, the value of [tex]\alpha=0.01[/tex] and [tex]\alpha/2 =0.005[/tex], 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 [tex]t_{\alpha/2}=3.25[/tex]
Now we have everything in order to replace into formula (1):
[tex]78-3.25\frac{15.391}{\sqrt{10}}=62.182[/tex]
[tex]78-3.25\frac{15.391}{\sqrt{10}}=93.818[/tex]
So on this case the 99% confidence interval would be given by (62.182;93.818)
