Respuesta :
Answer:
[tex]6.76-2.01\frac{2.55}{\sqrt{50}}=6.03[/tex]
[tex]6.76+2.01\frac{2.55}{\sqrt{50}}=7.49[/tex]
The 95% confidence interval would be given by (6.03;7.49)
Step-by-step explanation:
Notation
[tex]\bar X[/tex] represent the sample mean
[tex]\mu[/tex] population mean (variable of interest)
s represent the sample standard deviation
n=50 represent the sample size
Solution
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)
We can calculate the mean and the sample deviation we can use the following formulas:
[tex]\bar X= \sum_{i=1}^n \frac{x_i}{n}[/tex] (2)
[tex]s=\sqrt{\frac{\sum_{i=1}^n (x_i-\bar X)}{n-1}}[/tex] (3)
The mean calculated for this case is [tex]\bar X=6.76[/tex]
The sample deviation calculated [tex]s=2.55[/tex]
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=50-1=49[/tex]
We assume a standard confidence level of 0.95 or 95%, the value of [tex]\alpha=0.05[/tex] and [tex]\alpha/2 =0.025[/tex], and we can use excel, a calculator or a tabel to find the critical value. The excel command would be: "=-T.INV(0.025,49)".And we see that [tex]t_{\alpha/2}=2.01[/tex]
Now we have everything in order to replace into formula (1):
[tex]6.76-2.01\frac{2.55}{\sqrt{50}}=6.03[/tex]
[tex]6.76+2.01\frac{2.55}{\sqrt{50}}=7.49[/tex]
The 95% confidence interval would be given by (6.03;7.49)
