Respuesta :
Answer:
[tex]3.3048-1.83\frac{0.132}{\sqrt{10}}=3.2284[/tex]
[tex]3.3048 +1.83\frac{0.132}{\sqrt{10}}=3.3812[/tex]
So on this case the 90% confidence interval would be given by (3.2284;3.3812)
Step-by-step explanation:
Previous concepts
A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".
The margin of error is the range of values below and above the sample statistic in a confidence interval.
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
[tex]\bar X[/tex] represent the sample mean for the sample
[tex]\mu[/tex] population mean (variable of interest)
s represent the sample standard deviation
n represent the sample size
Data: 3.087 3.131 3.241 3.241 3.270 3.353 3.400 3.411 3.437 3.477
Solution to the problem
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 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=3.3048[/tex]
The sample deviation calculated [tex]s=0.132[/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=10-1=9[/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(0.05,9)".And we see that [tex]t_{\alpha/2}=1.83[/tex]
Now we have everything in order to replace into formula (1):
[tex]3.3048-1.83\frac{0.132}{\sqrt{10}}=3.2284[/tex]
[tex]3.3048 +1.83\frac{0.132}{\sqrt{10}}=3.3812[/tex]
So on this case the 90% confidence interval would be given by (3.2284;3.3812)
