Respuesta :
Answer:
[tex]163.83-4.03\frac{20.094}{\sqrt{6}}=130.77[/tex]
[tex]163.83+4.03\frac{20.094}{\sqrt{6}}=196.89[/tex]
So on this case the 99% confidence interval would be given by (130.77;196.89)
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".
Data: 175 177 175 180 138 138
We can calculate the mean and the deviation from these data with the following formulas:
[tex]\bar X= \frac{\sum_{i=1}^n x_i}{n}[/tex]
[tex]s=\sqrt{\frac{\sum_{i=1}^n (x_i -\bar X)^2}{n-1}}[/tex]
[tex]\bar X=163.83[/tex] represent the sample mean for the sample
[tex]\mu[/tex] population mean (variable of interest)
s=20.093 represent the sample standard deviation
n=6 represent the sample size
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=6-1=5[/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(0.005,5)".And we see that [tex]t_{\alpha/2}=4.03[/tex]
Now we have everything in order to replace into formula (1):
[tex]163.83-4.03\frac{20.094}{\sqrt{6}}=130.77[/tex]
[tex]163.83+4.03\frac{20.094}{\sqrt{6}}=196.89[/tex]
So on this case the 99% confidence interval would be given by (130.77;196.89)
