Respuesta :
Answer:
[tex]182.167-2.03\frac{114.05}{\sqrt{36}}=143.580[/tex]
[tex]182.167+2.03\frac{114.05}{\sqrt{36}}=220.754[/tex]
So on this case the 95% confidence interval would be given by (143.580;220.754)
Step-by-step explanation:
Assuming the following dataset:
77, 349,417,349, 167 , 225, 265, 360,205
145,335,40,139, 177,108, 163, 202, 22
123,439, 125,135, 86,43, 217,49, 156
119,178, 151, 61, 350, 312, 91, 89,89
We can calculate the sample mean with the followinf formula:
[tex] \bar X = \frac{\sum_{i=1}^n X_i}{n}= 182.167[/tex]
And the sample deviation with:
[tex] s = \sqrt{\frac{\sum_{i=1}^n (X_i-\bar X)^2}{n-1}}=114.05[/tex]
The sample size on this case is n =36.
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=182.167[/tex] represent the sample mean
[tex]\mu[/tex] population mean (variable of interest)
s=114.05 represent the sample standard deviation
n=36 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)
The point estimate of the population mean is [tex]\hat \mu = \bar X =182.167[/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=36-1=35[/tex]
Since the Confidence is 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 table to find the critical value. The excel command would be: "=-T.INV(0.025,35)".And we see that [tex]t_{\alpha/2}=2.03[/tex]
Now we have everything in order to replace into formula (1):
[tex]182.167-2.03\frac{114.05}{\sqrt{36}}=143.580[/tex]
[tex]182.167+2.03\frac{114.05}{\sqrt{36}}=220.754[/tex]
So on this case the 95% confidence interval would be given by (143.580;220.754)
