Respuesta :
Answer:
The 95% confidence interval would be given by (0.0825;0.1395)
Step-by-step explanation:
1) 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: 0.07, 0.11, 0.15, 0.13, 0.12, 0.07, 0.13
We can calculate the sample mean and deviation 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)^n}{n-1}}[/tex]
[tex]\bar X=0.111[/tex] represent the sample mean for the sample
[tex]\mu[/tex] population mean (variable of interest)
s=0.0308 represent the sample standard deviation
n=7 represent the sample size
2) Confidence interval
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=7-1=6[/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(0.025,6)".And we see that [tex]t_{\alpha/2}=2.45[/tex]
Now we have everything in order to replace into formula (1):
[tex]0.111-2.45\frac{0.0308}{\sqrt{7}}=0.0825[/tex]
[tex]0.111+2.45\frac{0.0308}{\sqrt{7}}=0.1395[/tex]
So on this case the 95% confidence interval would be given by (0.0825;0.1395)
