Respuesta :
Answer:
[tex]\frac{(7)(0.442)^2}{16.012} \leq \sigma^2 \leq \frac{(7)(0.442)^2}{1.690}[/tex]
[tex] 0.0855 \leq \sigma^2 \leq 0.8099[/tex]
Now we just take square root on both sides of the interval and we got:
[tex] 0.2924 \leq \sigma \leq 0.89995[/tex]
Lower endpoint = 0.2924
Upper endpoint= 0.89995
Step-by-step explanation:
Data given and notation
Data: 3.4, 3.1, 3.6, 3.3, 2.7, 2.8, 2.4, 3.6
We can calculate the sample deviation with the following formula:
[tex] s = \sqrt{\frac{\sum_{i=1}^n (X)i -\bar X)^2}{n-1}}[/tex]
s=0.442 represent the sample standard deviation
[tex]\bar x[/tex] represent the sample mean
n=8 the sample size
Confidence=95% or 0.95
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 mean or variance 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.
The Chi Square distribution is the distribution of the sum of squared standard normal deviates .
Calculating the confidence interval
The confidence interval for the population variance is given by the following formula:
[tex]\frac{(n-1)s^2}{\chi^2_{\alpha/2}} \leq \sigma^2 \leq \frac{(n-1)s^2}{\chi^2_{1-\alpha/2}}[/tex]
The next step would be calculate the critical values. First we need to calculate the degrees of freedom given by:
[tex]df=n-1=8-1=7[/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 tabel to find the critical values.
The excel commands would be: "=CHISQ.INV(0.025,7)" "=CHISQ.INV(0.975,7)". so for this case the critical values are:
[tex]\chi^2_{\alpha/2}=16.012[/tex]
[tex]\chi^2_{1- \alpha/2}=1.690[/tex]
And replacing into the formula for the interval we got:
[tex]\frac{(7)(0.442)^2}{16.012} \leq \sigma^2 \leq \frac{(7)(0.442)^2}{1.690}[/tex]
[tex] 0.0855 \leq \sigma^2 \leq 0.8099[/tex]
Now we just take square root on both sides of the interval and we got:
[tex] 0.2924 \leq \sigma \leq 0.89995[/tex]
Lower endpoint = 0.2924
Upper endpoint= 0.89995
