Answer :
Answer:
[tex]\frac{(7)(0.378)^2}{14.067} \leq \sigma^2 \leq \frac{(7)(0.378)^2}{2.167}[/tex]
[tex] 0.0711 \leq \sigma^2 \leq 0.462[/tex]
And rounded we got:
[tex] 0.07 \leq \sigma^2 \leq 0.46[/tex]
Step-by-step explanation:
Data given and notation
s represent the sample standard deviation
[tex]\bar x[/tex] represent the sample mean
n=8 the sample size
Confidence=90% or 0.90
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]
On this case we need to find the sample standard deviation with the following formula:
[tex]s=sqrt{\frac{\sum_{i=1}^8 (x_i -\bar x)^2}{n-1}}
And in order to find the sample mean we just need to use this formula:
[tex]\bar x =\frac{\sum_{i=1}^n x_i}{n}[/tex]
The sample mean obtained on this case is [tex]\bar x= 12.7625[/tex] and the deviation s=0.378
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 0.90 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 values.
The excel commands would be: "=CHISQ.INV(0.05,7)" "=CHISQ.INV(0.95,7)". so for this case the critical values are:
[tex]\chi^2_{\alpha/2}=14.067[/tex]
[tex]\chi^2_{1- \alpha/2}=2.167[/tex]
And replacing into the formula for the interval we got:
[tex]\frac{(7)(0.378)^2}{14.067} \leq \sigma^2 \leq \frac{(7)(0.378)^2}{2.167}[/tex]
[tex] 0.0711 \leq \sigma^2 \leq 0.462[/tex]
And rounded we got:
[tex] 0.07 \leq \sigma^2 \leq 0.46[/tex]