Respuesta :
Answer:
[tex]\frac{(78)(2.37)^2}{104.316} \leq \sigma^2 \leq \frac{(78)(2.37)^2}{55.466}[/tex]
[tex] 4.254 \leq \sigma^2 \leq 8.000[/tex]
And for the deviation we just need to take the square root of the variance and we got:
[tex] 2.06 \leq \sigma^2 \leq 2.83[/tex]
Step-by-step explanation:
Data given and notation
s=2.37 represent the sample standard deviation
[tex]\bar x[/tex] represent the sample mean
n=79 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=79-1=78[/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 values.
The excel commands would be: "=CHISQ.INV(0.025,78)" "=CHISQ.INV(0.975,78)". so for this case the critical values are:
[tex]\chi^2_{\alpha/2}=104.316[/tex]
[tex]\chi^2_{1- \alpha/2}=55.466[/tex]
And replacing into the formula for the interval we got:
[tex]\frac{(78)(2.37)^2}{104.316} \leq \sigma^2 \leq \frac{(78)(2.37)^2}{55.466}[/tex]
[tex] 4.254 \leq \sigma^2 \leq 8.000[/tex]
And for the deviation we just need to take the square root of the variance and we got:
[tex] 2.06 \leq \sigma^2 \leq 2.83[/tex]
