Respuesta :
Answer:
[tex]\frac{(99)(10)^2}{123.225} \leq \sigma^2 \leq \frac{(99)(10)^2}{77.046}[/tex]
[tex] 80.341 \leq \sigma^2 \leq 128.495[/tex]
Now we just take square root on both sides of the interval and we got:
[tex] 8.963 \leq \sigma \leq 11.336[/tex]
Step-by-step explanation:
Data given and notation
s=10 represent the sample standard deviation
[tex]\bar x[/tex] represent the sample mean
n=100 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]
The next step would be calculate the critical values. First we need to calculate the degrees of freedom given by:
[tex]df=n-1=100-1=99[/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(0.05,99)" "=CHISQ.INV(0.95,99)". so for this case the critical values are:
[tex]\chi^2_{\alpha/2}=123.225[/tex]
[tex]\chi^2_{1- \alpha/2}=77.046[/tex]
And replacing into the formula for the interval we got:
[tex]\frac{(99)(10)^2}{123.225} \leq \sigma^2 \leq \frac{(99)(10)^2}{77.046}[/tex]
[tex] 80.341 \leq \sigma^2 \leq 128.495[/tex]
Now we just take square root on both sides of the interval and we got:
[tex] 8.963 \leq \sigma \leq 11.336[/tex]
