Respuesta :
Answer:
And on this case we have a 0.0303% of chance that the sample variance would be higher than 20. So on this case its not reasonable that the sample variance would be higher than 20.
Step-by-step explanation:
Notation and previous concepts
A chi-square test is "used to test if the variance of a population is equal to a specified value. This test can be either a two-sided test or a one-sided test. The two-sided version tests against the alternative that the true variance is either less than or greater than the specified value"
[tex]n=20[/tex] represent the sample size
[tex]s^2 =20 [/tex] represent the sample variance obtained
[tex]\sigma^2 =8[/tex] represent the population variance
We can calculate the probability that [tex]S^2[/tex] would be higher than 20 and we will have an idea how to solve the problem.
For this case we can use the following statistic:
[tex]\chi^2 =\frac{n-1}{\sigma^2} s^2[/tex]
And this statistic is distributed chi square with n-1 degrees of freedom. We have eveything to replace.
[tex]\chi^2 =\frac{20-1}{8} 20 =47.5[/tex]
Calculate the probability
We can calculate the probability that [tex]S^2[/tex] would be higher than 20. The degrees of freedom are given by n-1=20-1=19
[tex]P(S^2 >20) =P(\chi^2_{19} >47.5)=0.000303[/tex]
And hte reason of this is this one:
[tex]P(S^2 >20)= P(\frac{\chi^2 \sigma^2}{n-1}>20)=P(\chi^2 >\frac{20(n-1)}{\sigma^2})=P(\chi^2 >\frac{20*19}{8})=P(\chi^2 >47.5)[/tex]
In order to calculate this probability we can use the following code in excel:
"==1-CHISQ.DIST(47.5,19,TRUE)"
And on this case we have a 0.0303% of chance that the sample variance would be higher than 20. So on this case its not reasonable that the sample variance would be higher than 20.
