Answer:
Step-by-step explanation:
Hello!
You need to construct a 95% CI for the population variance of the forces the safety helmets transmit to wearers.
The variable of interest is X: Force a helmet transmits its wearer when an external force is applied (pounds)
Assuming this variable has a normal distribution, the manufacturer expects it to have a mean of μ= 800 pounds and a standard deviation of σ= 40 pounds
A test sample of n=40 was taken and the resulting mean and variance are:
X[bar]= 825 pounds
S²= 2350 pounds²
To estimate the population variance per confidence interval you have to use the following statistic:
[tex]X^2= \frac{(n-1)S^2}{Sigma^2} ~~X^2_{n-1}[/tex]
And the CI is calculated as:
[[tex]\frac{(n-1)S^2}{X^2_{n-1;1-\alpha /2}}[/tex];[tex]\frac{(n-1)S^2}{X^2_{n-1;\alpha /2}}[/tex]]
[tex]X^2_{n-1;1-\alpha /2}= X^2_{39;0.975}= 58.1[/tex]
[tex]X^2_{n-1;\alpha /2}= X^2_{39;0.025}= 23.7[/tex]
[[tex]\frac{39*2350}{58.1}[/tex];[tex]\frac{39*2350}{23.7}[/tex]]
[1577.45; 3867.09] pounds²
Using a confidence level of 95% you'd expect that the interval [1577.45; 3867.09] pounds² contains the value of the population variance of the force the safety helmets transmit to their wearers when an external force is applied.
I hope this helps!
