Answer
Plan: We will use the Central Limit Theorem to estimate the probability.
Solve: Here, we have n = 10000, $μ = 125$, σ = 300, x̅ = 135
$σ_{x̅} = \frac{σ}{\sqrt n} = \frac{300 }{\sqrt 10000} = 3$
We need to calculate the z score for the probability that x̅ takes a value between 112 and 118 mg/dL.
$z = \frac{x̅ - μ_{x̅}}{σ_{x̅}} = \frac{135-125}{3} = 3.33$
P(z < 3.33) = 0.9996
Conclusion: We can be 99.96% sure that the losses will not exceed $135 per policy on average.
Work Step by Step
Plan: We will use the Central Limit Theorem to estimate the probability.
Solve: Here, we have n = 10000, $μ = 125$, σ = 300, x̅ = 135
$σ_{x̅} = \frac{σ}{\sqrt n} = \frac{300 }{\sqrt 10000} = 3$
We need to calculate the z score for the probability that x̅ takes a value between 112 and 118 mg/dL.
$z = \frac{x̅ - μ_{x̅}}{σ_{x̅}} = \frac{135-125}{3} = 3.33$
P(z < 3.33) = 0.9996
Conclusion: We can be 99.96% sure that the losses will not exceed $135 per policy on average.
