Answer to Question #295170 in Statistics and Probability for john k

Question

(a) The central limit theorem is perhaps the most important result in statistics. State and prove it.

(b) The times spent by customers coming to a Kalingalinga Total gas station to fill up can be viewed as independent random variable with mean 3 minutes and variance 1.5 minutes. Approximate the chance that a random sample of 75 customers in this gas station will spend a total time less than 3 hours.

(c) Let 𝑋1,𝑋2, … , 𝑋𝑛 be a random sample from a population with mean 𝜇 and variance 𝜎². Consider the sample variance

𝑆 2 = 1 1 − 𝑛 ∑(𝑋𝑖 − 𝑋̅) 2 𝑛 𝑖=1

Show that 𝐸[𝑆 2 ] = 𝜎 2 .

(d) Suppose 𝑋₁,𝑋₂, … 𝑋₁₀ is a random sample from a standard normal distribution. Determine number 𝛼 and 𝛽 such that

𝑃 (𝛼 ≤ ∑𝑋𝑖 2 10 𝑖=1 ≤ 𝛽) = 0.95