A six-sided die is rolled 100times. Using the normal approximation, find the probabilitythat the face showing a six turns up between 15 and 20 times. Find the probability that the sum of the face values of the 100trials is less than 300.
What I know:
(a) Xi ~ i. i.d. Bernoulli (p = 1/6), μ = p = 1/6 and σ2 = p(1-p) and n = 100. Since np > 10, our rule-of-thumb for the CLT to hold forBinomial r. v.'s applies, so if
100
X = Σ Xi , by the CLT, X is approximatelyN(np, np(1-p)). Now use the CLT and Normal tables to
i=1
approximate P( 15X20). The answer should be around 0.49 (without thecontinuity correction)
(b) Now Yi = face value on the ithroll, so μ = E[Yi] = 3.5 and σ2 =Var[Yi] = 2.92 (you must verify both thesecalculations directly from the fact the Yi is discreteuniform on the outcome values). Then
n=100
Sn = Σ Yiand from here we are ready to apply the CLT again to thissum.
i=1