Question

Medicare Hospital Insurance The average yearly Medicare Hospital Insurance benefit per person was $4064 in a recent year. Suppose the benefits are normally distributed with a standard deviation of $460. Round the final answers to at least four decimal places and intermediate z-value calculations to two decimal places. Part 1 of 2 Find the probability that the mean benefit for a random sample of 28 patients is less than $3870. p(x < 3870) - LE dl. Part 2 of 2 Find the probability that the mean benefit for a random sample of 28 patients is more than $4160 p(x>4160) -

Answer 1

Facts : 1) If X, Z be the normal and standard normal distibuted random variables respectively. To find the probabilites for X we convert X to Z by using Z X- o Where u = mean(X) o = standard deviation(X) X = The insurance benefit (in dollars) of a patient. Given that X follows an unknown distribution with mean(u) = 4064 and standard deviation() = 460. X is sample mean(Sample average) random variable. Sample size = n = 28 Mg = u = 4064 03 460 V28 86.9318 vn

a) Now we have to find PX <3870). P(I <3870) = P(X – už <3870 – 4064) X-up 3870-4064 OX 86.9318 = P(Z < -2.23) = 0.0129 :: P(X < 3870) = 0.0129

b) Now we have to find P(x > 4160). P( > 4160) = P(1 – už > 4160 – 4064) 4160-4064 = P(XMB) 07 86.9318 = P(Z > 1.10) = 1- P(Z < 1.10) = 1-0.8643 = 0.1357 :: P(X> 4160) = 0.1357

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