Question

It is well documented that a typical washing machine can last anywhere between 5 to 20 years. Let the life of a washing machine be represented by a lognormal variable, Y = eX where X is normally distributed. In addition, let the mean and standard deviation of the life of a washing machine be 15 and half years and 6 years, respectively. [You may find it useful to reference the z table.] a. Compute the mean and the standard deviation of X. (Round your intermediate calculations to at least 4 decimal places and final answers to 4 decimal places.) b. What proportion of the washing machines will last for more than 17 years? (Round your intermediate calculations to at least 4 decimal places, “z” value to 2 decimal places, and final answer to 4 decimal places.) c. What proportion of the washing machines will last for less than 11 years? (Round your intermediate calculations to at least 4 decimal places, “z” value to 2 decimal places, and final answer to 4 decimal places.) d. Compute the 75th percentile of the life of the washing machines

Answer 1

here we have,

mean = 15.5

standard deviation = 6

a) mean of X,

mean = ln(μ/(√((σ²/μ²)+1))) = ln(15.5/(√((6²/15.5²)+1))) = ln(14.4549) = 2.6710

standard deviation = √(ln((σ²/μ²)+1)) = √(ln((6²/15.5²)+1)) = 0.3736

b) P(X > 17) = P(Z > (ln(17) - 2.6710) / 0.3736) = 0.4336

c) P(X < 11) = P(Z < (ln(11) - 2.6710) / 0.3736) = P(Z < -0.7310) = 0.2324

d) for 75th percentile, Z = 0.675 (you can calculate this from Z - table or online calculator)

hence the corresponding value = e^{2.6710 + 0.675*0.3736} = 18.6 = ~ 18

please revert if any doubts