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 9 and half years and 7 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 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.)

c. What proportion of the washing machines will last for less than 4 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 90th percentile of the life of the washing machines. (Round your intermediate calculations to at least 4 decimal places, “z” value to 3 decimal places, and final answer to the nearest whole number.)

Answer 1

a. The mean of X is approximately 1.7918 and the standard deviation of X is approximately 0.7778. b. The proportion of washing machines lasting more than 11 years is approximately 0.3872. c. The proportion of washing machines lasting less than 4 years is approximately 0.2389. d. The 90th percentile of the life of washing machines is approximately 19 years.

Step-by-step explanation:

a. Mean and standard deviation of X:

The transformation Y = eX follows a lognormal distribution. To find the mean and standard deviation of X, we need to use the properties of the lognormal distribution.

Mean of X = ln(mean of Y) - [(1/2) * (std deviation of Y)^2]

Mean of X = ln(9) - [(1/2) * (7)^2]

Mean of X ≈ 1.7918

Standard deviation of X = (std deviation of Y) / (mean of Y)

Standard deviation of X = 7 / 9

≈ 0.7778

b. Proportion of washing machines lasting more than 11 years:

To find this proportion, we need to find the area to the right of 11 years under the standard normal distribution curve.

Z-value for 11 years = (X - mean) / standard deviation

= (11 - 9) / 7

≈ 0.2857

Proportion = 1 - normalcdf(0.2857)

≈ 1 - 0.6128

≈ 0.3872

c. Proportion of washing machines lasting less than 4 years:

To find this proportion, we need to find the area to the left of 4 years under the standard normal distribution curve.

Z-value for 4 years = (X - mean) / standard deviation

= (4 - 9) / 7

≈ -0.7143

Proportion = normalcdf(-0.7143) ≈ 0.2389

d. 90th percentile of the life of washing machines:

To find the 90th percentile, we need to find the z-value corresponding to the 90th percentile and then convert it back to the original scale.

Z-value for 90th percentile = invNorm(0.9) ≈ 1.2816

90th percentile of X = mean + (Z-value * standard deviation)

= 9 + (1.2816 * 7)

≈ 18.97

≈ 19 years