TUTORIAL SHEET 5

Lognormal, Exponential and Weibull Distributions

Retrieve the six_lifetimes.mtw Minitab worksheet to use in questions 1-3.

1. In tutorial 3 it was shown that the lognormal distribution was a good fit for the lifetime of component E. Use the probability plot to estimate the location and scale parameters. (These are the mean and variance of the underlying normal distribution.) Hence use normal tables to find:

(i) the probability that the component will cease working before 50 hours of operation

(ii) the lifetime that is exceeded by 95% of components.

2. It was also shown that the exponential distribution was a good fit for the lifetime of component B. From the data, calculate the sample mean lifetime and use it to estimate the rate l (). Confirm this with the estimate provided on the probability plot. Use this value for l in the cumulative distribution function (cdf) to calculate:

(i) the probability that this component will still be working after 2500 hours

(ii) the percentage of components of this type with lifetimes between 1000 and 2000 hours

(iii) the lifetime that is exceeded by only 5% of components

(iv) the median lifetime.

Compare the calculated median lifetime in (iv) with the median of the sample of 100 lifetimes.

3. In a similar fashion, the Weibull distribution was a good model for the lifetimes of component D once the outlier is removed.

(i) Look at the shape of the histogram of the data for this component and compare it with the shapes on page 92 of the lecture notes. Write down your guess at the value of the shape parameter b.

(ii) Estimate the parameters a and b from the probability plot.

(iii) Use these estimates and the cdf to calculate the reliability of this component at 400 and at 1400 hours of operation.

(iv) Use the appropriate distribution under Calc®Probability Distributions to confirm your answers in (iii).

4. A system which operates continuously develops faults at random with an average of 14 faults per week. The system fails at midday on a Monday.

(i) What is the probability that it will fail again before midnight?

(iii) What is the probability that it will operate without faults for at least 2 days?

5. The time to failure (in hours) of a bearing in a mechanical shaft is satisfactorily modelled by a Weibull distribution with a = 5000 hours and b = 0.5. Find the percentage of bearings that will fail before 6000 hours of operation.