Reliability and Hazard Rate Analysis of Aircraft Fuel Pumps Using the Weibull Distribution

Reliability and Hazard Rate Analysis of Aircraft Fuel Pumps Using the Weibull Distribution

Reliability

Doing this in Minitab, we get the following output:

Weibull with shape = 1 and scale = 30

x	P( X ≤ x )
10	0.283469

This means that the hazard rate function for the pumps in 10 years is 0.2835.

ANOVA

Question 1

After running the analysis of variance (ANOVA) of the data for the five different types of oils in Minitab, we have the following output:

Analysis of Variance

Source	DF	Adj SS	Adj MS	F-Value	P-Value
Factor	4	948.0	237.00	5.47	0.003
Error	25	1084.0	43.36
Total	29	2032.0

The analysis of variance (ANOVA) was conducted to assess whether the quantity of oil that onion rings absorb during frying varies significantly across five different kinds of oil (corn, olive, peanut, sunflower, and soybean). The data from six batches of onion rings were considered. Based on the output, the p-value for the factor (type of oil) is 0.003, which is less than the significance level of 0.05. Accordingly, this means that the null hypothesis stating that oil absorption in frying onion rings is similar for all five kinds of oils can be rejected. Further, the F-value of 5.47 also indicates that there is a statistically significant difference in the mean absorption of oil among the different kinds of oil. As a result, one can conclude that the quantity of oil that onion rings absorb is dependent on the type of oil used for frying. The analysis suggests that at least one of the oil types results in a significantly different amount of oil absorption compared to the others.

Question 2

The Minitab ANOVA analysis of the treatment done on the types of gasoline is as follows:

Analysis of Variance

Source	DF	Adj SS	Adj MS	F-Value	P-Value
Factor	3	68.20	22.73	1.28	0.314
Error	16	283.60	17.72
Total	19	351.80

The ANOVA output above shows that the p-value for the factor (type of gasoline) is 0.314, which is greater than the typical significance level of 0.05. This indicates that we fail to reject the null hypothesis that the mean octane numbers are equal across the five different types of gasoline. The F-value of 1.28 is also not statistically significant. Therefore, based on the data from the randomized complete block design (RCBD) experiment, there is no evidence to suggest that the octane numbers differ significantly among the five gasolines. However, it is important to note that the power of the test may be limited because of the small sample size and large experimental error (as indicated by the relatively high mean square error of 17.72). A larger sample size or more precise measurements may be needed to detect potential differences in octane numbers between the gasoline types.

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Suppose that the lifetime in years of fuel pumps used in an aircraft gas turbine engine is modelled by the Weibull distribution, with threshold parameter 0, shape parameter, and scale parameter. Find the reliability and the hazard rate function for these pumps at 10 years.

ANOVA

1. Five different types of oil (olive, soybean, corn, peanut, and sunflower) are often used for frying onion rings. It is not known whether the amount of oil absorbed by the onion rings depends on the type of oil. For five types of oil, certain batches of equal size (6) of onion rings are prepared. The experiment was carried out in random order. The data in the Table below shows the amount of oil (in grams) absorbed per batch. We want to test a hypothesis at the 5% level of significance that the absorption of oil in frying onion rings is the same for all five types of oil.

2. The quality of gasoline is usually determined by its octane number. An experimenter determines the octane numbers of five gasolines using four different methods. Since “Methods” is a nuisance variable, the experimenter decided to use an RCB design. The experiment in each block was carried out in random order. The data obtained are shown in the Table below. Analyze these data using MINITAB.