Statistics - distribution function and median lifetime

[jameswill1am]

Homework Statement

given distribution function F(t)=1-(1+t)^-c, 0<_t<_infinity, c>0. And that a company claim the median lifetime of a component is at least 7 years and that a component fails after 2.22. Using only this information test the claim that the median lifetime is in fact shorter than 7 years.

Homework Equations

The Attempt at a Solution

Setting F(t) equal to a half and solving for t i get the median lifetime is 2^(1/c)-1 years. After that however I'm unsure to proceed. What is an appropriate test in this case involving median?

 

[mighty2000]

I would approach this problem by conducting a hypothesis test. The null hypothesis would be that the median lifetime is equal to 7 years, and the alternative hypothesis would be that the median lifetime is less than 7 years. I would then use the given information, specifically that a component fails after 2.22 years, to calculate the probability of observing a value less than or equal to 2.22 under the assumption that the null hypothesis is true. This probability can be calculated using the distribution function F(t).

If this probability is very low, then it would suggest that the null hypothesis is not supported by the data and that the alternative hypothesis is more likely to be true. In this case, it would indicate that the median lifetime is indeed shorter than 7 years. On the other hand, if the probability is not low, then it would suggest that the null hypothesis is still plausible and that there is not enough evidence to reject it.

In addition to this, I would also consider other factors such as the sample size and the reliability of the data in making a conclusion. It is important to have a sufficient sample size and reliable data in order to make accurate conclusions about the population. I would also consider conducting further experiments or collecting more data to provide more evidence for or against the claim.