Final answer:
The Weibull distribution calculates probabilities related to the lifetimes of insulating materials under stress. A t-test determines the reasonableness of a company's claim about battery lifespan. The exponential distribution, with its rate parameter, models time to failure for products such as automobile batteries.
Step-by-step explanation:
Analysis of Weibull Distribution for Solid Insulating Materials
The Weibull distribution is used to model the time to failure for insulating materials under stress, such as AC voltage. Given the Weibull distribution parameters β = 2.7 and α = 220, we can calculate the probabilities for different time-to-failure scenarios. The probability that a specimen's lifetime is at most 250 hours or less than 250 hours requires integration of the Weibull probability density function up to 250 hours. To find the probability of a specimen lasting more than 300 hours, one would integrate the probability density function from 300 hours to infinity. In practice, such calculations are often done using statistical software or a calculator with the capability to handle Weibull functions.
To address the statistics class question about NeverReady batteries, we need to determine if it is likely to obtain a sample mean life span of 16.7 hours or less from 30 randomly selected batteries, given the population mean of 17 hours and a standard deviation of 0.8 hours. This can be calculated using a t-test for the sample mean. Our null hypothesis would be that the sample mean is the same as the claimed average life span. Failure to reject the null hypothesis suggests the company's claim is reasonable.
For the exponential distribution, which is often used to model time to failure for products like automobile batteries, if m = 10, both the mean and standard deviation are 10. The probability density function for an exponential distribution with rate λ is given by f(x) = λ e^{-λ x}.
When conducting hypothesis testing for mean life spans in a county, one might use a t-test to compare the means from two independent samples (white and nonwhite death records). This test would determine if the observed differences in sample means are statistically significant.