One of the most widely used quantitative estimation of radiation sensitivity of a population of experimental animals is the median lethal dose, LD50, within a specified period. The LD50 is the dosage that would kill 50% of a population, i.e. half of the population will die when subjected to the LD50 dose. A 30-day observation period is commonly used to tabulating mortality and the dose necessary to kill 50 per cent of the animals.
The choice of 50% lethality as a benchmark avoids the potential for ambiguity of making measurements in the extremes and reduces the amount of testing required. However, this also means that LD50 is not the lethal dose for all subjects; some may be killed by much less, while others survive doses far higher than the LD50. In practice, the LD50 is an interpolated value based on the response of subsamples to graded doses of radiation.
Assume we can model the radiation lethal dose for a particular mouse strain using a normal distribution with population mean of 700 rad and population standard deviation of 50 rad. I.e., let X be a random variable representing the radiation dose (in rad) required to kill within 30 days, X ∼ N(700, 502).
(a) What is the LD50 in this scenario. Provide a one sentence justification. (Hint: what is the median of the random variable X?)
(b) What proportion of mice would die when exposed to a dose of 800 rad or more?
(c) What dosage is required to kill 99% of mice? (Hint: this is asking for LD99.)
(d) Researchers have identified a particular gene they think makes mice more resistant to radiation. A new strain of mice has been bred that has the radiation resistant gene. The researchers claim that 90% of these radiation resistant mice would survive when administered a dose of 700 rad. To test this claim, a random sample of 25 radiation resistant mice is administered a dose of 700 rad. If the claim is true, what is the distribution of the number of mice who survive for 30 days? Explain your reasoning.
(e) Of the 25 radiation resistant mice, 19 survive for 30 days. What is the probability of getting at most 19 survivers in a random sample of 25 if the researchers claim is true?
(f) Based on your result in (e), do you believe the researchers claim? Why or why not?
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Sample size, n = 25
The distribution of the number of mice who survive for 30 days will follow B (90%, 25) for x = 0, 1, 2, …, n
All assumptions of binominal distribution are satisfied in this case:
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