Markov’s inequality says that for any distribution with nonnegative values, and any positive real number a, at most E(X) / a of the data is greater than a. (Here E(X) is the mean or expected value of the distribution). We need not know anything about the shape of the distribution. All that is required to use Markov’s inequality is to know that we have positive values in the distribution.
The following are solutions to practice problems related to Markov’s inequality. A version of the worksheet with solutions is located here.
Markov’s Inequality Problems
- A class of fourth graders has mean height of five feet. At most, what percent of the class is 5 and a half feet or taller?
- Light bulbs from a particular company are found to last on average four years before burning out. At most what percent of the light bulbs last for more than twelve years?
- Bacteria in a culture live for a mean time of two hours. At most what fraction of the bacteria live for ten hours or more?
- In a class of 120 the average score on a statistics test was 65%. At most how many students scored at or above 80%?
- In a class of 200 the average score on a biology test was 58%. At most how many students scored at or above 50%?
- Assume that incomes are all nonnegative. If the mean income reported for a country is $40,000, what can we say about the percent of the population with income greater than $1,000,000?
- Given a distribution with nonnegative values and a mean of 18. What is the smallest value of a so that at most 15% of the data of a distribution is greater than or equal to a?