The incomes of all families in a particular suburb can be...

The incomes of all families in a particular suburb can be represented by a continuous random variable. It is known that the median income for all families in this suburb is $$\$ 60,000$$ and that $40 \%$ of all families in the suburb have incomes above $$\$ 72,000$$ a. For a randomly chosen family, what is the probability that its income will be between $$\$ 60,000$$ and $$\$ 72,000 ?$$ b. Given no further information, what can be said about the probability that a randomly chosen family has an income below $\$ 65,000 ?$

The incomes of all families in a particular suburb can be represented by a continuous random variable. It is known that the median income for all families in this suburb is $$\$ 60,000$$ and that $40 \%$ of all families in the suburb have incomes above $$\$ 72,000$$
a. For a randomly chosen family, what is the probability that its income will be between $$\$ 60,000$$ and $$\$ 72,000 ?$$
b. Given no further information, what can be said about the probability that a randomly chosen family has an income below $\$ 65,000 ?$

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Key Concepts

Continuous Random Variable

A continuous random variable is one that can take on any value within a certain interval, and its probabilities are described by a probability density function. The probability of the variable taking on any single exact value is zero, so probabilities are determined by integrating over intervals.

Median

The median of a distribution is the value that separates the distribution into two equal halves, meaning that 50% of the outcomes lie below the median and 50% lie above. In a continuous context, it is the value for which the cumulative distribution function (CDF) equals 0.5.

Quantiles and Percentiles

Quantiles are cut points that divide the range of a probability distribution into contiguous intervals with equal probabilities, while percentiles are quantiles expressed as percentages. Knowing that a certain percentage of outcomes lie above or below a particular value provides insight into the distribution's structure.

Interpreting Cumulative Probabilities

Understanding cumulative probabilities involves analyzing the values of the CDF at various points. For intervals, the probability is found by taking the difference between the CDF values at the interval’s endpoints, giving a measure of the area under the probability density function between those points.

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