Question

Suppose that the probability is \(.1\) that any given citrus tree will show measurable damage when the temperature falls to \(30^{\circ} \mathrm{F}\). If the temperature does drop to \(30^{\circ} \mathrm{F}\), what is the expected number of citrus trees showing damage in orchards of 2000 trees? What is the standard deviation of the number of trees that show damage?

Short answer

The expected number of citrus trees showing damage is 200 and the standard deviation is approximately 13.42.

Step-by-step solution

Find the Expected Number

Use the formula for the expectation of a binomial random variable, which is \(E(X) = np\). Here, \(n = 2000\) (the number of trees) and \(p = .1\) (the probability that any given tree shows measurable damage). Substituting these values into the formula, you get \(E(X) = np = 2000 * .1 = 200\). So the expected number of trees showing damage is 200.

Find the Standard Deviation

Use the formula for the standard deviation of a binomial random variable, which is \(\sigma = \sqrt{np(1-p)}\). Here, \(n = 2000\), \(p = .1\), and \(1 - p = .9\). Substituting these values into the formula, you get \(\sigma = \sqrt{np(1-p)} = \sqrt{2000 * .1 * .9} = \sqrt{180} \approx 13.42\). So the standard deviation of the number of trees that show damage is approximately 13.42.