Chapter 8

The speeds of cars on a road are approximately normally distributed with a mean \(\mu=58 \mathrm{~km} / \mathrm{hr}\) and standard deviation \(\sigma=4 \mathrm{~km} / \mathrm{hr}\) (a) What is the probability that a randomly selected car is going between 60 and \(65 \mathrm{~km} / \mathrm{hr}\) ? (b) What fraction of all cars are going slower than 52 \(\mathrm{km} / \mathrm{hr} ?\)

A person who travels regularly on the \(9: 00 \mathrm{am}\) bus from Oakland to San Francisco reports that the bus is almost always a few minutes late but rarely more than five minutes late. The bus is never more than two minutes early, although it is on very rare occasions a little early. (a) Sketch a density function, \(p(t)\), where \(t\) is the number of minutes that the bus is late. Shade the region under the graph between \(t=2\) minutes and \(t=4\) minutes. Explain what this region represents. (b) Now sketch the cumulative distribution function \(P(t)\). What measurement(s) on this graph correspond to the area shaded? What do the inflection point(s) on your graph of \(P\) correspond to on the graph of \(p ?\) Interpret the inflection points on the graph of \(P\) without referring to the graph of \(p\).

The distribution of IQ scores can be modeled by a normal distribution with mean 100 and standard deviation \(15 .\) (a) Write the formula for the density function of IQ scores. (b) Estimate the fraction of the population with IQ between 115 and 120 .

Let \(p(t)=-0.0375 t^{2}+0.225 t\) be the density function for the shelf life of a brand of banana which lasts up to 4 weeks. Time, \(t\), is measured in weeks and \(0 \leq t \leq 4\). Find the median shelf life of a banana using \(p(t) .\) Plot the median on a graph of \(p(t)\). Does it look like half the area is to the right of the median and half the area is to the left?

Let \(p(t)=-0.0375 t^{2}+0.225 t\) be the density function for the shelf life of a brand of banana which lasts up to 4 weeks. Time, \(t\), is measured in weeks and \(0 \leq t \leq 4\). Find the mean shelf life of a banana using \(p(t) .\) Plot the mean on a graph of \(p(t)\). Does it look like the mean is the place where the density function balances?

Suppose that \(x\) measures the time (in hours) it takes for a student to complete an exam. All students are done within two hours and the density function for \(x\) is $$ p(x)=\left\\{\begin{array}{ll} x^{3} / 4 & \text { if } 0

Let \(p(t)=0.1 e^{-0.1 t}\) be the density function for the waiting time at a subway stop, with \(t\) in minutes, \(0 \leq t \leq 60\). (a) Graph \(p(t)\). Use the graph to estimate visually the median and the mean. (b) Calculate the median and the mean. Plot both on the graph of \(p(t)\). (c) Interpret the median and mean in terms of waiting time.

Suppose \(F(x)\) is the cumulative distribution function for heights (in meters) of trees in a forest. (a) Explain in terms of trees the meaning of the statement \(F(7)=0.6\). (b) Which is greater, \(F(6)\) or \(F(7) ?\) Justify your answer in terms of trees.

Let \(P(x)\) be the cumulative distribution function for the household income distribution in the US in 2006 . Values of \(P(x)\) are in the following table: $$ \begin{array}{|l|c|c|c|c|c|c} \hline \text { Income } x \text { (thousand \$) } & 20 & 40 & 60 & 80 & 100 & 150 \\ \hline P(x)(\%) & 21.7 & 45.4 & 63.0 & 75.8 & 84.0 & 94.0 \\ \hline \end{array} $$ (a) What percent of the households made between \(\$ 40,000\) and \(\$ 60,000\) ? More than \(\$ 150,000\) ? (b) Approximately what was the median income? (c) Is the statement "More than one-third of households made between \(\$ 40,000\) and \(\$ 80,000^{\prime \prime}\) true or false?

Let \(p(t)=-0.0375 t^{2}+0.225 t\) be the density function for the shelf life of a brand of banana, with \(t\) in weeks and \(0 \leq t \leq 4\). See Figure \(8.21\). Find the probability that a banana will last (a) Between 1 and 2 weeks. (b) More than 3 weeks. (c) More than 4 weeks.

A group of people have received treatment for cancer. Let \(t\) be the survival time, the number of years a person lives after the treatment. The density function giving the distribution of \(t\) is \(p(t)=C e^{-C t}\) for some positive constant \(C .\) What is the practical meaning of the cumulative distribution function \(P(t)=\int_{0}^{t} p(x) d x\) ?

Graph a possible density function representing crop yield (in kilograms) from a field under the given circumstance. All yields from 0 to \(100 \mathrm{~kg}\) are equally likely; the field never yields more than \(100 \mathrm{~kg}\).

The probability of a transistor failing between \(t=a\) months and \(t=b\) months is given by \(c \int_{a}^{b} e^{-c t} d t\), for some constant \(c\). (a) If the probability of failure within the first six months is \(10 \%\), what is \(c\) ? (b) Given the value of \(c\) in part (a), what is the probability the transistor fails within the second six months?

While taking a walk along the road where you live, you accidentally drop your glove, but you don't know where. The probability density \(p(x)\) for having dropped the glove \(x\) kilometers from home (along the road) is $$ p(x)=2 e^{-2 x} \quad \text { for } x \geq 0 . $$ (a) What is the probability that you dropped it within 1 kilometer of home? (b) At what distance \(y\) from home is the probability that you dropped it within \(y \mathrm{~km}\) of home equal to \(0.95\) ?

Graph a possible density function representing crop yield (in kilograms) from a field under the given circumstance. A drought makes low yields most common, and there is no yield greater than \(30 \mathrm{~kg}\).

Which of the following functions makes the most sense as a model for the probability density representing the time (in minutes, starting from \(t=0\) ) that the next customer walks into a store? (a) \(p(t)=\left\\{\begin{array}{ll}\cos t & 0 \leq t \leq 2 \pi \\ e^{t-2 \pi} & t \geq 2 \pi\end{array}\right.\) (b) \(p(t)=3 e^{-3 t}\) for \(t \geq 0\) (c) \(p(t)=e^{-3 t}\) for \(t \geq 0\) (d) \(p(t)=1 / 4\) for \(0 \leq t \leq 4\)