Optimal goal target in soccer. When attempting to score a goal in soccer, where should you aim your shot? Should you aim for a goalpost (as some soccer coaches teach), the middle of the goal, or some other target? To answer these questions, Chance (Fall 2009) utilized the normal probability distribution. Suppose the accuracy $x$ of a professional soccer player's shots follows a normal distribution with a mean of 0 feet and a standard deviation of 3 feet. (For example, if the player hits his target, $x=0$; if he misses his target 2 feet to the right, $x=2$; and if he misses 1 foot to the left, $x=-1$.) Now, a regulation soccer goal is 24 feet wide. Assume that a goalkeeper will stop (save) all shots within 9 feet of where goalkeeper who stands in the middle of the goal. If the player aims for halfway between the right goalpost and the outer limit of the goalkeeper's reach, what is the probability that he will score?

Optimal goal target in soccer. When attempting to score a goal in soccer, where should you aim your shot? Should you aim for a goalpost (as some soccer coaches teach), the middle of the goal, or some other target? To answer these questions, Chance (Fall 2009) utilized the normal probability distribution. Suppose the accuracy $x$ of a professional soccer player's shots follows a normal distribution with a mean of 0 feet and a standard deviation of 3 feet. (For example, if the player hits his target, $x=0$; if he misses his target 2 feet to the right, $x=2$; and if he misses 1 foot to the left, $x=-1$.) Now, a regulation soccer goal is 24 feet wide. Assume that a goalkeeper will stop (save) all shots within 9 feet of where goalkeeper who stands in the middle of the goal. If the player aims for the right goalpost, what is the probability that he will score?

Ambulance response time. Ambulance response time is measured as the time (in minutes) between the initial call to emergency medical services (EMS) and when thepatient is reached by ambulance. Geographical Analysis (Vol. 41, 2009) investigated the characteristics of ambulance response time for EMS calls in Edmonton, Alberta. For a particular EMS station (call it Station A), ambulance response time is known to be normally distributed with $\mu=7.5$ minutes and $\sigma=2.5$ minutes. A randomly selected EMS call in Edmonton has an ambulance response time of 2 minutes. Is it likely that this call was serviced by Station A? Explain.

Ambulance response time. Ambulance response time is measured as the time (in minutes) between the initial call to emergency medical services (EMS) and when the patient is reached by ambulance. Geographical Analysis (Vol. 41, 2009) investigated the characteristics of ambulance response time for EMS calls in Edmonton, Alberta. For a particular EMS station (call it Station A), ambulance response time is known to be normally distributed with $\mu=7.5$ minutes and $\sigma=2.5$ minutes. Regulations require that $90 \%$ of all emergency calls be reached in 9 minutes or less. Are the regulations met at EMS Station A? Explain.

Consider the probability distribution for the random variable x shown here and locate $\mu$ and the interval $\mu \pm 2 \sigma$ on your graph. What is the probability that $x$ will fall within the interval $\mu \pm 2 \sigma ?$

Consider the probability distribution for the random variable x shown here and graph $p(x)$.

Consider the possibility distribution for the random variable x shown here and find $\mu, \sigma^2$, and $\sigma$.

Whistle-blowing among federal employees. Whistleblowing refers to an employee's reporting of wrongdoing by co-workers. A survey found that about $5 \%$ of employees contacted had reported wrongdoing during the past 12 months. Assume that a sample of 25 employees in one agency are contacted and let $x$ be the number who have observed and reported wrongdoing in the past 12 months. Assume that the probability of whistle-blowing is $.05$ for any federal employee over the past 12 months. If 5 of the 25 contacted have been whistle-blowers over the past 12 months, what would you conclude about the applicability of the $5 \%$ assumption to this agency? Use your answer to part b to justify your conclusion.

Whistle-blowing among federal employees. Whistleblowing refers to an employee's reporting of wrongdoing by co-workers. A survey found that about $5 \%$ of employees contacted had reported wrongdoing during the past 12 months. Assume that a sample of 25 employees in one agency are contacted and let $x$ be the number who have observed and reported wrongdoing in the past 12 months. Assume that the probability of whistle-blowing is $.05$ for any federal employee over the past 12 months. Write the event that at least 5 of the employees are whistle-blowers in terms of $x$. Find the probability of the event.

Whistle-blowing among federal employees. Whistleblowing refers to an employee's reporting of wrongdoing by co-workers. A survey found that about $5 \%$ of employees contacted had reported wrongdoing during the past 12 months. Assume that a sample of 25 employees in one agency are contacted and let $x$ be the number who have observed and reported wrongdoing in the past 12 months. Assume that the probability of whistle-blowing is $.05$ for any federal employee over the past 12 months. Find the mean and standard deviation of $x$. Can $x$ be equal to its expected value? Explain.

Detecting a computer virus attack. Chance (Winter 2004) presented basic methods for detecting virus attacks (e.g., Trojan programs or worms) on a network computer that are sent from a remote host. These viruses reach the network through requests for communication (e.g., e-mail, Web chat, or remote log-in) that are identified as "packets." For example, the "SYN flood" virus ties up the network computer by "flooding" the network with multiple packets. Cybersecurity experts can detect this type of virus attack if at least one packet is observed by a network sensor. Assume that the probability of observing a single packet sent from a new virus is only .001. If the virus actually sends 150 packets to a network computer, what is the probability that the virus is detected by the sensor?

Errors in measuring truck weights. To help highway planners anticipate the need for road repairs and design future construction projects, data are collected on the estimated volume and weight of truck traffic on specific roadways (Transportation Planning Handbook, 2016) using specialized "weigh-in-motion" equipment. In an experiment involving repeated weighing of a 27,907-pound truck, it was found that the weights recorded by the weigh-in-motion equipment were approximately normally distributed with a mean of 27,315 and a standard deviation of 628 pounds (Minnesota Department of Transportation). It follows that the difference between the actual weight and recorded weight, the error of measurement, is normally distributed with a mean of 592 pounds and a standard deviation of 628 pounds. It is possible to adjust (or calibrate) the weigh-in motion equipment to control the mean error of measurement. At what level should the mean error be set so the equipment will understate the weight of a 27,907 -pound truck $50 \%$ of the time? Only $40 \%$ of the time?

Errors in measuring truck weights. To help highway planners anticipate the need for road repairs and design future construction projects, data are collected on the estimated volume and weight of truck traffic on specific roadways (Transportation Planning Handbook, 2016) using specialized "weigh-in-motion" equipment. In an experiment involving repeated weighing of a 27,907-pound truck, it was found that the weights recorded by the weigh-in-motion equipment were approximately normally distributed with a mean of 27,315 and a standard deviation of 628 pounds (Minnesota Department of Transportation). It follows that the difference between the actual weight and recorded weight, the error of measurement, is normally distributed with a mean of 592 pounds and a standard deviation of 628 pounds. What is the probability that the error in the weight recorded by the weigh-in-motion equipment for a 27,907-pound truck exceeds 400 pounds?

Errors in measuring truck weights. To help highway planners anticipate the need for road repairs and design future construction projects, data are collected on the estimated volume and weight of truck traffic on specific roadways (Transportation Planning Handbook, 2016) using specialized "weigh-in-motion" equipment. In an experiment involving repeated weighing of a 27,907-pound truck, it was found that the weights recorded by the weigh-in-motion equipment were approximately normally distributed with a mean of 27,315 and a standard deviation of 628 pounds (Minnesota Department of Transportation). It follows that the difference between the actual weight and recorded weight, the error of measurement, is normally distributed with a mean of 592 pounds and a standard deviation of 628 pounds. If a 27,907-pound truck was driven over the weigh in-motion equipment 100 times, approximately how many times would the equipment overstate the truck's weight?