On average, an American professional football game lasts about three hours, even though the ball is actually in play only 11 minutes (www.sportsgrid.com, January 14, 2014). Assume that game times are normally distributed with a standard deviation of $0.4$ hour.

a. Find the probability that a game lasts less than $2.5$ hours.

b. Find the probability that a game lasts either less than $2.5$ hours or more than $3.5$ hours.

c. Find the maximum value for the game time that will place it in the bottom $1 \%$ of the distribution.

A new car battery is sold with a two-year warranty whereby the owner gets the battery replaced free of cost if it breaks down during the warranty period. Suppose an auto store makes a net profit of $\$ 20$ on batteries that stay trouble-free during the warranty period; it makes a net loss of $\$ 10$ on batteries that break down. The life of batteries is known to be normally distributed with a mean and a standard deviation of 40 and 16 months, respectively.

a. What is the probability that a battery will break down during the warranty period?

b. What is the expected profit of the auto store on a battery?

c. What is the expected monthly profit on batteries if the auto store sells an average of 500 batteries a month?

The time required to assemble an electronic component is normally distributed with a mean and a standard deviation of 16 minutes and 4 minutes, respectively.

a. Find the probability that a randomly picked assembly takes between 10 and 20 minutes.

b. It is unusual for the assembly time to be above 24 minutes or below 6 minutes. What proportion of assembly times fall in these unusual categories?

Sixty percent of a firm's employees are men. Suppose four of the firm's employees are randomly selected.

a. What is more likely, finding three men and one woman or two men and two women?

b. Do you obtain the same answer as in part a if $70 \%$ of the firm's employees had been men?

An estimated $1.8$ million students take on student loans to pay ever-rising tuition and room and board (The New York Times, April 17, 2009). It is also known that the average cumulative debt of recent college graduates is about $\$ 22,500$. Let the cumulative debt among recent college graduates be normally distributed with a standard deviation of $\$ 7,000$. Approximately how many recent college graduates have accumulated student loans of more than $\$ 30,000$ ?

The historical returns on a balanced portfolio have had an average return of $8 \%$ and a standard deviation of $12 \%$. Assume that returns on this portfolio follow a normal distribution.

a. What percentage of returns were greater than $20 \%$ ?

b. What percentage of returns were below $-16 \%$ ?

The margarita is one of the most common tequila-based cocktails, made with tequila mixed with triple sec and lime or lemon juice, often served with salt on the glass rim. A common ratio for a margarita is $2: 1: 1$, which includes $50 \%$ tequila, $25 \%$ triple sec, and $25 \%$ fresh lime or lemon juice. A manager at a local bar is concerned that the bartender uses incorrect proportions in more than $50 \%$ of margaritas. He secretly observes the bartender and finds that he used the correct proportions in only 10 out of 30 margaritas. Test if the manager's suspicion is justified at $\alpha=0.05$.

An investor considers investing $\$ 10,000$ in the stock market. He believes that the probability is $0.30$ that the economy will improve, $0.40$ that it will stay the same, and $0.30$ that it will deteriorate. Further, if the economy improves, he expects his investment to grow to $\$ 15,000$, but it can also go down to $\$ 8,000$ if the economy deteriorates. If the economy stays the same, his investment will stay at $\$ 10,000$.

a. What is the expected value of his investment?

b. Should he invest the $\$ 10,000$ in the stock market if he is risk neutral?

c. Is the decision clear-cut if he is risk averse? Explain.

Suppose that the miles-per-gallon (mpg) rating of passenger cars is a normally distributed random variable with a mean and a standard deviation of $33.8 \mathrm{mpg}$ and $3.5$ mpg, respectively.

a. What is the probability that a randomly selected passenger car gets at least $40 \mathrm{mpg}$ ?

b. What is the probability that a randomly selected passenger car gets between 30 and $35 \mathrm{mpg}$ ?

c. An automobile manufacturer wants to build a new passenger car with an mpg rating that improves upon 99% of existing cars. What is the minimum mpg that would achieve this goal?

Suppose that the miles-per-gallon (mpg) rating of passenger cars is a normally distributed random variable with a mean and a standard deviation of $33.8$ and $3.5 \mathrm{mpg}$, respectively.

a. What is the probability that a randomly selected passenger car gets more than $35 \mathrm{mpg}$ ?

b. What is the probability that the average mpg of four randomly selected passenger cars is more than $35 \mathrm{mpg}$ ?

c. If four passenger cars are randomly selected, what is the probability that all of the passenger cars get more than $35 \mathrm{mpg}$ ?

Suppose that the miles-per-gallon (mpg) rating of passenger cars is normally distributed with a mean and a standard deviation of $33.8$ and $3.5 \mathrm{mpg}$, respectively.

a. What is the probability that a randomly selected passenger car gets more than $35 \mathrm{mpg}$ ?

b. What is the probability that the average mpg of four randomly selected passenger cars is more than $35 \mathrm{mpg}$ ?

c. If four passenger cars are randomly selected, what is the probability that all of the passenger cars get more than $35 \mathrm{mpg}$ ?

(Use computer) Suppose 40% of recent college graduates plan on pursuing a graduate degree. Fifteen recent college graduates are randomly selected.

a. What is the probability that no more than four of the college graduates plan to pursue a graduate degree?

b. What is the probability that exactly seven of the college graduates plan to pursue a graduate degree?

c. What is the probability that at least six but no more than nine of the college graduates plan to pursue a graduate degree?

There is a growing public support for marijuana law reform, with polls showing more than half the country is in favor of some form of marijuana legalization. However, opinions on marijuana are divided starkly along political party lines. The results of the Pew Research Center survey conducted in 2016 are shown in the following table. In addition, assume that $27 \%$ of Americans identify as Republicans, $30 \%$ as Democrats, and $43 \%$ as independents.

$$\begin{array}{|l|c|} \hline \text { Political Party } & \text { Support } \\ \hline \text { Republican } & 41 \% \\ \hline \text { Democrat } & 66 \% \\ \hline \text { Independent } & 63 \% \\ \hline \end{array}$$

b. Calculate the probability that a randomly selected American adult supports marijuana legalization and is a Democrat.

Scores on a marketing exam are known to be normally distributed with a mean and a standard deviation of 60 and 20 , respectively.

a. Find the probability that a randomly selected student scores between $50$ and $80$ .

b. Find the probability that a randomly selected student scores between $20$ and $40$ .

c. The syllabus suggests that the top $15 \%$ of the students will get an $\mathrm{A}$ in the course. What is the minimum score required to get an $\mathrm{A}$ ?