The length of a certain kind of Colorado brook trout is normally distributed with a mean of 12.5 inches and a standard deviation of 1.2 inches. What minimum size limit should the Department of Natural Resources set if it wishes to allow people to keep 80 percent of the trout they catch?

In a certain microwave oven on the high power setting, the time it takes a randomly chosen kernel of popcorn to pop is normally distributed with a mean of 140 seconds and a standard deviation of 25 seconds. What percentage of the kernels will fail to pop if the popcorn is cooked for

(a) 2 minutes?

(b) Three minutes?

(c) If you wanted 95 percent of the kernels to pop, what time would you allow?

(d) If you wanted 99 percent to pop?

If the weight (in grams) of cereal in a box of Lucky Charms is $N$ (470,5), what is the probability that the box will contain less than the advertised weight of 453 gm?

A study found that the mean waiting time to see a physician at an outpatient clinic was 40 minutes with a standard deviation of 28 minutes. Use Excel to find each probability.

(a) What is the probability of more than an hour’s wait?

(b) Less than 20 minutes?

(c)At least 10 minutes. (Data are from J. C. Bennett and D. J. Worthington, “An Example of Good but Partially Successful OR Engagement: Improving Outpatient Clinic Operations,” Interfaces 28, no. 5 [September-October 1998], pp. 56–69.)

The fracture strength of a certain type of manufactured glass is normally distributed with a mean of 579 MPa with a standard deviation of 14 MPa. (a) What is the probability that a randomly chosen sample of glass will break at less than 579 MPa? (b) More than 590 MPa? (c) Less than 600 MPa?

Find the standard normal area for each of the following, showing your reasoning clearly and indicating which table you used.

$$P(1.22<Z<2.15)$$

Assume the weight of a randomly chosen American passenger car is a uniformly distributed random variable ranging from 2,500 pounds to 4,500 pounds. (a) What is the mean weight of a randomly chosen vehicle? (b) The standard deviation? (c) What is the probability that a vehicle will weigh less than 3,000 pounds? (d) More than 4,000 pounds? (e) Between 3,000 and 4,000 pounds?

A rock concert promoter has scheduled an outdoon concert on July $4\text{th}$. If it does not rain, the promoter will make $\$ 30,000$. If it does rain, the promoter will lose $\$ 15,000$ in guarantees made to the band and other expenses. The probability of rain on the $4\text{th}$ is $.4$.

In order to break even, how much should an insurance company charge to insure the promoter’s full losses? Explain your answer.

On hot, sunny, summer days, Jane rents inner tubes by the river that runs through her town. Based on her past experience, she has assigned the following probability distribution to the number of tubes she will rent on a randomly selected day.

(a) Calculate the expected value and standard deviation of this random variable X by using the PDF shown.

(b) Describe the shape of this distribution.

On hot, sunny, summer days, Jane rents inner tubes by the river that runs through her town. Based on her past experience, she has assigned the following probability distribution to the number of tubes she will rent on a randomly selected day.

(a) Find P(X = 75).

(b) Find P(X $\le$ 75).

(c) Find P(X > 50).

(d) Find P(X < 100).

(e) Which of the probability expressions in parts (a)–(d) is a value of the CDF?

On hot, sunny, summer days, Jane rents inner tubes by the river that runs through her town. Based on her past experience, she has assigned the following probability distribution to the number of tubes she will rent on a randomly selected day. (a) Calculate the expected value and standard deviation of this random variable $X$ by using the PDF shown. (b) Describe the shape of this distribution.

$$\begin{array}{cccccc} \hline \boldsymbol{x} & 25 & 50 & 75 & 100 & \text { Total } \\ \boldsymbol{P}(\boldsymbol{x}) & .20 & .40 & .30 & .10 & 1.00 \\ \hline \end{array}$$

Consider the following situations for Shocker: a. On November 28, 2018, Shocker receives a $4,500 payment from a customer for services to be rendered evenly over the next three months. Deferred Revenue is credited. b. On December 1, 2018, the company pays a local radio station$2,700 for 30 radio ads that were to be aired, 10 per month, throughout December, January, and February. Prepaid Advertising is debited. c. Employee salaries for the month of December totaling $8,000 will be paid on January 7, 2019. d. On August 31, 2018, Shocker borrows$70,000 from a local bank. A note is signed with principal and 9% interest to be paid on August 31, 2019. Record the necessary adjusting entries for Shocker at December 31, 2018. No adjusting entries were made during the year.

The amount of fill in a half-liter (500 ml) soft drink bottle is normally distributed. The process has a standard deviation of 5 ml. The mean is adjustable.

(a) Where should the mean be set to ensure a 95 percent probability that a half-liter bottle will not be underfilled?

(b) A 99 percent probability?

(c) A 99.9 percent probability? Explain.

Write the Excel binomial formula for each probability.

a. Three successes in 20 trials with a 30 percent chance of success.

b. Seven successes in 50 trials with a 10 percent chance of success.

c. Six or fewer successes in 80 trials with a 5 percent chance of success.

d. At least 30 successes in 120 trials with a 20 percent chance of success.