Exam 4: Random Variables and Probability Distributions

A certain baseball player hits a home run in 3% of his at-bats. Consider his at-bats as independent events. How many home runs do we expect the baseball player to hit in 600 at-bats?

A lab orders a shipment of 100 frogs each week. Prices for the weekly shipments of frogs follow the distribution below: Price \ 10.00 \ 12.50 \ 15.00 Probability 0.3 0.4 0.3 How much should the lab budget for next year's frog orders assuming this distribution does not change? (Hint: Find the expected price and assume 52 weeks per year.)

Transportation officials tell us that 80% of drivers wear seat belts while driving. What is the probability of observing 518 or fewer drivers wearing seat belts in a sample of 700 drivers?

According to a published study, 1 in every 8 men has been involved in a minor traffic accident. Suppose we have randomly and independently sampled twenty-five men and asked each whether he has been involved in a minor traffic accident. How many of the 25 men do we expect to have never been involved in a minor traffic accident? Round to the nearest whole number.

A paint machine dispenses dye into paint cans to create different shades of paint. The amount of dye dispensed into a can is known to have a normal distribution with a mean of 5 milliliters (ml) and a standard deviation of 0.4 ml. Answer the following questions based on this information. Find the dye amount that represents the 9th percentile of the distribution.

The number of goals scored at each game by a certain hockey team follows a Poisson distribution with a mean of 5 goals per game. Find the probability that the team will score more than three goals during a game.

Determine if it is appropriate to use the normal distribution to approximate a binomial distribution when $n = 5 \text { and } p = 0.1$

A recent study suggested that 70% of all eligible voters will vote in the next presidential election. Suppose 20 eligible voters were randomly selected from the population of all eligible voters. Which of the following is necessary for this problem to be analyzed using the binomial random variable? I. There are two outcomes possible for each of the 20 voters sampled. II. The outcomes of the 20 voters must be considered independent of one another. III. The probability a voter will actually vote is 0.70, the probability they won't is 0.30.

The age of customers at a local hardware store follows a uniform distribution over the interval from 18 to 60 years old. Find the probability that the next customer who walks through the door exceeds 50 years old. Round to the nearest ten-thousandth.

The mean of the standard normal distribution is 1 and the standard deviation is 0.

The time between arrivals at an ATM machine follows an exponential distribution with $\theta = 10$ minutes. Find the probability that between 15 and 25 minutes will pass between arrivals.

Transportation officials tell us that 90% of drivers wear seat belts while driving. What is the probability that between 525 and 530 drivers in a sample of 600 drivers wear seat belts?

Suppose the number of babies born each hour at a hospital follows a Poisson distribution with a mean of 3. Some people believe that the presence of a full moon increases the number of births that take place. Suppose during the presence of a full moon, the hospital experienced eight consecutive Hours with more than four births each hour. Based on this fact, comment on the belief that the full moon increases the number of births.

$\text { Compute } \left( \begin{array} { l } 5 \\4\end{array} \right)$

A recent article in the paper claims that business ethics are at an all-time low. Reporting on a recent sample, the paper claims that 44% of all employees believe their company president possesses low ethical standards. Assume that responses were randomly and independently collected. A president of a local company that employs 1,000 people does not believe the paper's claim applies to her company. If the claim is true, how many of her company's employees believe that she possesses low ethical standards?

The price of a gallon of milk follows a normal distribution with a mean of $3.20 and a standard deviation of $0.10. Find the price for which 12.3% of milk vendors exceeded.

Nearly 100% of the observed occurrences of a random variable x that is normally distributed will fall within three standard deviations of the mean of the distribution of x.

A certain baseball player hits a home run in 3% of his at-bats. Consider his at-bats as independent events. Find the probability that this baseball player hits at most 10 home runs in 600 at-bats?

A loan officer has 76 loan applications to screen during the next week. If past record indicates that she turns down $19 \%$ of the applicants, what is the $z$ -value associated with 71 or more of the 76 applications being rejected?