Exam 4: Random Variables and Probability Distributions

Suppose that x has an exponential distribution with $\theta = 5 . \text { Find } P ( x \leq 10 )$

Classify the following random variable according to whether it is discrete or continuous. The number of pills in a container of vitamins

The time between equipment failures (in days) at a particular factory is exponentially distributed with a mean of 4.5 days. A machine just failed and was repaired today. a. Find the probability that another machine will fail within the next day. b. Find the probability that there will be no more equipment failures in the next week.

The diameters of ball bearings produced in a manufacturing process can be described using a uniform distribution over the interval 6.5 to 8.5 millimeters. What is the probability that a randomly selected ball bearing has a diameter greater than 7.4 millimeters?

Classify the following random variable according to whether it is discrete or continuous. The blood pressures of a group of students the day before the final exam

Suppose $x$ is a random variable best described by a uniform probability distribution with $c = 20$ and $d = 40$ . Find $P ( x < 30 )$ .

The probability that an individual is left-handed is 0.13. In a class of 70 students, what is the mean and standard deviation of the number of left-handed students? Round to the nearest hundredth When necessary.

The time between customer arrivals at a furniture store has an approximate exponential distribution with mean $\theta = 8.5$ 8.5 minutes. If a customer just arrived, find the probability that the next customer will not arrive for at least 20 minutes.

IQ test scores are normally distributed with a mean of 104 and a standard deviation of 17. An individual's IQ score is found to be 115. Find the z-score corresponding to this value.

Classify the following random variable according to whether it is discrete or continuous. The number of cups of coffee sold in a cafeteria during lunch

The time between customer arrivals at a furniture store has an approximate exponential distribution with mean $\theta = 8.5$ 8.5 minutes. If a customer just arrived, find the probability that the nex customer will arrive in the next 5 minutes.

After a particular heavy snowstorm, the depth of snow reported in a mountain village followed a uniform distribution over the interval from 15 to 22 inches of snow. Find the standard deviation of the snowfall amounts.

Suppose that $88 \%$ of the stocks listed on a particular exchange increased in value yesterday. Let $x$ be the number of stocks that increased in value yesterday in a random of 72 stocks listed on the exchange. Find the mean and standard deviation of $x$ .

Before a new phone system was installed, the amount a company spent on personal calls followed a normal distribution with an average of $500 per month and a standard deviation of $50 per month. Refer to such expenses as PCE's (personal call expenses). Find the probability that a randomly selected month had PCE's below $350.

Farmers often sell fruits and vegetables at roadside stands during the summer. One such roadside stand has a daily demand for tomatoes that is approximately normally distributed with a mean of 131 tomatoes and a standard deviation of 30 tomatoes. How many tomatoes must be available on any given day so that there is only a 1.5% chance that all tomatoes will be sold?

50 students were randomly sampled and asked questions about their exercise habits. One of the questions they were asked concerned the frequency of exercise, defined to be the number of times they exercised in a week. This variable would be characterized as which type of random variable?

The diameters of ball bearings produced in a manufacturing process can be described using a uniform distribution over the interval 2.5 to 4.5 millimeters. What is the mean diameter of ball bearings produced in this manufacturing process?

Assume that $x$ is a binomial random variable with $\mathrm { n } = 100$ and $p = 0.60$ . Use a normal approximation to find $P ( x < 65 )$ .

An airline has requests for standby flights at half of the usual one-way air fare. Past experience has shown that these passengers have about a 1 in 5 chance of getting on the standby flight. When they fail to get on a flight as a standby, the only other choice is to fly first class on the next flight out. Suppose that the usual one-way air fare to a certain city is $158 and the cost of flying first class is $385. Should a passenger who wishes to fly to this city opt to fly as a standby? [Hint: Find the expected cost of the trip for a person flying standby.]

A physical fitness association is including the mile run in its secondary-school fitness test. The time for this event for boys in secondary school is known to possess a normal distribution with a mean of 470 seconds and a standard deviation of 60 seconds. The fitness association wants to recognize the fastest 10% of the boys with certificates of recognition. What time would the boys need to beat in order to earn a certificate of recognition from the fitness association?