Q&A Hero

Q: Sampling error is the difference between the sample statistic and the population parameter.

Q: The actual mean fill volume for all bottles of a soft drink product that were filled on a Tuesday is 11.998 ounces. A sample of 64 bottles was randomly selected and the sample mean fill volume was 12.004 ounces. Based upon this information, the sampling error is .006 ounce.

Q: Recently the State Fish and Game planted several thousand tagged fish in a local river. The mean length of these fish, which constitute a population, is 12.6 inches. Yesterday, fishermen caught 100 of these tagged fish. You could expect that the mean length for these fish would be 12.6 inches as well since they come from the population.

Q: Taking a larger sample size will always result in less sampling error but costs more money and takes more time.

Q: The size of the sampling error that comes from a random sample depends on both the variation in the population and the size of the sample being selected.

Q: The sample mean is a parameter.

Q: Suppose the time it takes for a customer to be served at a fast-food chain business is thought to be uniformly distributed between 3 and 8 minutes, then the probability that a customer is served in less than 3 minutes is 0.

Q: One of the basic differences between a uniform probability distribution and a normal probability distribution is that the uniform is symmetrical but the normal is skewed depending on the value of the standard deviation.

Q: For a normal distribution, the probability of a value being between a positive z-value and its population mean is the same as that of a value being between a negative z-value and its population mean.

Q: Any normal distribution can be converted to a standard normal distribution.

Q: The vehicle speeds on a city street have been determined to be normally distributed with a mean of 33.2 mph and a variance of 16. Based on this information, the probability that if three randomly selected vehicles are monitored and that two of the three will exceed the 35 mph speed limit is slightly greater than 0.18.

Q: A seafood shop sells salmon fillets where the weight of each fillet is normally distributed with a mean of 1.6 pounds and a standard deviation of 0.3 pounds. Based on this information we can conclude that 90 percent of the fillets weight more than 1.0 pound.

Q: The Varden Packaging Company has a contract to fill 50-gallon barrels with gasoline for use by the U.S. Army. The machine that Varden uses has an adjustable device that allows the average fill per barrel to be adjusted as desired. However, the actual distribution of fill volume from the machine is known to be normally distributed with a standard deviation equal to 0.5 gallons. The contract that Varden has with the military calls for no more than 2 percent of all barrels to contain less than 49.2 gallons of gasoline. Suppose Varden managers are unwilling to set the mean fill at any level higher than 50 gallons. Given that, in order to meet the requirements, they will need to increase the standard deviation of fill volume.

Q: The Varden Packaging Company has a contract to fill 50 gallon barrels with gasoline for use by the U.S. Army. The machine that Varden uses has an adjustable device that allows the average fill per barrel to be adjusted as desired. However, the actual distribution of fill volume from the machine is known to be normally distributed with a standard deviation equal to 0.5 gallons. The contract that Varden has with the military calls for no more than 2 percent of all barrels to contain less than 49.2 gallons of gasoline. In order to meet this requirement, Varden should set the mean fill to approximately 49.92 gallons.

Q: Watersports Rental at Flathead Lake rents jet skis and power boats for day use. Each piece of equipment has a clock that records the time that it was actually in use while rented. The company has observed over time that the distribution of time used is normally distributed with a mean of 3.6 hours and a standard deviation equal to 1.2 hours. Watersports management has decided to give a rebate to customers who use the equipment for only a short amount of time. They wish to grant a rebate to no more than 10 percent of all customers. Based on the information provided, the amount of time that should be set as the cut-off between getting the rebate and not getting the rebate is approximately 2.06 hours.

Q: The State Department of Forests has determined that annual tree growth in a particular forest area is normally distributed with a mean equal to 17 inches and a standard deviation equal to 6 inches. If 2 trees are randomly chosen, the probability that both trees will have grown more than 20 inches during the year is approximately .037.

Q: The time it takes a parent to assemble a children's bicycle has been shown to be normally distributed with a mean equal to 295 minutes with a standard deviation equal to 45 minutes. Given this information, the probability that it will take a randomly selected parent more than 220 minutes is about 0.0475.

Q: The standard normal distribution has a mean of 0 and a standard deviation of 1.0.

Q: The standard normal distribution table provides probabilities for the area between the z-value and the population mean.

Q: The actual weight of 2-pound sacks of salted peanuts is found to be normally distributed with a mean equal to 2.04 pounds and a standard deviation of 0.25 pounds. Given this information, the probability of a sack weighing more than 2.40 pounds is 0.4251.

Q: The parameters of a normal distribution are the mean and the standard deviation.

Q: All symmetric distributions can be assumed normally distributed.

Q: When a single die is rolled, each of the six sides are equally likely. This is an example of a uniform distribution.

Q: If the mean, median and mode are all equal for a continuous random variable, then the random variable is normally distributed.

Q: A continuous random variable approaches normality as the level of skewness increases.

Q: For a continuous distribution the total area under the curve is equal to 100.

Q: When graphed, the probability distribution for a discrete random variable looks like a histogram.

Q: The probability distribution for a continuous random variable is represented by a probability density function that defines a curve.

Q: One example of a difference between discrete random variables and continuous random variables is that in a discrete distribution P(x > 2) = P(x ≥ 3) while in a continuous distribution P(x > 2) is treated the same as P(x ≥ 2).

Q: The number of defects manufactured by workers in a small engine plant is an example of a discrete random variable.

Q: Typically, a continuous random variable is one whose value is determined by measurement instead of counting.

Q: The normal distribution is one of the most frequently used discrete probability distributions.

Q: In comparing a uniform distribution with a normal distribution where both distributions have the same mean and the same range, explain which distribution will have the larger standard deviation.

Q: Answer: Since both products have the same mean and are both normally distributed, the one with the largest standard deviation will provide the higher probability of a heavy sack. Since potatoes have a standard deviation of 2 pounds compared to 0.50 pounds for onions, you would be more apt to see a very heavy sack of potatoes than onions.

Q: What is the difference between a normal distribution and the standard normal distribution?

Q: If a continuous random variable is said to be exponentially distributed, what would be the easiest way to reduce the standard deviation?

Q: A random variable is normally distributed with a mean of 25 and a standard deviation of 5. If an observation is randomly selected from the distribution, what value will 15% of the observations be below?A) 19.8B) 16.2C) 18.7D) 17.2

Q: A randomly selected value from a normal distribution is found to be 2.1 standard deviations above its mean. What is the probability that a randomly selected value from the distribution will be less than 2.1 standard deviations from the mean? A) 0.9488 B) 0.9821 C) 0.9976 D) 0.9712

Q: For the normal distribution with parameters μ = 0, σ = 3; calculate P(x > 1). A) 0.5812 B) 0.1214 C) 0.3707 D) 0.4412

Q: For the normal distribution with parameters μ = 5, σ= 2; calculate P(0 < x < 8). A) 0.8023 B) 0.4152 C) 0.9270 D) 0.8845

Q: A random variable, x, has a normal distribution with μ = 13.6 and σ = 2.90. Determine a value, x0, so that P(μ - x0≤ x ≤ μ + x0) = 0.95. A) 7.916 B) 4.535 C) 3.178 D) 9.425

Q: Consider a random variable, z, that has a standardized normal distribution. Determine P(z > -1). A) 0.8413 B) 0.1251 C) 0.1512 D) 0.2124

Q: For a standardized normal distribution, determine a value, say z0, so that P(z ≤ z0) = 0.01. A) -2.33 B) -1.96 C) 2.33 D) 1.96

Q: For a standardized normal distribution, calculate P(1.78 < z < 2.34). A) 0.0124 B) 0.0341 C) 0.0412 D) 0.0279

Q: For a standardized normal distribution, calculate P(-1.00 < z < 1.00). A) 0.6826 B) 0.6667 C) 0.4572 D) 0.5521

Q: For a standardized normal distribution, calculate P(z ≥ 0.85). A) 0.8033 B) 0.1977 C) 0.2340 D) 0.7660

Q: For a standardized normal distribution, calculate P(z < 1.5). A) 0.9332 B) 0.0668 C) 0.333 D) 0.667

Q: The transportation manager for the State of New Jersey has determined that the time between arrivals at a toll booth on the state's turnpike is exponentially distributed with λ = 4 cars per minute. Based on this information, what is the probability that the time between any two cars arriving will exceed 11 seconds?A) Approximately 0.47B) About 0.199C) About 0.747D) About 0.801

Q: It is assumed that the time failures for an electronic component are exponentially distributed with a mean of 50 hours between consecutive failures. If one extra component is installed as a backup, what is the probability of at least one of the two components working for at least 60 hours?A) About 0.51B) About 0.09C) About 0.06D) About 0.70

Q: It is thought that the time between customer arrivals at a fast food business is exponentially distributed with λ equal to 5 customers per hour. Given this information, what is the mean time between arrivals?A) 12 minutesB) 5 minutesC) 5 hoursD) 2 minutes

Q: Employees at a large computer company earn sick leave in one-minute increments depending on how many hours per month they work. They can then use the sick leave time any time throughout the year. Any unused time goes into a sick bank account that they or other employees can use in the case of emergencies. The human resources department has determined that the amount of unused sick time for individual employees is uniformly distributed between 0 and 480 minutes. The company has decided to give a cash payment to any employee that returns over a specified amount of sick leave minutes. Assuming that the company wishes no more than 5 percent of all employees to get a cash payment, what should the required number of minutes be? A) 24 minutes B) 419 minutes C) 456 minutes D) 470 minutes

Q: It is assumed that the time customers spend in a record store is uniformly distributed between 3 and 12 minutes. Based on this information, what is the probability that a customer will be exactly 7.50 minutes in the record store? A) 0.1250 B) 0.05 C) Essentially zero D) 0.111

Q: Which of the following probability distributions would most likely be used to describe the time between failures for electronic components? A) Binomial distribution B) Exponential distribution C) Uniform distribution D) Normal distribution

Q: A store sells 6 different models of cell phones and have found that they sell an equal number of each model. The probability distribution that would describe this random variable is called: A) uniform distribution. B) Poisson distribution. C) continuous distribution. D) relative frequency distribution.

Q: A professor noted that the grades of his students were normally distributed with a mean of 75.07 and a standard deviation of 11.65. If only 10 percent of the students received grades of A, what is the minimum score needed to receive an A?A) 80.00B) 85.00C) 90.00D) 95.00

Q: In a standard normal distribution, the probability P(-1.00< z < 1.20) is the same as: A) P(1< z < 1.20) - P(0 < z < 1.00). B) P(1< z < 1.20) - 2*P(0 < z < 1.00). C) 2 ∗ P(1< z < 1.20) - P(0 < z <1.00). D) P(1 < z < 1.20) + 2 ∗ P(0 < z <1.00).

Q: In a standard normal distribution, the probability that z is greater than 0 is: A) 0.5 B) equal to 1 C) at least 0.5 D) 1.96

Q: A major cell phone service provider has determined that the number of minutes that its customers use their phone per month is normally distributed with a mean equal to 445.5 minutes with a standard deviation equal to 177.8 minutes. The company is thinking of charging a lower rate for customers who use the phone less than a specified amount. If it wishes to give the rate reduction to no more than 12 percent of its customers, what should the cut-off be?A) About 237 minutesB) About 654 minutesC) About 390 minutesD) About 325 minutes

Q: Suppose that it is believed that investor returns on equity investments at a particular brokerage house are normally distributed with a mean of 9 percent and a standard deviation equal to 3.2 percent. What percent of investors at this brokerage house earned at least 5 percent? A) 89.44 percent B) 10.56 percent C) 39.44 percent D) 100 percent

Q: A recent study showed that the length of time that juries deliberate on a verdict for civil trials is normally distributed with a mean equal to 12.56 hours with a standard deviation of 6.7 hours. Given this information, what is the probability that a deliberation will last between 10 and 15 hours?A) Approximately 0.29B) Nearly 0.75C) About 0.48D) About 0.68

Q: A major U.S. automaker has determined that the city mileage for one of its new SUV models is normally distributed with a mean equal to 15.2 mpg. A report issued by the company indicated that 22 percent of the SUV model vehicles will get more than 17 mpg in the city. Given this information, what is the city mileage standard deviation for this SUV model?A) 0.77 mpgB) Approximately 2.34 mpgC) 1.8 mpgD) Approximately 3.1 mpg

Q: The makers of Sweet-Things candy sell their candy by the box. Based on company policy, the mean target weight of all boxes is 2.0 pounds. To make sure that they are not putting too much in the boxes, the manager wants no more than 3 percent of all boxes to contain more than 2.10 pounds of candy. In order to do this, what should the mean fill weight be set to if the fill standard deviation is 0.13 pounds? Assume that the box weights are normally distributed. A) Just over 2 pounds B) Approximately 2.33 pounds C) Nearly 1.27 pounds D) Approximately 1.86 pounds

Q: The manager of a computer help desk operation has collected enough data to conclude that the distribution of time per call is normally distributed with a mean equal to 8.21 minutes and a standard deviation of 2.14 minutes. What is the probability that three randomly monitored calls will each be completed in 4 minutes or less?A) 0.4756B) Approximately 0.1076C) About 0.00001D) Can't be determined without more information.

Q: The manager at a local movie theater has collected data for a long period of time and has concluded that the revenue from concession sales during the first show each evening is normally distributed with a mean equal to $336.25 and a standard deviation equal to $80. Based on this information, what are the chances that the revenue on the first show will be between $300 and $500?A) About 0.3062B) Approximately 0.6534C) 0.1736D) Approximately 0.4798

Q: Students who have completed a speed reading course have reading speeds that are normally distributed with a mean of 950 words per minute and a standard deviation equal to 220 words per minute. If two students were selected at random, what is the probability that they would both read at less than 400 words per minute?A) 0.4938B) 0.0062C) 0.00004D) 0.2438

Q: Assuming that the change in daily closing prices for stocks on the New York Stock Exchange is a random variable that is normally distributed with a mean of $.35 and a standard deviation of $.33. Based on this information, what is the probability that a randomly selected stock will close up $.75 or more?A) 0.0202B) 0.5207C) 0.4798D) 0.9798

Q: Which of the following probability distributions could be used to describe the distribution for a continuous random variable? A) Exponential distribution B) Normal distribution C) Uniform distribution D) All of the above

Q: Which of the following is not a characteristic of the normal distribution? A) Symmetric B) Mean = median = mode C) Bell-shaped D) Equal probabilities at all values of x

Q: Which of the following probability distributions can be used to describe the distribution for a continuous random variable? A) Normal distribution B) Binomial distribution C) Poisson distribution D) Hypergeometric

Q: A study of cars arriving at a parking structure at the local airport shows that the time between arrivals is 1.2 minutes and is exponentially distributed. The probability that more than 2 minutes will elapse between the arrivals of cars is about 0.81.

Q: An electronics repair shop has determined that the time between failures for a particular electronic component part is exponentially distributed with a mean time between failures of 200 hours. Based on this information, the probability that a part will not fail in the first 200 hours is 0.50.

Q: Service time for customers at a drive-through coffee shop has been shown to be uniformly distributed between 2 and 10 minutes. Customers will complain when service time exceeds 7.5 minutes. Based on this information, the probability of getting a complaint based on service time is 0.3125.

Q: An assembly process takes between 20 and 40 minutes to complete with the distribution of time thought to be uniformly distributed. Based on this, the percentage of assemblies that require less than 25 minutes is 0.05.

Q: The amount of drying time for the paint applied to a plastic component part is thought to be uniformly distributed between 30 and 60 minutes. Currently, the automated process selects the part from the drying bin after the part has been there for 50 minutes. The probability that none of three parts picked are still wet when they are selected is approximately 0.04.

Q: It has been determined the weight of bricks made by the Dillenger Stone Company is uniformly distributed between 1 and 1.5 pounds. Based on this information, the probability that two randomly selected bricks will each weigh more than 1.3 pounds is 0.16.

Q: If a uniform distribution and normal distribution both have the same mean and the same range, the normal distribution will have a larger standard deviation than the uniform distribution

Q: If the time it takes for a customer to be served at a fast-food chain business is thought to be uniformly distributed between 3 and 8 minutes, then the probability that the time it takes for a randomly selected customer to be served will be less than 5 minutes is 0.40.

Q: The manager of a fast food store realizes that his staffing problems are a result of the variation in the number of customers that arrive at the store. If the same number of customers came each hour, she would know exactly how many servers to have working. It turns out that the Poisson distribution works well to describe the arrivals of customers in any given hour. Explain why the manager has more trouble staffing the store during those hours when the average arrival rate is higher.

Q: Under what conditions is the binomial distribution symmetric?

Q: The Swanson Auto Body business repaints cars that have been in an accident or which are in need of a new paint job. Its quality standards call for an average of 1.2 paint defects per door panel. Explain why there is a difference between the probability of finding exactly 1 defect when 1 door panel is inspected and finding exactly 2 defects when 2 doors are inspected.