Exam 6: Random Variables and Probability Distributions

In the old Roman Coliseum, two horses would be placed in tandem (side by side) and hitched to a chariot. Since the fast chariots needed to be able to pass the slow chariots, it was of some importance that the horses have room to run, but also they should not be too large. For chariot hitching purposes, the widest measure across a horse occurs in the rump area. The mean rump width is approximately 27" with a standard deviation of about 2". Let random variable w = width in inches across the rump of a randomly selected horse a)The usual length measure as it applies to horses is the "hand," which is by definition equal to 4". Define random variable h = width in hands across the rump of a randomly selected horse. What are the mean and standard deviation of h? b)Suppose horses are randomly chosen for a particular Roman chariot. Define random variable c = h₁ + h₂ to be the rump width of two randomly selected horses. What are the mean and standard deviation of c? c)Suppose that in the original chariot design a 16 inch separation of the horses is specified so that the horses have room to avoid each other. This leads to random variable a = 16 + w₁ + w₂. Describe how this addition of 16 inches of "wiggle room" would change the mean and standard deviation in part (b). Do not recalculate the mean and standard deviation.

One common method used to forecast the results of elections is to take an "exit poll," asking people who they supported as they come out of the voting place. Suppose Smith and Jones are running for City Dogcatcher. In point of fact, 60% of the voting public in the neighborhood of the polling place support Smith. Assuming that the candidate of choice for a voter leaving the polling place is independent of the candidate of choice for the next person to leave, what is the probability that the first 5 voters to leave the polling place would have this sequence for candidate of choice? Smith, Smith, Smith, Jones, Smith

What is a random variable?

Using the notation C = continuous and D = discrete, indicate whether each of the random variables are discrete or continuous. a)The number of stair steps to the principal's office b)The average night time temperature in July at the top of Long's Peak in Colorado c)The number of forks in your kitchen d)The number of sentences in a short story e)The circumference of a California redwood tree f)The number of grains of sand it takes to fill a quart jar

Briefly describe how one would decide which of two transformations was better for transforming skewed data to obtain a distribution that was approximately normal.

In the somewhat less than towering ski slopes of northeastern Iowa new skiers come to learn to ski. On Big Bunny Slope skiers will fail to make the turn at Big Bend. On Little Bunny Slope, skiers will sometimes tumble at Little Hill. The ski instructors send the new skiers down the two slopes in groups of 25, wait a few moments, and then send the Ski Patrol Ambulance down after them, stopping at Big Bend and then Little Hill. The skiers descend the slopes far enough apart that they don't run into each other, so their spills are all independent. a)The probability that a random new skier in the group will need to be carried to the First Aid Station after a spill at Big Bend is 0.3. If we define the random variable B = number of new skiers needing to be driven to the First Aid Station from Big Bend, we can model this situation as a binomial chance experiment. What is the mean and standard deviation of B? b)The probability that a random new skier in the group will need to be carried to the First Aid Station after a spill at Little Hill is 0.1. If we define the random variable L = number of new skiers needing to be driven to the First Aid Station from Little Hill, what is the mean and standard deviation of L? c)The total number of injuries requiring the Ambulance, T, is a random variable formed by calculating B + L. What are the mean and standard deviation of the random variable T? d)When the 25 skiers are sent down from the top of Big Bunny Slope, they are sent one at a time. What is the probability that the first one needing the Ambulance is the 7th one to be sent down the slope?

In a study of the reaction times of left- and right-handers' reaction times to tones delivered to the right ear, the right-handers' scores were approximately normally distributed with a mean of 240 milliseconds and standard deviation of 32 milliseconds. The mean score for right-handers was 210 ms. a)Sketch a normal distribution that describes left-handers' reaction times, and locate the mean reaction time for right-handers in this distribution. b)About what proportion of left-handers reaction times would be "better" (i.e. smaller) than the mean reaction time for right-handers?

What information about a probability distribution do the mean and standard deviation of a random variable provide?

The time that it takes a randomly selected employee to perform a certain task is approximately normally distributed with a mean value of 120 seconds and a standard deviation of 20 seconds. The slowest 10% (that is, the 10% with the longest times) are to be given remedial training. What times qualify for the remedial training?

The owners of the Burger Emporium are looking for new supplier of onions for their famous hamburgers. It is important that the onion slice be roughly the same diameter as the hamburger patty. After careful analysis, they determine that they can only use onions with diameters between 9 and 10 cm. Company A provides onions with diameters that are approximately normally distributed with mean 10.3 cm and standard deviation of 1.2 cm. Company B provides onions with diameters that are approximately normally distributed with mean 10.6 cm and standard deviation of 0.9 cm. Which company provides the higher proportion of usable onions? Justify your choice with an appropriate statistical argument.

The standard normal distribution has a mean of 1 and standard deviation of 0.

Briefly describe how the correlation coefficient can be used to check for normality.