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Homework answers / question archive / 1)The higher the value of the density function f(x), the less likely the value x            c

1)The higher the value of the density function f(x), the less likely the value x            c

Statistics

1)The higher the value of the density function f(x),

    1. the less likely the value x            c. the less likely the distribution is normal
    2. the more likely the value x          d. None of the above

 

 

 

 

  1. Tossing a coin is an example of a (n)
    1. binomial distribution       c. exponential distribution
    2. normal distribution         d. Poisson distribution

 

 

 

 

  1. We assume that the outcomes of successive trials in a binomial experiment are:
    1. probabilistically independent      c. identical from trial to trial
    2. probabilistically dependent         d. random number between 0 and 1

 

 

 

 

  1. One reason for standardizing random variables is to measure variables with:
    1. different means and standard deviations on a non-standard scale
    2. different means and standard deviations on a single scale
    3. dissimilar means and standard deviations in like terms
    4. similar means and standard deviations on two scales

 

 

 

 

 

  1. If the value of the standard normal random variable Z is positive, then the original score is where in relationship to the mean?
    1. equal to the mean           c. to the right of the mean
    2. to the left of the mean    d. None of the above

 

 

 

 

 

  1. The normal distribution is:
    1. a discrete distribution with two parameters
    2. a binomial distribution with only one parameter
    3. a density function of a discrete random variable
    4. a continuous distribution with two parameters

 

 

 

 

 

  1. The standard deviation  of a probability distribution is a:

measure of variability of the distribution            c. measure of relative likelihood

    1. measure of central location         d. measure of skewness of the distribution

 

 

 

  1. The mean  of a probability distribution is a:
    1. measure of variability of the distribution            c. measure of relative likelihood
    2. measure of central location         d. measure of skewness of the distribution

 

 

 

 

 

  1. If we plot a continuous probability distribution f(x), the total probability under the curve is
    1. -1            c. 1
    2. 0             d. 100

 

 

 

 

  1. A continuous probability distribution is characterized by:
    1. a list of possible values   c. an array of individual values
    2. counts     d. a continuum of possible values

 

 

 

 

  1. Which of the following equations shows the process of standardizing?
    1.  

       

      c. 
    2. d.

 

 

 

 

  1. A Poisson distribution is:
    1. relevant when we sample from a population with only two types of members.
    2. relevant when we perform a series of independent, identical experiments with only two possible outcomes.
    3. usually relevant when we are interested in the number of events that occur over a given interval of time
    4. the cornerstone of statistical theory
    5. All of the above

 

 

 

 

  1. The standard normal distribution has a mean and a standard deviation respectively equal to
    1. 0 and 0   c. 1 and 0
    2. 1 and 1   d. 0 and 1

 

 

 

 

  1. Sampling done without replacement means that
    1. only certain members of the population can be sampled
    2. each member of the population can be sampled repeatedly
    3. each member of the population can be sampled only once
    4. each member of the population can be sampled twice

 

 

 

 

  1. The Poisson random variable is a:
    1. discrete random variable with infinitely many possible values
    2. discrete random variable with finite number of possible values
    3. continuous random variable with infinitely many possible values
    4. continuous random variable with finite number of possible values

 

 

 

 

 

  1. Which probability distribution applies to the number of events occurring within a specified period of time or space
    1. Binomial distribution      c. Any discrete probability distribution
    2. Poisson distribution        d. Any continuous probability distribution

 

 

 

 

 

  1. Which of the following might not be appropriately modeled with a normal distribution?
    1. The daily low temperature in Anchorage, Alaska
    2. The returns on a stock
    3. The daily change in inventory at a computer manufacturer
    4. The salaries of employees at a large company

 

 

 

 

 

  1. The variance of a binomial distribution for which n = 100 and p = 0.20 is:
    1. 100         c. 20
    2. 80           d. 16

 

 

 

 

 

  1. The binomial probability distribution is used with
    1. a discrete random variable
    2. a continuous random variable
    3. either a discrete or a continuous random variable, depending on the variance
    4. either a discrete or a continuous random variable, depending on the sample size

 

 

 

 

  1. Given that Z is a standard normal random variable, P(-1.0

    Z

    1.5) is
    1. 0.7745    c. 0.0919
    2. 0.8413    d. 0.9332

 

 

 

 

 

  1. Given that Z is a standard normal variable, the value z for which P(Z  z) = 0.2580 is
    1. 0.70        c. -0.65
    2. 0.758      d. 0.242

 

 

 

  1. If the random variable X is exponentially distributed with parameter = 3, then P(X  2) , up to 4 decimal places , is
    1. 0.3333    c. 0.6667
    2. 0.5000    d. 0.0025

 

 

 

 

  1. If the random variable X is exponentially distributed with parameter = 1.5, then P(2

    X

     

     4), up to

4 decimal places, is

    1. 0.6667    c. 0.5000
    2. 0.0473    d. 0.2500

 

 

 

 

 

  1. If X is a normal random variable with a standard deviation of 10, then 3X has a standard deviation equal to
    1. 10           c. 30
    2. 13           d. 90

 

 

 

 

 

  1. Given that the random variable X is normally distributed with a mean of 80 and a standard deviation of

10, P(85

 

X

 

90) is

 

    1. 0.5328    c. 0.1915
    2. 0.3413    d. 0.1498

 

 

 

 

 

  1. The Poisson and Exponential distributions are commonly used in which of the following applications
    1. Inventory models            c. Failure analysis models
    2. Financial models             d. All of these options

 

 

 

 

 

 

  1. Which of the following distributions is appropriate to measure the length of time between arrivals at a grocery checkout counter?
    1. Uniform c. Exponential
    2. Normal   d. Poisson

 

 

 

 

 

  1. If the mean of an exponential distribution is 2, then the value of the parameter  is
    1. 4             c. 1
    2. 2             d. 0.5

 

 

 

 

 

TRUE/FALSE

 

  1. The total area under the normal distribution curve is equal to one.

 

 

 

 

  1. The number of loan defaults per month at a bank is Poisson distributed.

 

 

 

 

  1. The variance of a binomial distribution is given by the formula, where n is the number of trials, and p is the probability of success in any trial.

 

 

 

 

  1. If the random variable X is normally distributed with mean  and standard deviation , then the random variable Z defined by  is also normally distributed with mean 0 and standard deviation 1.

 

 

 

 

  1. Much of the study of probabilistic inventory models, queuing models, and reliability models relies heavily on the Poisson and Exponential distributions.

 

 

 

 

  1. Using the standard normal distribution, the Z-score representing the 5th percentile is 1.645.

 

 

 

 

  1. The Poisson probability distribution is one of the most commonly used continuous probability distributions.

 

 

 

 

  1. Poisson distribution is appropriate to determine the probability of a given number of defective items in a shipment.

 

 

 

 

  1. The binomial distribution is a continuous distribution that is not far behind the normal distribution in order of importance.

 

  1. The binomial distribution is a discrete distribution that deals with a sequence of identical trials, each of which has only two possible outcomes.

 

 

 

 

  1. A binomial distribution with n number of trials, and probability of success p can be approximated well by a normal distribution with mean np and variance  if np > 5 and n(1-p) > 5.

 

 

 

 

  1. For a given probability of success p that is not too close to 0 or 1, the binomial distribution tends to take on more of a symmetric bell shape as the number of trials n increases.

 

 

 

 

  1. The Poisson distribution is characterized by a single parameter, which must be positive.

 

 

 

  1. An exponential distribution with parameter = 0.2 has mean and standard deviation both equal to 5.

 

 

 

  1. The binomial random variable represents the number of successes that occur in a specific period of time.

 

 

 

 

  1. A random variable X is standardized when each value of X has the mean of X subtracted from it, and the difference is divided by the standard deviation of X.

 

 

 

 

  1. Using the standard normal distribution, the Z- score representing the 99th percentile is 2.326.

 

 

 

 

  1. The mean and standard deviation of a normally distributed random variable which has been "standardized" are zero and one, respectively.

 

 

 

 

  1. Using the standard normal curve, the Z- score representing the 75th percentile is 0.674.

 

 

 

 

  1. A random variable X is normally distributed with a mean of 175 and a standard deviation of 50. Given that X = 150, its corresponding Z- score is –0.50.

 

 

 

 

  1. The binomial distribution deals with consecutive trials, each of which has two possible outcomes.

 

 

 

  1. The Poisson distribution is applied to events for which the probability of occurrence over a given span of time, space, or distance is very small.

 

 

 

 

  1. The Poisson random variable is a discrete random variable with infinitely many possible values.

 

 

 

 

  1. The variance of a binomial distribution for which n = 50 and p = 0.20 is 8.0.

 

 

 

 

  1. Using the standard normal curve, the Z- score representing the 10th percentile is 1.28.

 

 

 

SHORT ANSWER

 

 

  1. What is the probability that a randomly selected customer will spend less than $15?

 

 

 

 

  1. What is the probability that a randomly selected customer will spend $20 or more?

 

 

 

  1. What is the probability that a randomly selected customer will spend $30 or more?

 

 

 

  1. What is the probability that a randomly selected customer will spend between $20 and $35?

 

 

 

  1. What two dollar amounts, equidistant from the mean of $30, such that 90% of all customer purchases are between these values?

 

 

 

  1. What two dollar amounts, equidistant from the mean of $30, such that 98% of all customer purchases are between these values?

 

 

 

 

 

  1. There is a 5% chance that GM will sell more than what number of cars during the          next year?

 

 

 

  1. What is the probability that GM will sell between 2.0 and 2.3 million cars during the next year?

 

 

 

 

 

  1. What is the probability that exactly half the male adults will be less than 62 inches tall?

 

 

 

  1. Let Y be the number of the 12 male adults who are less than 62 inches tall.  Determine the mean and standard deviation of Y.

 

 

 

  1. Calculate the mean, variance, and standard deviation for the entire year (assume 52 weeks in the year).

 

 

 

  1. There is a 1% chance that this company will sell more than what number of cars during the next year?

 

 

 

  1. What is the probability that this company will sell more than 2 million cars next year?

 

 

 

 

  1. What is the probability that this company will sell between 2.0 and 2.15 million cars next year?

 

 

 

  1. What number of cars, equidistant from the mean, such that 90% of car sales are between these values?

 

 

 

  1. What number of cars, equidistant from the mean, such that 98% of car sales are between these values?

 

 

 

 

 

 

  1. How many hamburgers must the restaurant stock to be 99% sure of not running out on a given day?

 

 

 

  1. How many chicken sandwiches must the restaurant stock to be 99% sure of not running out on a given day?

 

 

 

  1. If the restaurant stocks 600 hamburgers and 150 chicken sandwiches for a given day, what is the probability that it will run out of hamburgers or chicken sandwiches (or both) that day? Assume that the demands for hamburgers and chicken sandwiches are probabilistically independent.

 

 

  1. Why is the independence assumption in Question 74 probably not realistic? Using a more realistic assumption, do you think the probability in Question 74 would increase or decrease?

 

 

 

 

 

  1. What percentage of students scored between 81 and 89 on this exam?

 

 

 

  1. What is the probability of getting a score higher than 85 on this exam?

 

 

 

  1. Only 5% of the students taking the test scored higher than what value?

 

 

 

 

  1. What type of probability distribution will most likely be used to analyze warranty repair needs on new microwaves in this situation?

 

 

  1. What is the probability that none of the 20 new microwaves sold will require a warranty repair in the first 90 days?

 

 

 

  1. What is the probability that exactly two of the 20 new microwaves sold will require a warranty repair in the first 90 days?

 

 

 

  1. What is the probability that less than two of the 20 new microwaves sold will require a warranty repair in the first 90 days?

 

 

 

  1. What is the probability that at most two of the 20 new microwaves sold will require a warranty repair in the first 90 days?

 

 

 

  1. What is the probability that only one of the 20 new microwaves sold will require a warranty repair in the first 90 days?

 

 

  1. What is the probability that more than one of the 20 new microwaves sold will require a warranty repair in the first 90 days?

 

 

 

  1. What is the probability that at least one of the 20 new microwaves sold will require a warranty repair in the first 90 days?

 

 

 

  1. What is the probability that between two and four (inclusive) of the 20 new microwaves sold will require a warranty repair in the first 90 days?

 

 

 

  1. What is the probability that between three and six (exclusive) of the 20 new microwaves sold will require a warranty repair in the first 90 days?

 

 

 

  1. What is the expected number of the new microwaves sold that will require a warranty repair in the first 90 days?

 

 

 

  1. What is the standard deviation of the number of the new microwaves sold that will require a warranty repair in the first 90 days?

 

 

 

  1. Find the probability distribution of X.

 

 

 

 

 

  1. Find P(X < 3).

 

 

 

  1. Find P(2

    X

    4).

 

 

 

  1. Find the mean and the variance of X.

 

 

 

 

 

  1. Describe the probability distribution of X.

 

 

 

  1. Find P(X = 10).

 

 

 

  1. Find P(4 < X < 9).

 

 

 

  1. Find P(X = 2).

 

 

  1. Find P(3

    X

    6).

 

 

 

  1. Suppose that an highway patrol officer can obtain radar readings on 500 vehicles during a typical shift. How many traffic violations would be found in a shift?

 

 

 

 

 

 

  1. (A) Using the binomial distribution, find the probability that 6 or more of the 30 students taking this course in a given semester will withdraw from the class.

 

      1. Using the normal approximation to the binomial, find the probability that 6 or more of the 30 students taking this course in a given semester will withdraw from the class.

 

      1. Compare the results obtained in (A) and (B). Under what conditions will the normal approximation to this binomial probability become even more accurate?

 

 

 

 

 

  1. (A) Assuming the supplier’s claim is true, compute the mean and the standard deviation of the number of defective DVDs in the sample.

 

      1. Based on your answer to (A), is it likely that as many as six DVDs would be found to be defective, if the claim is correct?

 

      1. Suppose that six DVDs are indeed found to be defective. Based on your answer to (A), what might be a reasonable inference about the manufacturer’s claim for this shipment of 10,000 DVDs?

 

 

 

 

  1. What is the probability that at least 25 customers arrive at this checkout counter in a given hour?

 

 

 

  1. What is the probability that at least 20 customers, but fewer than 30 customers arrive at this checkout counter in a given hour?

 

 

 

  1. What is the probability that fewer than 33 customers arrive at this checkout counter in a given hour?

 

 

 

  1. What is the probability that the number of customers who arrive at this checkout counter in a given hour will be between 30 and 35 (inclusive)?

 

 

 

  1. What is the probability that the number of customers who arrive at this checkout counter in a given hour will be greater than 35?

 

 

 

 

 

 

  1. Find the probability that the number of arrivals between 3:00 and 5:00 P.M. is at least 10.

 

 

 

  1. Find the probability that the number of arrivals between 3:30 and 4:00 P.M. is at least 10.

 

 

 

  1. Find the probability that the number of arrivals between 4:00 and 5:00 P.M. is exactly two.

 

 

 

 

  1. What is the probability that he will have to wait at least 8 days before making another sale?

 

 

 

  1. What is the probability that he will have to wait between 6 and 10 days before making another sale?

 

 

 

 

 

 

  1. What is the probability density function for the time it takes a technician to fix a computer problem?

 

 

 

  1. What is the probability that it will take a technician less than 10 minutes to fix a computer problem?

 

 

  1. What is the variance of the time it takes a technician to fix a computer problem?

 

 

 

  1. What is the probability that it will take a technician between 10 to 15 minutes to fix a computer problem?

 

 

 

 

 

  1. What is the distribution of X and what are the parameters?

 

 

 

  1. Find the mean and standard deviation of X.

 

 

 

  1. What is the probability that X is between 1 and 3?

 

 

  1. What is the probability that X is at most 2?

 

 

 

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