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Random variables of repeated events

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Random Variables

Events repeat themselves, such as going to work, producing a widget from machine X, selling widgets, etc. In the events there are outcomes that can be quantified or measured; e.g. the time it takes to drive to work, defective widgets, number of widgets sold each month. But these outcomes are not the same. The time it takes to drive to work can vary each day depending on a lot of factors. And the same can be said of the other outcomes. Since we are not able to know the exact amount for these VARIABLES, we call them Random Variables. This is in contrast to known or fixed amounts, such as your weekly pay check, the size of your shoes, etc. Random Variables are random because there are factors at work in the process that affect the outcome. Yet, if we collect the data over time, we can calculate statistics about these variables, such as the average (mean) and variance.

What are some random variables that you experience in your daily life? Identify three of them and discuss how you would go about collecting data to determine the mean and variance.

Solution Preview

REALITY-BASED RANDOM VARIABLES: DATA COLLECTION AND STATISTICAL TREATMENT

Several events in our lives happen to us at random or by chance. Despite our care and strategies in avoiding these to happen, these events occur because of factors that may be beyond our control or at first were within our control but turned out to be beyond control because of some changes in the environment.
What will be discussed in this report are three among the random events that take place in relation to family and work, with the corresponding data collection and functional statistical procedures done in order to handle them, as indicated below:
1. Spending beyond the budget. This happens both at home and at work.
At home, this is a situation when consumption expenditures exceed aggregate family disposable income. This is a critical event that needs to be addressed, otherwise, may result to debts or the so-called dissavings.
To address the issue, data that include the following must be gathered: earnings from all family sources that include base compensation, commissions, and profits from business. On the other hand, consumption expenditures on food, education, transportation, clothing, health, social commitments, and others must be obtained.
To analyze or give ...

Solution Summary

The random variable of repeated events is determined.

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Quantitative Methods (Stats) Multiple Choice & True/False Probs

1. Random numbers generated by a mathematical process instead of a physical process are pseudorandom numbers. (Points: 4)
True
False

2. Experimental outcomes must occur as numerical values in order to define their probability distribution.
True
False

3. Sample information with an efficiency rating of 100% is perfect information.

True
False

4. States of nature should be defined so that one and only one will actually occur.
True
False

5. Identification and definition of a problem (Points: 4)
cannot be done until alternatives are proposed.
is the first step of decision making.
is the final step of problem solving.
requires consideration of multiple criteria.

6. In a multicriteria decision problem, (Points: 4)
it is impossible to select a single decision alternative.
the decision maker must evaluate each alternative with respect to each criterion.
successive decisions must be made over time.
each of the above is true.

7. Which of the following is not a valid representation of a probability? (Points: 4)
35%
0
1.04
3/8

8. In the set of all past due accounts, let the event A mean the account is between 31 and 60 days past due and the event B mean the account is that of a new customer. The union of A and B is (Points: 4)
all new customers.
all accounts fewer than 31 or more than 60 days past due.
all accounts from new customers and all accounts that are from 31 to 60 days past due.
all new customers whose accounts are between 31 and 60 days past due.

9. A numerical description of the outcome of an experiment is
a normal variable.
a discrete variable.
a random variable.
an experimental variable.

10. Which of the following are continuous random variables?
I. the weight of an elephant
II. the time to answer a questionnaire
III. the number of floors in a skyscraper
IV. the square feet of countertop in a kitchen (Points: 4)
I and II only
I and III only
I, II and IV
I, II, III, and IV

11. A statement that matches the values of a random variable with the probabilities of those values is
the expected value.
the variation of the random variable.
an experiment.
a probability distribution.

12. In order to measure the dispersion of a random variable, look at its
standard deviation.
mean.
expected value.
average.

13. Experiments with repeated independent trials will be described by the binomial distribution if
the trials are continuous.
each trial result influences the next.
the time between trials is constant.
each trial has exactly two outcomes whose probabilities do not change.

14. The options from which a decision maker chooses a course of action are
called the decision alternatives.
under the control of the decision maker.
not the same as the states of nature.
each of the above is true.

15. States of nature (Points: 4)
can describe uncontrollable natural events such as floods or freezing temperatures.
can be selected by the decision maker.
cannot be enumerated by the decision maker.
each of the above is true.

16. A payoff
is always measured in profit.
is always measured in cost.
exists for each pair of decision alternative and state of nature.
exists for each state of nature.

17. A decision tree
presents all decision alternatives first and follows them with all states of nature.
presents all states of nature first and follows them with all decision alternatives.
alternates the decision alternatives and states of nature.
arranges decision alternatives and states of nature in their natural chronological order.

18. Time series methods
discover a pattern in historical data and project it into the future.
include cause-effect relationships.
are useful when historical information is not available.
each of the above is true.

19. Gradual shifting of a time series over a long period of time is called
periodicity.
cycle.
regression.
trend.

20. Seasonal components
cannot be predicted.
are regular repeated patterns.
are long runs of observations above or below the trend line.
reflect a shift in the series over time.

21. In assigning random numbers to probabilistic events in a simulation,
several events are associated with the same random number
every random number is associated with a particular event
every event is associated with the same random number
all of the above

22. A researcher wants to simulate sunny and rainy days in her town for a 3-week period. What is the minimum number of digits the student must obtain from a random number table for each observation if it rained on two-fifths of the days over the past several years at this time of the year? Assume that days can be classified historically as either sunny or rainy.
1
2
3
all of the above.

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