1. Briefly, what is probability (include in this the 3 "approaches" to probability discussed in the early part of chapter 5).

Classical approach to Probability

Suppose there are outcomes for a random experiment, which are equally likely, mutually exclusive and exhaustive. If of the outcomes are favorable to an event , then the probability of the event is defined as .

Frequency approach to probability

Let the experiment be repeated times. Suppose the event occurs f times and does not occur in times. Then is called the frequency of the event in repetitions and is called the relative frequency. The limit of frequency ration as becomes larger and larger and tends to infinity is defined as the probability of the event

Axiomatic approach to probability.

Consider the collection of all events. Then the function defined for every event is a probability function if it satisfies the following three axioms.

Axiom 1 (Axiom of nonnegativity)

If A is any event, then P(A)≥0

Axiom 2(axiom of certainty)

Let S be the sample space. Then P(S)=1

Axiom 3EAxiom of additivity)

If A and b are mutually exclusive events, then P(A =P(A)+P(B).

2. Please (1) work the following problems, (2) tell me what rule or principle you used to solve them and (3) then look at others' answers to check yourself and that person -- let's see if we can come to consensus on them!

  1. Ninety students will graduate from Lima Shawnee High School this spring. Of the 90 students, 50 are planning to attend college. Two students are to be picked at random to carry flags at graduation.
    a. What is the probability both of the selected students plan to attend college?

Total number of ways in which 2 students can be selected from 90 students =

Number of ways in which 2 students can be selected from 50 students

Probability that both of the selected students are planning to attend college

b. What is the probability one of the two selected students plans to attend college?

Total number of ways in which 2 students can be selected from 90 students =

Number of ways in which 1 student can be selected from the group of 50 students and 1 student from the remaining group of 40 students

Probability that one of the selected students plans to attend college

  1. A survey of undergraduate students in the School of Business at Northern University revealed the following regarding the gender and majors of the students:

Major

Gender Accounting Management Finance total

Male 100 150 50 300

Female 1005050200

total 200 200 100 500

Let us define the following events.

Let denote the event a randomly selected student is a female.

Let M denote the event a randomly selected student is a male.

Let A denote the event a randomly selected student is a accounting major

Let B denote the event a randomly selected student is a management major.

Let C denote the event a randomly selected student is a finance major.

  1. What is the probability of selecting a female student?

Total number of students = 500

Number of female students = 200

Probability that a randomly selected student is a female

  1. What is the probability of selecting a finance or accounting major?

Total number of students = 500

Number of finance major students = 100

Probability that a randomly selected student is a finance major student

Number of accounting major students = 200

Probability that a randomly selected student is a accounting major student

Then the event that a randomly selected student is finance major or accounting major is .

A student cannot be both accounting major and finance major simultaneously. So the events and C are mutually exclusive. By the axiom of additivity,

The probability that a randomly selected student is finance major or accounting major = 0.6

c. What is the probability of selecting a female or an accounting major? Which rule of

addition did you apply?

The required probability is

By the addition theorem of probability,

Then

  1. Are gender and major independent? Why?

Gender and major are independent if and only if all of the following conditions is satisfied.

If any one of the conditions is not satisfied, then gender and major are not independent.

A / B / C / Total
M / 100 / 150 / 50 / 300
F / 100 / 50 / 50 / 200
Total / 200 / 200 / 100 / 500

From the above table

Clearly

Therefore the events and are not independent.

Gender and major are not independent.

  1. What is the probability of selecting an accounting major, given that the person selected is a male?

The required probability is the conditional probability of the event given that the event has occurred. By the definition of conditional probability,

  1. Suppose two students are selected randomly to attend a lunch with the president of the

university. What is the probability that both of those selected are accounting majors?

Total number of ways in which 2 students can be selected from 500 students

Number of ways in which 2 accounting major students can be selected from 200 accounting major students

Probability that the 2 students are accounting major

  1. Reynolds Construction Company has agreed not to erect all "look-alike" homes in a new subdivision. Five exterior designs are offered to potential home buyers. The builder has standardized three interior plans that can be incorporated in any of the five exteriors. How many different ways can the exterior and interior plans be offered to potential home buyers?

There are 5 exterior designs and there are 3 interior designs.

Number of ways in which 1 exterior design from 5 exterior designs 1 interior design fro 3 interior design can be selected

  1. A puzzle in the newspaper presents a matching problem. The names of 10 in one U.S. presidents are listed column, and their vice presidents are listed in random order in the second column. The puzzle asks the reader to match each president with his vice president. If you make the matches randomly, how many matches are possible? What is the probability all 10 of your matches are correct?

The total number ways in which 10 vice presidents can be arranged against the 10 presidents is

Of the 3628800 arrangements only is in the correct order.

Therefore, the required probability is

  1. What is a continuous probability distribution (use the information from Applied Statistics in Business and Economics, Chapters 7) as compared with a discrete probability distribution which is discussed in chapter 6 (same text)?

Let be a random variable. If the variable assumes a finite number of values or a countably infinite number of values, then it is known as a discrete random variable. The number of students in a randomly selected school or the number of children in a randomly selected family are examples of discrete random variables.

If the variable assumes a continuum of values it is called a continuous random variable. A continuous variable assumes all values in a given interval. The height of randomly selected student, the weight of a new born baby etc are examples of continuous random variables.

  1. How does the Empirical Rule apply to continuous discrete probability distributions (Chapter 6)? Does the Empirical Rule always apply to discrete probability distributions? Why or why not?
    Does the Empirical Rule apply to continuous probability distributions (Chapter 7)? If so, how and when?

If the distribution is normal 68% of the data lie within 1 standard deviation from the mean, 99% lie within 2 standard deviations from the mean and 99.7% lie within 3 standard deviations. If the distribution is approximately normal, thenapproximately 68% of the data lie within 1 standard deviation from the mean,approximately 99% lie within 2 standard deviations from the mean and approximately 99.7% lie within 3 standard deviations. This the empirical rule. Since normal distribution is a continuous distribution, the empirical rule is applied to continuous distributions.

  1. Use Chapter 7 (Applied Statistics in Business and Economics) to address the following 2 problems:
  2. The mean starting salary for college graduates in the spring of 2004 was $36,280. Assume that the distribution of starting salaries follows the normal distribution with a standard deviation of $3,300. What percent of the graduates have starting salaries:
  3. Between $35,000 and $40,000?
  4. More than $45,000?
  5. Between $40,000 and $45,000?

Let denote the starting salary of a randomly selected college graduate. Then X follows normal distribution with mean 36,280 and standard deviation 3,300. Then Z = follows standard normal distribution.

  • The price of shares of Bank of Florida at the end of trading each day for the last year followed the normal distribution. Assume there were 240 trading days in the year. The mean price was $42.00 per share and the standard deviation was $2.25 per share.
  • What percent of the days was the price over $45.00? How many days would you estimate?
  • What percent of the days was the price between $38.00 and $40.00?
  • What was the stock's price on the highest 15 percent of days?

Let denote the price of shares of Bank of Florida on at the end of a randomly selected trading day. Then follows normal distribution with mean 42.00 and standard deviation 2.25.

The price was above 45.00 in 9.12% of days which is

The price was between 38 and 40 in 14.93% of the days.

  1. From tables of standard normal distribution

The corresponding value of is

The stock’s price on the highest 15 percent of days was above $44.33

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Statistical Symbols and Definitions Matching Assignment

Match the letter of the definition on the right to the appropriate symbol on the left.

Symbols / Definitions
  • 1. S (Uppercase Sigma) ____
/
  • a. Null hypothesis

  • 2. m (Mu) ____
/
  • b. Summation

  • 3. s (Lowercase Sigma) ____
/
  • c. Factorial

  • 4. p (Pi) ____
/
  • d. Nonparametric hypothesis test

  • 5. e (Epsilon) ____
/
  • e. Population standard deviation

  • 6. c2 (Chi Square) ____
/
  • f. Alternate hypothesis

  • 7. ! ____
/
  • g. Maximum allowable error

  • 8. H0 ____
/
  • h. Population mean

  • 9. H1 ____
/ i. Probability of success in a binomial trial
Symbol / Definition
/ Summation
/ Population mean
/ Population standard deviation
/ Probability of success in binomial trial
/ Maximum allowable error
/ Nonparametric hypothesis test
/ Factorial
/ Null hypothesis
/ Alternate hypothesis

Match the letter of the term on the right to the definition of that term on the left.

Definitions / Terms
  • 1. The average of the squared deviation scores from a distribution mean. ____
/
  • a. Reliability

  • 2. Midpoint in the distribution of numbers. ____
/
  • b. Mode

  • 3. It has to do with the accuracy and precision of a measurement procedure. ____
/
  • c. Generalization

  • 4. Examines if an observed causal relationship generalizes across persons, settings, and times. ____
/
  • d. Variance

  • 5. The difference between the largest and smallest score in a distribution. ____
/
  • e. Median

  • 6. The arithmetic average. ____
/
  • f. External validity

  • 7. Refers to the extent to which a test measures what we actually wish to measure. ____
/
  • g. Mean

  • 8. The most frequently occurring value in a set of numbers. ____
/
  • h. Internal validity

  • 9. The conclusion from research conducted on a sample population to the population as a whole. ____
  • 10. Examines whether the conclusion that we draw about a demonstrated experimental relationship truly implies cause. ____
  • 11. Determines how far away the data values are from the average. ____
/
  • i. Range
  • j. Standard deviation
  • k. Validity

Definitions / Terms
The average of the squared deviation scores from a distribution mean. ____ / Variance
. Midpoint in the distribution of numbers. ____ / Median
It has to do with the accuracy and precision of a measurement procedure. ____ / Reliability
. Examines if an observed causal relationship generalizes across persons, settings, and times. ____ / External validity
The difference between the largest and smallest score in a distribution. ____ / Range
The arithmetic average. ____ / Mean
. Refers to the extent to which a test measures what we actually wish to measure. ____ / Validity
The most frequently occurring value in a set of numbers. ____ / Mode
The conclusion from research conducted on a sample population to the population as a whole. ____ / Generalization
Examines whether the conclusion that we draw about a demonstrated experimental relationship truly implies cause. ____ / Internal validity
Determines how far away the data values are from the average. ____ / Standard deviation