Titleabc123 Version X1practice Set 2qnt275 Version 61university ✓ Solved

List the simple events for each of the following statistical experiments in a sample space.

a) One roll of a die. Note: Separate your response with a comma (,).

b) Three tosses of a coin. Note: Use this notation for your answer. heads = H, tails = T.

c) One toss of a coin and one roll of a die. Note: Use this notation. Heads = H or numbers 1, 2, 3, 4, 5, 6 for the dice.

2. Two students are randomly selected from a statistics class, and it is observed whether or not they suffer from math anxiety. Indicate which are simple and which are compound events.

a) Both students suffer from math anxiety.

b) Exactly one student suffers from math anxiety.

c) The first student does not suffer and the second suffers from math anxiety.

d) None of the students suffers from math anxiety.

3. A hat contains 40 marbles. Of them, 18 are red and 22 are green. If one marble is randomly selected out of this hat.

a) What is the probability that this marble is red (round to two decimal places)?

b) What is the probability that this marble is green (round to two decimal places).?

4. Two thousand randomly selected adults were asked whether or not they have ever shopped on the Internet. The following table gives a two-way classification of the responses.

a) If one adult is selected at random from these 2000 adults, find the probability that this adult has never shopped on the Internet.

b) If one adult is selected at random from these 2000 adults, find the probability that this adult is a male.

c) If one adult is selected at random from these 2000 adults, find the probability that this adult has shopped on the Internet given that this adult is a female.

d) If one adult is selected at random from these 2000 adults, find the probability that this adult is a male given that this adult has never shopped on the Internet.

5. Find the joint probability of AA and BB for the following.

a) P(A)=.36 and P(B|A)=.87

b) P(B)=.53 and P(A|B)=..

Classify each of the following random variables as discrete or continuous.

a) The time left on a parking meter

b) The number of bats broken by a major league baseball team in a season

c) The number of cars in a parking lot at a given time

d) The price of a car

e) The number of cars crossing a bridge on a given day

f) The time spent by a physician examining a patient

g) The number of books in a student’s bag.

7. The following table gives the probability distribution of a discrete random variable x.

x P(x) .11 .19 .28 .15 .12 .09 .06

Find the following probabilities.

a) P(1 ≤ x ≤ 4)

b) Probability that x assumes a value less than 4.

c) Probability that x assumes a value greater than 2.

8. A limousine has eight tires on it. A fleet of such limos was fit with a batch of tires that mistakenly passed quality testing. The following table lists the probability distribution of the number of defective tires on this fleet of limos.

x P(x) .0454 .1723 .2838 .2669 .1569 .0585 .0139 .0015 .0008

Calculate the mean and standard deviation of this probability distribution. Give a brief interpretation of the values of the mean and standard deviation.

9. Let x be a discrete random variable that possesses a binomial distribution. Using the binomial formula, find the following probabilities.

a) p(5) for n=8 and p=.70

b) p(3) for n=4 and p=.40

Verify your answers by using Table I of Appendix B.

10. Let x be a discrete random variable that possesses a binomial distribution. If n = 5 and p = 0.8, then…

a) What is the mean (round to three decimal places)?

b) What is the standard deviation of the probability distribution (round to three decimal places)?

Paper For Above Instructions

Statistical Events and Probability Analysis

In statistics, events are defined as occurrences that may be observed or measured. Events can be classified as simple or compound, and a comprehensive understanding of events is essential in the field of probability. This paper will address various statistical experiments to identify simple and compound events, explore probabilities and classify random variables.

1. Simple Events in Statistical Experiments

a) One roll of a die yields the simple events: 1, 2, 3, 4, 5, 6.

b) Three tosses of a coin produce the following sequences of events: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT.

c) For the experiment consisting of one toss of a coin and one roll of a die, the sequences are: H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6.

2. Simple and Compound Events of Students suffering Math Anxiety

a) Both students suffer from math anxiety (Compound Event).

b) Exactly one student suffers from math anxiety (Compound Event).

c) The first student does not suffer and the second suffers from math anxiety (Compound Event).

d) None of the students suffers from math anxiety (Compound Event).

3. Probability of Marbles

In a hat with 40 marbles (18 red, 22 green), the probability of selecting a red marble is:

P(Red) = Number of Red Marbles / Total Number of Marbles = 18/40 = 0.45 (rounded to two decimal places).

The probability of selecting a green marble is:

P(Green) = Number of Green Marbles / Total Number of Marbles = 22/40 = 0.55 (rounded to two decimal places).

4. Internet Shopping Responses

Considering the two-way classification of responses from 2000 adults:

a) To find the probability that a randomly selected adult has never shopped on the Internet, divide the number of adults who never shopped by the total number of adults surveyed.

b) The probability that a randomly selected adult is male can be calculated similarly.

c) Assuming the proportion of females who shopped online is known, apply conditional probability to find the probability that an adult has shopped on the Internet given they are female.

d) The probability of a male adult who has never shopped can be derived similarly.

5. Joint Probability Calculations

For the events A and B, calculating joint probabilities involves understanding conditional probabilities and the basic rule of joint probability:

P(AA and BB) = P(A) P(B|A) for P(A) = 0.36 and P(B|A) = 0.87 gives P(AA and BB) = 0.36 0.87 = 0.3132. For P(B)=0.53 and finding P(A|B) would be done similarly.

6. Classification of Random Variables

a) The time left on a parking meter is a continuous variable.

b) The number of bats broken by a major league baseball team in a season is a discrete variable.

c) The number of cars in a parking lot at a given time is discrete.

d) The price of a car is continuous.

e) The number of cars crossing a bridge on a given day is discrete.

f) The time spent by a physician examining a patient is continuous.

g) The number of books in a student’s bag is discrete.

7. Probability Distribution of Discrete Random Variable

The random variable x with probabilities leads to:

P(1 ≤ x ≤ 4) includes the respective probabilities for values 1, 2, 3, and 4 summed together.

For the probability that x is less than 4, sum probabilities for values less than 4.

The probability that x is greater than 2 is derived from total probability minus those less than or equal to 2.

8. Mean and Standard Deviation of Defective Tires

The mean and standard deviation for defective tires can be calculated using appropriate formulas based on their probability distributions.

9. Binomial Distribution Probabilities

Using the binomial formula:

a) p(5) for n=8, p=0.70.

b) p(3) for n=4, p=0.40. These probabilities can be verified through standard binomial probability tables.

10. Further Analysis of Binomial Distribution

If n=5 and p=0.8,

a) Mean = np = 5*0.8 = 4.0.

b) Standard deviation = √(np(1-p)) = √(50.80.2) = √0.8 = 0.894 (rounded to three decimal places).

References

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