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Importance of statistics and probability.

 

1. Importance of statistics and probability.In your own words, explain why random variables are important to statistics and probability.
2. What is the difference between a continuous and discrete random variable?
3. Would you consider the temperature outside a discrete random variable? Why or why not?
4. Suppose a piggy bank contains 100 coins (25 pennies, 25 dimes, 25 nickels, and 25 quarters). Let the random variable X represent the total value of five randomly selected coins. Is X a discrete or continuous random variable?
5. What are the properties of a continuous distribution?
6. What are the properties of a discrete distribution?
7. Give an example of a discrete and a continuous random variable.
8. Give examples of three different discrete random variables.
9. Give examples of three different continuous random variables.
10. Can you describe a random variable Y that is both discrete and continuous?

 

1. Suppose you have a dataset and define the random variable X as the number of students at your university with the same last name. what type of random variable is this?
2. Give two examples of variable that are not random.
3. What are the major differences between discrete and continuous random variables?
4. In each of the following situations, indicate whether the random variable is discrete or continuous:
   1. The number of Twitter followers for each student in your statistic class
   2. The amount of rainfall in 2020 in each U.S. state
   3. The amount of time it takes each student in your statistics class to travel to campus
   4. The number of text message students on a college campus received today
   5. The GPA of the first-year students at your college

 

• Section 5.2: pp. 114-115, problems #1- #9
  1. For discrete random variables, what type of plot would you use to graphically display the probability of distribution? What provides the probability for a specific outcome?

 

Use the following to answer questions 2-5:

Let’s define an experiment in which a fair coin is tossed three times. Let the random variable X represent the number of times the coin  lands on tails in the three flips.

2. What is the probability that two tails are observed?
3. Describe two formulations that you could use to calculate the probability that at least one tail is observed and then solve either.
4. What is the probability that no tails are observed?
5. Create a probability histogram that graphically displays the probability distribution of X.
6. For continuous random variables, what type of plot would you use to graphically display the probability distribution? How is the plot used to find the probability of an event?
7. When visually examining graphs of continuous distributions, what features of the graph are important to notice? What can we note about the area under the entire graph?
8. Consider the continuous probability distribution shown in the graph below. What features of the distribution do you observe?
9. Consider the continuous probability distribution shown in the graph below. What features of the distribution do you observe?

• Section 5.3: pp. 117-118, problems #1- #10

Use the following to answer questions 1-5:

Assume the distribution of a random variable Y is defined in the table below:

1. Is the distribution a valid probability model? Explain why or why not.
2. Does the probability model describe a continuous or discrete random variable?
3. Compute the central tendency of the probability distribution.
4. What is the variance of the probability distribution?
5. What is the standard deviation of the probability distribution?

 

Use the following to answer 6-10:

Construct your own valid probability model that describes a discrete random variable X. Assume the random variable X can take on five different values. The random variable X should have an expected value of 12.

6. Fill in the table below to show the values and probabilities for your model.
7. How do you know that you constructed a valid probability distribution?

 

8. What is the variance of your probability distribution?

 

9. What is the standard deviation of your probability distribution?

 

10. Plot the probability histogram.

 

• Section 5.4: pp. 121, problems #1- #15

1. Is the geometric distribution discrete or continuous?
2. Is the Poisson distribution discrete or continuous?
3. Suppose you flip a coin 10 times and count the number of times the coin lands on heads. Is this a setting that would allow the use of a Poisson distribution?
4. Netflix^TM is a popular streaming service that allows you to watch movies and TV shows for a subscription price. Suppose you count the number of episodes each student in your class watched last night of any show. Would this be a setting appropriate for the Poisson distribution? Explain why or why not.
5. Give an example of a dataset that could be modeled using a Poisson distribution.
6. Suppose a couple would like to start having children. The random variable X represents the number of children they have until the first girl. Is this setting for a geometric distribution? Explain why or why not.
7. Give an example of a dataset that could be modeled using the geometric distribution.

Use the following to answer questions 8-10

We are going to use the Poisson distribution to model the number of customers at a popular fast-food chain restaurant. Suppose the mean number of people at the restaurant is 98 during any given hour.Importance of statistics and probability.

8. What is the probability that there will be over 100 customers at the restaurant from p.m. -6 p.m. today?
9. What is the probability that there will be exactly 10 customers at the restaurant for the hour starting at 1 p.m.?
10. What is the variance for this distribution?

Use the following to answer questions 11-13.

The University of Maryland has approximately 40,000 students from all over the world. As a new student on campus, you are interested in meeting students from your home town of Paris, France. Suppose that when you meet someone, the probability that they are from Paris is .04. Use the geometric distribution to answer the questions below.

11. What is the probability of success, p, and the probability of failure, q, for this distribution?
12. What is the probability you have to ask random students on campus where they are from before you find a student who is from Paris?
13. What is the mean and variance of this distribution?
14. Stephen Curry, a member of the Golden State Warriors professional basketball has one of the three best free throw shooting percentages of all time. Using the fact that his free throw shooting percent is 90.56%, calculate the probability that it takes more than 10 shots before Curry would miss a shot.
15. Babe Ruth, known as The Sultan of Swat, is often referred to as the greatest baseball player of all time. Ruth was known for hitting home runs. He had a career home run percent of 8.5%. Design an appropriate probability model, and then calculate (a) the expected number of at bats to hit a homerun and (b) the probability that Ruth goes more than 15 at bats in a row without a home run.Importance of statistics and probability.