Each question is 75-150 words
1. What is the textbook’s definition of probability?
2. If there are 8 green marbles and 4 red marbles in a bag, and you were asked to draw one without looking, what is the probability of drawing one green marble?
3. If you don’t put that green marble that you drew in #2 back in the bag, now what is the probability of drawing another green marble?
In Â§4.4 we are introduced to the concept of conditional probability. The notation P(A | B) denotes the probability of event A occurring given that we know event B has occurred. Thatâ€™s the definition.
Now consider conditional probability in the Monty Hall Problem which is introduced in the following videos:
4. After watching these videos, we know that if we are given the option to switch doors that, probabilistically speaking, it is in our best interest to switch. Why is the probability of winning NOT 50/50 when the contestant is given the opportunity to switch? There are only two doors left, so why is it not 50/50?
5. Find the answers to the following. Given that there are 3 yellow marbles and 4 blue marbles in a bag,
A. What is the probability of drawing a blue marble given that a yellow one was already drawn?
B. What is the probability of drawing a yellow marble, not replacing it, and then drawing a blue marble?
C. Which one, A or B, exemplifies conditional probability, and why?
D. If you solved A and B, you should have different answers. What are they?
The Normal Probability Distribution
In Â§6.2 we are introduced to the Normal Probability Distribution and the special case of the Normal Probability Distribution, the Standard Normal Probability Distribution, which is a Normal Probability Distribution with mean (u) zero and variance (Ïƒ2) one.
Watch these videos to get some background information.
1. What are the properties of a standard, normal probability distribution?
2. What is a z score? What is the purpose of a z score?
3. Look at a picture of the normal probability distribution. This looks like the graph of a mountain.
Where on the graph is the mean?
If a z score were found to be -3.2, where on the graph would it be? Is this a rare or a common score? Why?
One way to find probabilities from a Standard Normal Distribution is to use probability tables, which are in your book or in online tables.
A hint from Dr. Klotz: This is my favorite online z table. http://www.z-table.com/ Don’t lose that link!
We are now making sure you can read the table so that you can do your homework.
4. According to the table, what is the probability when z â‰¤ -2.46? The probability when z â‰¤ 2.46?
5. According to the table, what is the probability when z â‰¤ -0.91? The probability when z â‰¤ 0.91?
6. Add each of the pairs in #4 and #5 (show your work). What is the total each time? Why is that the total?
7. Elenore says that her probability found from the z table was -2.74. How do you know she made an error?
Sampling with a Pair of Dice
Go to the link: http://www.random.org/dice/ (or any other online dice rolling link)
- Roll the 2 virtual dice and calculate the sum of the pair of virtual dice. Do this 3 times. List your rolls. Show your calculations.
- Then after you have rolled the virtual pair dice 3 times and calculated 3 sums, calculate the average of these 3 sums. Show your calculations.
- Conduct this experiment again, showing your list and average, but this time roll the virtual pair of dice 20 times. Calculate the 20 sums, and then find the average of these 20 sums. Show your work.
- Explain the Central Limit Theorem.
- How does these exercises relate to this weekâ€™s lesson, particularly the Central Limit Theorem?
- If we took everyoneâ€™s averages for the 20 rolls and made a histogram, what would we expect to see, and why?
Hint from Dr. Klotz: I strongly recommend recording the dice rolls and calculations in Excel because you need them again next week. It will save you time if you have the spreadsheet ready to go.
Constructing Confidence Intervals
Watch this video
- What is a confidence interval?
- Go to the Notations and Symbols area for Week 5 and look at the equations for confidence intervals. Why will there be two numbers in every confidence interval? Which one is given first?
- In that same Notations and Symbols area, notice that the title of every confidence interval contains the words “population mean” or “population proportion.” This is a “game changer” in terms of your Project. What doorway has been opened in your discussion of ROI? (Hint: let’s say the person you were talking to said, “Your data means nothing because I am not going to any of THAT SAMPLE of 20 colleges.”)
- In the discussion for week 4, you rolled a pair of dice 3 times and calculated the average sum of your rolls. Then you did the same thing with 20 rolls. Use your results from the week 4 discussion for the average of 3 rolls and for the average of 20 rolls to construct 95% confidence intervals for the true mean of the sum of a pair of dice (assume Ïƒ = 2.41). You will need a formula from the Notations and Symbols. We want to try this here so that the assignment / project are easier to do.
- What do you notice about the length of the interval for the mean of 3 rolls versus the mean of 20 rolls? Did you expect this? Why or why not?
- Using your mean for 20 rolls, calculate the 90% confidence interval and the 99% confidence interval. Look at the width of the interval for 90%, 95%, and 99%. What is happening? Why?
Hints – these websites might help to find z or t sub alpha over 2. (you have to know which one)