Based on the boxplots you have created, does one type of track (wood or steel) tend to have faster roller coasters, or are they about the same?
Does one type of track tend to have more variability in the speeds of the roller coasters, or is the variability about the same?
Explain how you came to your conclusion using the boxplots. Is this result a surprise to you? Explain why this result is either expected or a surprise to you.
Pick one roller coaster and find an interesting or fun fact about that roller coaster and even a picture if you would like. Include this interesting fact in your report. Make sure you cite where you got your fact and picture, using any citation style you want.
2. (5 marks) A study on the occurrence of concussions for college athletes was conducted, where researchers looked at sex, sport, year, whether a concussion was sustained during a game (yes or no), and the number of games during which a concussion was sustained.
Summarize the data in a contingency table according to sport played (as the rows of your table) and whether or not a concussion was sustained (yes/no, as the columns of your table).
The body of your table should sum the number of games in each sport in which a concussion did/did not occur. (Note: this means you are essentially ignoring the columns of Sex and Year, which is expected). Make sure your table has appropriate row/column labels, and include it in your report.
Using the contingency table you created, determine the following probabilities:
What is the probability of a randomly selected game being either a basketball game or a softball/baseball game?
What is the probability that a concussion was sustained in a randomly selected game(i.e. regardless of the sport that was being played)?
Select one of the sports, and determine the probability a concussion was sustained given the randomly selected game was of the sport you selected.
Which sport has the highest concussion rate? Does this surprise you? Explain why or why not in your report.
Note that your report should include your work on how each calculation was obtained. You should also be sure to report your values in the context of some text, that is: do not simply list your calculated probabilities, but instead incorporate your answers as part of your write-up.
3. (10 marks) This question requires you to explore elements of probability.
(a) For the first question, you are required to create a probability distribution for a discrete random variable X, and use it to calculate various probabilities.
A tetrahedral die is a four-sided, pyramid shaped die which can display the numbers 1 through
4. If it is a fair tetrahedral die, it will display the numbers 1 through 4 with equal probability.
Suppose we roll two fair tetrahedral dice, and define our random variable X to be the sum of the values displayed.2.
Create a probability distribution for X, which displays all the values X can take on,
along with the corresponding probability for each value of X. You must show how you
derived your probabilities.
Using the probability distribution you created, determine:
The probability that the sum of the two tetrahedral dice will be at least 4.
The probability that the sum of the tetrahedral dice will be an even number.
Determine if the events “sum of at least 4” and “sum is an even number” are
independent events. Show all your work in your explanation of how you arrived at
your answer.
(b) For the second question, you are required to create a tree diagram to illustrate and solve conditional probabilities. Consider the following scenario:
For a certain public health unit (PHU), 80% of individuals in this PHU are considered fully vaccinated against COVID-19. When looking at the most recent COVID-19 positive cases (i.e. individuals who have tested positive for the virus), only 6.9% of cases are individuals who are fully vaccinated, while the remaining 93.1% of cases are individuals who have not been vaccinated.
Set up a tree diagram to illustrate the probabilities described above. Make sure you have clearly defined your notation.
You can use pen and paper to create your diagram, then take a picture of it and
import it into your report.
Make sure your writing is legible, and that your photo is nicely cropped so as to not
waste space.
Using the provided information and your tree diagram (if it is helpful), calculate:
The probability of having/getting COVID, given an individual has been vaccinated.
The probability of having/getting COVID, given an individual has not been vacci-
nated.
The probability of getting COVID, regardless of an individual’s vaccination status.
Show all your calculations for full marks. You should provide a small concluding
sentence for each of your calculations.
NOTE: This is not a political commentary on vaccine requirements. We are simply using probability rules to find probabilities of certain events, given knowledge of the probability of other events.
4. (5 marks) In this question, we will explore the concept of a sampling distribution for the sample mean, ̄X.
Suppose we have a random variable X, where X ∼N(μ,σ2).
Select a value for μ and σ. These can be any value you want under the restriction that σ > 0.3 State these values in your report.
Suppose repeated random samples of size n = 30 were sampled from the population of X, that is, suppose random samples of size n = 30 were repeatedly sampled from the N(μ,σ2) distribution, where you have specified a value for μ and σ. If for each repeated sample drawn from this population a sample mean, ̄X, is calculated, what would be the theoretical sampling distribution of ̄X.
This requires you to find and state in your report μ ̄Xand σ ̄X, as well as the shape of the distribution of ̄X.
Using Excel, draw a random sample of 1000 observations from each of the distributions for X and ̄X. To draw a random sample from a normal distribution in Excel, you can:
Start with cell A1 being your active cell. In the cell, type in the command“=norm.inv(rand(), mu, sigma)” where you have entered in appropriate values for “mu” and “sigma” accord- ing to the distribution for X. Hit enter. This has now generated your first observation from a N(μ,σ2) distribution.
To generate all 1000 observations, hover over the bottom right corner of A1 until you see the black ”+” sign. Click and hold on the bottom right corner, and drag your mouse down to cell A1000. You should now have 1000 observations.
You may want to use the “Copy” and “Paste Special −> Values” feature of Excel with
your data, so that your numbers do not change!
Repeat the steps above starting in cell B1, and using μ ̄Xand σ ̄Xin your “norm.inv()”command.
Create a side-by-side boxplot of the values generated from each distribution, and comment on what you see. Include your boxplot in your report. Is what you see expected? Explain why or why not in your report
