**Questions**

Questions 1-3 are based on this

Slack is a partnership between two traders: Simon and Jack.

Simon and Jack each invested money in different commodity futures. Simon’s coffee trades can lose 3.5million Euros with probability 1/3 or earn a profit of 1 million Euros with probability 2/3. Jacks’s cereal trades can lose 1 million Euros with probability 1/4 or earn a profit of 1 million Euros with probability 3/4.

1. What is the probability that Slack’s profits will be non-negative (larger than or equal to 0)?

2. What are Slack’s expected profits (in millions)?

3. If the coffee and cereal trades are statistically independent, what is the probability that Slack’s profits will be exactly equal to zero?

Questions 4-7 are based on this

**Speculate.com** is an internet trading company. The company is worried about the continual market slide; it seems to have resulted in a significant decline in both the total number of trades **Speculate.com** executes and the number of customers who actively trade on **Speculate.com**.

4. According to the results of a recent survey among 150 randomly selected **Speculate.com**customers, the average return on $1 invested among the 150 respondents was 96 cents (they lost 4 cents) with a sample variance of (25 cents)2. Based on the survey results, how confident can you be that **Speculate.com** customers are losing money on average?

5. A** Speculate.com** customer is considered “active” if s/he executes at least one trade per day. September 2003 records indicate that 20% of customers were “active” and that one-quarter of “active” customers outperformed S&P 500 index during that period. On the other hand, one-third of the “nonactive” customers outperformed S&P 500 index during that period. What percentage of customers who outperformed S&P 500 index in September 2003 were “active” customers in that month?

6. A sizeable proportion of **Speculate.com** customers seem to be day-traders (i.e., speculators in volatile stocks that close their positions at the end of every trading day). **Speculate.com** estimates that at the end of the day, the return on $1 invested in day-trading is normally distributed. **Speculate.com’s** trading records suggest that, on average, $1 invested in day-trading is worth only 95 cents at the end of the day (i.e., day-traders lose money as a group even though there might be some individual day-traders who make a fortune). The records also suggest a 45% chance that a $1 day-trading investment will turn out profitable at the end of the day. Using **Speculate.com’s** records and estimates, find the standard deviation of the return at the end of the day on $1 invested in day-trading.

7. During any given trading hour, the distribution of the number of complaints related to on-line access problems when placing a trade request has a mode of 0, a median of 2, and an expected value of 3.4. A sample indicating the number of complaints for each of 400 randomly selected one-hour trading intervals has been collected. What is the probability that the observed sample mean is less than 3.4?

Use the College.csv file for questions 8-10

8. Fit a linear model to explain Apps using variables Accept and Enroll. What is the equation of the fitted line?

9. Are the variables Accept and Enroll statistically significant at the 5% level?

10. Split the data set into a training set (the **first** 80% of the data) and a test set (the **remaining**20%). The response and explanatory variables are of a similar set up to that of Questions 1 and 2. Fit a linear model on the training set. What is the mean squared error (MSE) obtained on the test set?

Use BostonHousing.csv for questions 11-13

11. Consider fitting a multiple linear regression model where the response variable is medv, against the explanatory variables of rm, tax, and lstat. After fitting the multiple linear regression model, what is the reported adjusted R-squared?

12. Reconsider the same dataset as before. However, fit a model to this data using only the explanatory variables rm and tax. What is the 95% confidence interval on the coefficient of rm?

13. Using the model fitted in Q4, give a prediction of the on response variable medv when the explanatory variables rm=8.0 and tax=5.0.

14. **Backwards elimination in the context of regression is**

when the statistician starts by removing items from the end of their dataset | |

when the procedure starts with no predictors in the model, iteratively adds the most important predictors, and stops when you have an adequate model |

when the statistician inputs the variables in the reverse of their normal order | |

when the procedure starts with all predictors in the model, iteratively removes the least important predictors, and stops when you have an adequate model |

15. If the odds against Justin for defeating Bieber in chess is 3:1, what is the probability that Justin wins a game against Bieber

16. Two fair, six-sided die are rolled. Compute the probability that the sum of the outcomes of the two rolls is equal to 4.

17. You have a large jar containing 999 fair coins and one two-headed coin (i.e. a coin that is guaranteed to come up heads). Suppose you pick one coin at random out of the jar and flip it 10 times. What is the probability that the coin you chose is a two-headed one?

Bodysize dataset for question 19

Use this code to load it

read.csv(‘bodysize.csv’,header=TRUE)

18. Fit a logistic regression to predict survive using the explanatory variable bodysize. Based on this fitted model, what is the 95% confidence interval for the coefficient bodysize ?

Fict dataset for question 19

Use this code to load it

fict = read.table(‘fict.txt’,header=FALSE,col.names = c(‘x1’, ‘x2’, ‘x3’, ‘x4’, ‘x5’, ‘x6’, ‘x7’, ‘x8’, ‘y’))

19. **You built a regression model to regress y against all the explanatory variables, x1, x2,…,x8 and noticed that there might be model violations. Which of the following transformation should be performed to the response variable **(y)** to alleviate these violations?**

y3 | |

log(y) |

y2 | |

1/ y2 |

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