Write the sample model and distribution of the error term

Assignment Help Applied Statistics

Reference no: EM132250822

Question 1 - Ironman Triathlon

Luke and his long-time friend Paul are considering running the upcoming Ironman Triathlon race in Kona, Hawaii. An Ironman Triathlon is one of a series of long-distance triathlons consisting of a 2.4- mile swim, a 112-mile bicycle ride and a marathon 26.2-mile run, raced in that order and without a break. The Ironman event in Hawaii has a strict time limit of 17 hours (1020 minutes) to complete the race.

Luke has never run an Ironman race before, while Paul is an experienced Ironman racer. Paul's Ironman all-time record is 10 hours, 29 minutes (629 minutes). On the other hand, Luke is a good marathon runner, and he usually completes a marathon in 3 hours and 5 minutes (185 minutes).

Luke has a dataset available that contains marathon finish times and Ironman finish times for 40 different runners (all data is in minutes). With this data available, Luke runs a simple regression where MarathonTime is the independent variable and IronmanTime is the dependent variable. Note that both are measured in minutes.

Regression Statistics

R Square	Adj.RSqr	Std.Err.Reg.	# Cases
0.984	0.962	9.497	40

Summary Table

Variable	Coeff.	Std.Err.	t-Stat.	P-value	Lower95%	Upper95%
Intercept	113.779	16.560	6.871	0.000	80.255	147.303
MarathonTime	3.065	0.093	32.957	0.000	2.877	3.253

Forecasted : IronmanTime

	MarathonTime	Forecast	StErrFst
Fcst# 1	145	558.204	23.373
Fcst# 2	185	680.804	25.699
Fcst# 3	245	864.704	29.725
Fcst# 4	629	2041.554	61.533
Fcst# 5	285	987.304	32.664

Given the regression output above, answer the following questions.

(a) Write the sample model and the distribution of the error term.

(b) Compute a 99.7% confidence interval for the slope of MarathonTime.

(d) Build an exact 95% confidence interval for Luke's predicted Ironman finish time.

(e) What is the probability that Luke will beat Paul's all-time record during the Hawaii race?

(f) What is the probability that a runner that runs the marathon in 4 hours and 45 minutes (285 minutes) will not finish the Ironman race?

The Ironman finish time (IronmanTime) satisfies the following additive relation: IronmanTime = SwimTime + BikeTime + MarathonTime.

(g) Is it correct to interpret the coefficient of the intercept in the regression output as the expected time for performing the swim and the bike portion of the race? Justify your answer.

Question 2 - E-Visitors

Every day a number of visitors access Despair website, a website with "demotivation" messages instead of motivational messages. The number of visits per day is independent of each other, and you are told that each day the number of visitors is normally distributed with mean 5 thousand and standard deviation 1 thousand.

a) What is the probability that Despair website will receive more than 6 thousand visitors tomorrow?

Question 3 - Brazilian's Sandals

The manager of the sales division of a Brazilian company specialized in sandals is always collecting data on the purchase behavior of consumers and the performance of the different sandal styles. His boss requested him to choose one among two possible new product lines, one devoted to the World Cup, another devoted to the Olympics. These lines were being offered simultaneously but his boss observed substantial "cannibalization" between them (revenues from one line were essentially obtained by reducing revenues from the other). The manager noted that he could estimate the daily mean sales of each campaign looking at periods in which only one of them were being sold (i.e. the other was out of stock due to disruptions of the production supply chain created by recent street protests). The data for the World Cup line had 49 days with an average of 591 sandals sold per day with a sample standard deviation of 42. On the other hand, the data for the per day sales of the Olympics line, which was collected over 25 days, has a sample mean of 594 with a standard error of 3.

a) Build a 95% confidence interval for the mean sales per day of the World Cup line of sandals.

b) What was the sample standard deviation associated with the number of sandals sold per day in the data of the Olympics line?

Question 4 - Sign-up Commissions

Employees at a financial company earn annual commissions, measured in thousands of dollars, by signing up clients. The company is interested in identifying the characteristics of the employees who sign up many clients and earn large commissions. Three possible variables are considered as predictors of performance: Score on a training program (0 to 100 point scale), years of experience in the Industry (Yrs Experience), and the number of years of education post high school (Education). The company has 400 observations available, and obtains the following regression output.

Regression Statistics

R Square	Adj.RSqr	Std.Err.Reg.	# Cases
0.352	0.339	7.770	400

Summary Table

Variable	Coeff	Std.Err.	t-Stat.	P-value
Intercept	-8.177	7.244	-1.129	0.259
Score	0.542	0.072	7.506	0.000
YrsExperience	0.505	0.128	3.947	0.000
Education	1.672	0.369	4.538	0.000

Forecasted : Commissions

	Score	YrsExperience	Education	Forecast	StErrFst
Fcst# 1	50	10	9	39.042	11.282
Fcst# 2	60	0	4	31.057	11.497
Fcst# 3	70	0	4	36.481	11.788
Fcst# 4	70	0	9	44.842	11.817

Given the regression output above, answer the following questions.

(a) Write the sample model and the distribution of the error term.

(b) Provide a 95% confidence interval for the change in commissions if the number of years of post high school education increases by 2.

(c) Compute the expected change in commissions if the number of years of experience decreases by 10 and the number of years of post-high school education increases by 1.

(d) Now consider two employees hired as rookies in the last year (i.e. with zero years of experience). One of them has a 4-year college degree, while the other one has a 4-year college degree as well as a 5-year PhD. Both of them scored 70 during the training program. Given the data available, who earns the most commissions?

Reference no: EM132250822

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