STAT1070 Statistics For the Sciences Assignment

Question 1. [12 marks]
In the National Basketball Association (NBA), a North American professional basketball league,sometimes a team changes coaches early in the season. Usually, this is due to the team underperforming relative to expectations, with the hope that the new coach will improve the teams winningpercentage.To test whether teams tended to improve their winning percentage with the new coach, a randomsample of 20 teams that changed coaches early in the season was taken. Because the playoff format ofthe NBA changed in 1984, the sample only includes teams from 19842022. The file Coaches.omvcontains the resulting data, which includes the following variables:
Season: the season the team played
Team: the name of the team
First Coach: the coach during the early part of the season
First Coach Winning Percent: the percentage of games that the team won with the firstcoach
Second Coach: the coach during the later part of the season after the coaching change
Second Coach Winning Percent: the percentage of games that the team won with the secondcoach

As an example, the first team in the data set is the 1988 Washington Bullets, who began the seasonwith 8 wins and 19 losses (a 29.63% winning percentage) under the first coach Kevin Loughery, andfinished the season with 30 wins and 25 losses (a 54.55% winning percentage) under the second coachWes Unseld, representing an improvement of 54.55 ? 29.63 = 24.92%.

(a) [1 mark] Are the samples First Coach Winning Percent and Second Coach Winning Percentpaired or independent? Write a sentence justifying your choice.
(b) [6 marks] For NBA teams that change coaches early in the season, is there evidence thatthe average winning percentage improves under the second coach compared to the first coach?Conduct the appropriate test in jamovi at the 5% significance level and include all relevantoutput. Be sure to define any parameters you use, state the null and alternative hypotheses,observed test statistic, null distribution, p-value, decision and provide an appropriate conclusionin plain language.

(c) [2 marks] Report the 95% confidence interval for the average difference in the winning percentage between the second coach and the first coach for NBA teams that change coaches earlyin the season. Write a sentence interpreting this interval in plain language.
(d) [1 mark] Does your confidence interval from part (c) support the decision you made in part(b)?
(e) [2 marks] What are the assumptions of your analyses in parts (b) and (c)? State whethereach assumption is reasonable, with appropriate references to jamovi output where needed.

Question 2. [12 marks]
How do the train networks of Australias three msot populous cities compare in terms of the distancebetween consecutive stations? To explore this, a random sample of 40 routes between consecutivestations was taken for the Brisbane, Melbourne, and Sydney train networks, and the correspondinginterstation distances were recorded in the file Trains.omv.
The file includes the following variables:
Station 1: the name of the first station
Station 2: the name of the second station
City: the name of the city in which the route between the two stations is located
Interstation Distance: the distance between Station 1 and Station 2, in kilometres

(a) [6 marks] Is there evidence of a difference in average interstation distance among the threecities train networks? Conduct an appropriate hypothesis test at the 10% significance level. Besure to state the null and alternative hypotheses, test statistic, null distribution, p-value, decisionand an appropriate conclusion in plain language.
(b) [3 marks] If appropriate, perform post-hoc tests to determine which city train networks havesignificantly different average interstation distances. If post-hoc tests are not appropriate, explainthe purpose of a post-hoc test and why its not appropriate in this case.
(c) [3 marks] What are the assumptions of the analysis performed in part (a)? State whethereach assumption is reasonable with reference to appropriate jamovi output.

Question 3. [18 marks]
In Question 1c of Assignment 1, you explored the relationship between the absolute magnitude anddistance from Earth for 500 randomly selected stars within a distance of 30 parsecs from Earth.
The file Stars.omv contains observations of these 500 stars for the following variables:

ID: An identification for each star
Distance: Distance from Earth (in parsecs)
Visual magnitude: Visual magnitude, a measure of how bright a star appears from Earth
Absolute magnitude: Absolute magnitude, a measure of a stars true brightness
Spectral classification: Spectral classification, with classes A (bluish-white), F (white),
G (yellow), K (orange), and M (red)

Absolute magnitude and visual magnitude are both unitless measures, with the brigher the star, thelower the magnitude. These magnitudes are actually ratios, where a difference in magnitude of 1coincides with a ratio of 2.512 in brightness. That is, a star with an absolute magnitude of 2.0 is2.512 times brighter than a star with an absolute magnitude of 3.0.

(a) [3 marks] Write down the equation for the estimated regression line for predicting the absolutemagnitude of a star (y) based on its distance from Earth (x), and provide an interpretation of
the intercept and the slope coefficient.
(b) [1 mark] Predict the absolute magnitude for a star that is a distance of 20 parsecs from Earth.
(c) [6 marks] For the population of all stars within 30 parsecs from Earth, is there a statisticallysignificant linear relationship between the absolute magnitude of a star and its distance fromEarth? Conduct an appropriate test at the 5% significance level. Be sure to state the nulland alternative hypotheses, test statistic, null distribution, p-value, decision and an appropriateconclusion in plain language.
(d) [4 marks] State the assumptions necessary for your regression analysis in part (c) to beappropriate. State whether each of them is satisfied with a brief justification. This justificationmay refer to appropriate output from jamovi.
(e) [2 marks] Provide a 95% confidence interval for the slope of the population regression lineof the absolute magnitude of a star on its distance from Earth. Write an interpretation of thisinterval.
(f) [2 marks] Write down the R2 value for this regression and give an interpretation.