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 team’s winning percentage.To test whether teams tended to improve their winning percentage with the new coach, a random sample of 20 teams that changed coaches early in the season was taken. Because the playoff format of the NBA changed in 1984, the sample only includes teams from 1984–2022. The file Coaches.omv contains 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 first coach
• 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 second coach

As an example, the first team in the data set is the 1988 Washington Bullets, who began the season with 8 wins and 19 losses (a 29.63% winning percentage) under the first coach Kevin Loughery, and finished the season with 30 wins and 25 losses (a 54.55% winning percentage) under the second coach Wes 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 Percent” paired or independent? Write a sentence justifying your choice.
(b) [6 marks] For NBA teams that change coaches early in the season, is there evidence that the 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 relevant output. 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 conclusion in 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 early in 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 whether each assumption is reasonable, with appropriate references to jamovi output where needed.

Question 2. [12 marks]
How do the train networks of Australia’s three msot populous cities compare in terms of the distance between consecutive stations? To explore this, a random sample of 40 routes between consecutive stations was taken for the Brisbane, Melbourne, and Sydney train networks, and the corresponding interstation 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. Be sure to state the null and alternative hypotheses, test statistic, null distribution, p-value, decision and an appropriate conclusion in plain language.
(b) [3 marks] If appropriate, perform post-hoc tests to determine which city train networks have significantly different average interstation distances. If post-hoc tests are not appropriate, explain the purpose of a post-hoc test and why it’s not appropriate in this case.
(c) [3 marks] What are the assumptions of the analysis performed in part (a)? State whether each 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 and distance 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 star’s 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, the lower the magnitude. These magnitudes are actually ratios, where a difference in magnitude of 1 coincides with a ratio of 2.512 in brightness. That is, a star with an absolute magnitude of 2.0 is 2.512 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 absolute magnitude 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 statistically significant linear relationship between the absolute magnitude of a star and its distance from Earth? Conduct an appropriate test at the 5% significance level. Be sure to state the null and alternative hypotheses, test statistic, null distribution, p-value, decision and an appropriate conclusion in plain language.
(d) [4 marks] State the assumptions necessary for your regression analysis in part (c) to be appropriate. State whether each of them is satisfied with a brief justification. This justification may refer to appropriate output from jamovi.
(e) [2 marks] Provide a 95% confidence interval for the slope of the population regression line of the absolute magnitude of a star on its distance from Earth. Write an interpretation of this interval.
(f) [2 marks] Write down the R2 value for this regression and give an interpretation.