#spain_data <-read.csv("spanish_football_data.csv", header = TRUE)
#marathon_data <- read.csv("female_marathon.csv")
#### IF LOADING AN RSTUDIO DATA SPACE USE THIS COMMAND ###########
load("week2_workspace.Rdata")SR-M13 week 2 assignment 2026
This assignment is marked out of 32. It constitutes 8% of the total module mark.
Name: Carly Christensen
Student number: 2535701
Datasets for analysis
You will use data of goals scored in the Spanish football league and of the marathon race record time for female runners of different ages.
Hypothesis testing
This question requires you to manipulate the Spanish football dataset to select particular years or teams, to look at the distribution of the data, and to calculate the statistical probability of difference between data samples.
This code shows how to create a new dataset by selecting part of a larger dataset and how to calculate statistical metrics according to some criteria (e.g. calculate the mean for all values from the same year).
Q1. Create a new dataset that contains data for Racing Santander in the 1929 season. The dataset should contain ONLY the season, home team name, visiting team name and the number of home goals. Display the structure of your dataset using the head(name) function in your R-code. [4 MARKS]
RacingSantander1929 = spain_data[spain_data$home == 'Racing Santander' & spain_data$Season == '1929', c("Season", "home", "visitor", "hgoal")]
head(RacingSantander1929) Season home visitor hgoal
91 1929 Racing Santander Espanyol Barcelona 4
103 1929 Racing Santander Arenas de Getxo 2
114 1929 Racing Santander Atletico Madrid 3
130 1929 Racing Santander CE Europa 6
142 1929 Racing Santander Real Sociedad 2
151 1929 Racing Santander FC Barcelona 2Q2. Create a histogram of the home goals scored by Racing Santander in 1929 matches and display the mean and standard deviation of this metric.
(NOTE: make sure that the histogram bins are centred at integer values). [4 MARKS]
bins=(1:8)-0.5
hist(RacingSantander1929$hgoal,
breaks = bins,
main = "Home goals by Racing Santander",
xlab = "Home goals",
ylab = "Frequency",
col = "blue")
mean(RacingSantander1929$hgoal)[1] 3sd(RacingSantander1929$hgoal)[1] 1.414214Q3. Create a histogram of the mean number of home goals scored per season by Racing Santander, over the whole period covered by this Spanish football dataset. [6 MARKS]
homegoalmean = spain_data[spain_data$home == 'RacingSantander' & spain_data$Season]
homegoalmean = aggregate(hgoal ~ Season, data = spain_data, FUN = mean)
bins=(1:6)-0.5
hist(homegoalmean$hgoal,
main = "Racing Santander: Mean Home Goals per Season",
xlab = "Average Goals",
ylab = "Frequency (Number of Seasons)",
col = "darkgreen",
breaks = bins)
Q4. Use a one-sample Student’s-t test to determine if the mean homegoals scored by Santander in the 1929 season is statistically different to the average they scored per season over the whole data record. HINT: ask an AI engine: ‘how do i do a one-sample t-test in R?’ [5 MARKS]
all_santander_home = spain_data[spain_data$home == 'Racing Santander', ]
historical_mu = mean(all_santander_home$hgoal, na.rm = TRUE)
goals_1929 = all_santander_home$hgoal[all_santander_home$Season == 1929]
t_test_results = t.test(goals_1929, mu = historical_mu)
print(t_test_results)
One Sample t-test
data: goals_1929
t = 2.8859, df = 8, p-value = 0.02033
alternative hypothesis: true mean is not equal to 1.639551
95 percent confidence interval:
1.912939 4.087061
sample estimates:
mean of x
3 Delete one of the following statements:
The 1929 season IS statistically different to the average.
Q5. Why is it recommended to use the Student’s-t test in this case? [1 MARK]
The Student’s t-test was preferred over a Z-test because the sample size for the 1929 season is small and the true population standard deviation of Racing Santander’s scoring is unknown. The t-test provides a more conservative and accurate estimate of significance by accounting for the additional uncertainty inherent in small datasets.
Bivariate data
In this section you will perform a regression analysis to model the relationship between runner age and race time for the marathon. Dataset: marathon_data.
Q6. THE FINAL 2 QUESTIONS GO BEYOND WHAT I HAVE SHOWN YOU IN CLASS. You may want to complete these outside of the Wednesday session to give yourself more time. Remember - YOU CAN USE AN AI CHATBOT TO HELP YOU DEVELOP THE REQUIRED CODE.
Create a new dataset that contains marathon run times ONLY for runners aged between 30 and 50 years old. Create a fully-labeled plot of runtime vs. age for this dataset. [4 MARKS]
Use the R-function lm to calculate the coefficients of a 2nd order polynomial model for race time vs age.
i.e. model of the form: y = a + bx + cx2.
Add a line on your plot showing the regression line. [6 MARKS]
marathon_runtimes = marathon_data[marathon_data$age >= 30 & marathon_data$age <= 50, ]
plot(marathon_runtimes$age,marathon_runtimes$time,
main = "Marathon Runtime vs. age (30-50 Years Old)",
xlab = "Age (Years)",
ylab = "Runtime (Minutes)",
pch = 19,
col = "steelblue") 
model <- lm(time ~ age + I(age^2), data = marathon_runtimes)
summary(model)
Call:
lm(formula = time ~ age + I(age^2), data = marathon_runtimes)
Residuals:
Min 1Q Median 3Q Max
-3.6968 -0.4692 -0.1487 0.5606 4.4186
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 171.34281 22.90676 7.480 9e-07 ***
age -2.33757 1.16637 -2.004 0.0613 .
I(age^2) 0.04047 0.01460 2.772 0.0130 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.95 on 17 degrees of freedom
Multiple R-squared: 0.8977, Adjusted R-squared: 0.8857
F-statistic: 74.58 on 2 and 17 DF, p-value: 3.84e-09plot(marathon_runtimes$age, marathon_runtimes$time,
main = "Marathon Runtime vs. Age: Quadratic Fit",
xlab = "Age (Years)", ylab = "Runtime (Minutes)",
pch = 19, col = "steelblue")
age_range <- seq(30, 50, length.out = 100)
predicted_times <- predict(model, newdata = data.frame(age = age_range))
lines(age_range, predicted_times, col = "red", lwd = 3)
Comment on the accuracy of your regression model by comparing your predicted value to the actual measurement data in the marathon_data set. [2 MARKS]
The model is likely to be inaccurate for ages outside the 30–50 range. The actual runner was much faster (age 65.15, 187.8 minutes) in the original data set, than the predicted time, proving that the quadratic trend overestimates the impact of aging on performance for older athletes.
When asked to predict the time for a 65-year-old, you are assuming that the physiological “curve” of aging stays exactly the same for an extra 15 years. The model is likely to be inaccurate due to an extrapolation error; the quadratic trend established in the 30-50 age bracket overestimates the slowdown associated with aging, failing to account for the high performance levels possible in older, dedicated athletes.
A 2nd order polynomial is also very sensitive.
If your \(c\) coefficient (the \(age^2\) term) is positive, the curve starts to bend upward increasingly fast.
By the time it reaches age 65, the “bend” might be so steep that it predicts a much slower time than is realistic.