Problem Set 4

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library(tidyverse); theme_set(theme_classic())
library(readxl)
set.seed(2026)

Sample mean vs. a reference value

Scenario 1

A marine protected area (MPA) was established five years ago. Previous surveys from nearby fished reefs found a mean fish biomass of 12.2 kg 100m^-2.

Researchers survey 15 transects inside the MPA and want to test whether fish biomass is higher than the regional average of fished reefs.

The data are available from the “MPA” sheet in “H2SMS_practicalData.xlsx”. Load and check the data.

What is the response variable?

What is the null hypothesis?

What is the alternative hypothesis?

Evaluate the null hypothesis using \(\alpha = 0.05\). Perform an appropriate analysis (including assumption checks), and report the relevant results in written text.

In your own words, explain what the p-value represents.

Scenario 2

In New Zealand, Brewin et al. (2000) studied fecundity and spawning of the sea urchin Evechinus chloroticus, finding a mean annual gamete output of 0.068 g g^-1 yr^-1 between 1990-1994 at Dieffenbach Point. On my holiday, I decided to repeat that aspect of their experiment (securing all necessary permits and approvals).

The data are in the “UrchinGametes” sheet in “H2SMS_practicalData.xlsx”, already summarised to grams of gamete per grams of urchin per year. Load the data and perform appropriate checks.

What is the response variable?

What is the null hypothesis?

What is the alternative hypothesis?

Evaluate the null hypothesis using \(\alpha = 0.05\). Perform an appropriate analysis (including assumption checks), and report the relevant results in written text.

In your own words, interpret your results with regard to the null hypothesis and the p-value.

Confidence Intervals

The sheet “Plastics” in “H2SMS_practicalData.xlsx” contains observations of the number of plastic pellets per kilogram. Load the data.

Consider this the entire population of interest.

Write code to take a sample of 25 observations from this population and store it in a vector called obs.

Hint

sample()

How different is your sample mean from the true population mean?

What is the term that describes this difference?

Calculate 95% confidence intervals for your sample mean using only your sample. Do they include the true value?

Assuming each student starts with a different set.seed, what proportion of students would answer ‘yes’ to the previous question?

If you instead calculated 80% CIs, a) would you be more likely to capture the true mean or less? And, b) would the width of your confidence interval be larger or smaller?

Calculate the 80% CI for your sample and compare.

If you were to run your code again (without resetting set.seed), what is the probability that you would collect a sample with a 90% CI that includes the true value?