Set up

Refer back to the directions for Problem set 1 in the “Set up” section for formatting and completion guidelines.

Problems

1. Spread vs precision

Mathematically, how do SD and SE differ (in their formulas)?
Conceptually, how do SD and SE differ (i.e. how do they differ in what they represent)?

2. Hypothesis writing

What are the main characteristics of a:

biological,
statistical, and
mathematical hypothesis?

3. Hypothesis testing

a. Pickleweed

You read in a paper that mean pickleweed (Salicornia virginica) stem diameter in a slough in San Diego is 0.43 cm. You think that the pickleweed at Devereux Slough has a different height.

What are your null and alternative hypotheses in biological and statistical terms?
What test would you choose to test your hypothesis? Justify your answer.

b. Oak growth rates

You think that growth rate (measured in cm/week) of oak trees grown in the greenhouses will increase after a few weeks of nitrogen addition to the soil in which they are growing. You measure growth rate for two weeks before adding nitrogen, then measure growth rate for two weeks after adding nitrogen.

What are your null and alternative hypotheses in biological and statistical terms?
What test would you choose to test your hypotheses? Justify your answer.

4. Interpreting p-values

a. Mean monthly air temperature

You’re comparing mean daily air temperature between January and March at Toolik Field Station.

This is some summary information about the data:

# A tibble: 2 × 3
  month mean_monthly_airtemp     n
  <ord>                <dbl> <int>
1 Jan                  -20.5    31
2 Mar                  -12.9    31

And this is the result of your test:


    Welch Two Sample t-test

data:  mean_daily_airtemp by month
t = -3.5594, df = 40.139, p-value = 0.0009733
alternative hypothesis: true difference in means between group Jan and group Mar is not equal to 0
95 percent confidence interval:
 -11.899789  -3.280856
sample estimates:
mean in group Jan mean in group Mar 
        -20.52258         -12.93226

What kind of t-test is this?
What must be true to run this type of t-test?
Write a statement summarizing the results of this test in 1-2 sentences.

b. Crab carapace size

You’re comparing the carapace size (measured in mm) of crabs between Narragansett Bay National Estuarine Research Reserve and Cape Cod.

This is some summary information about the data:

# A tibble: 2 × 3
  name                  mean_size     n
  <chr>                     <dbl> <int>
1 Cape Cod                   16.8    27
2 Narragansett Bay NERR      17.1    29

This is the result of your test:


    Two Sample t-test

data:  size by name
t = -0.55802, df = 54, p-value = 0.5791
alternative hypothesis: true difference in means between group Cape Cod and group Narragansett Bay NERR is not equal to 0
95 percent confidence interval:
 -1.4934094  0.8430901
sample estimates:
             mean in group Cape Cod mean in group Narragansett Bay NERR 
                           16.81519                            17.14034

What kind of t-test is this?
What must be true to run this type of t-test?
Write a statement summarizing the results of this test in 1-2 sentences.

c. Trout weights

You’re comparing the weights of cutthroat trout (measured in g) between cascade vs pool habitats.

This is some summary information about the data:

# A tibble: 2 × 3
  unittype mean_weight     n
  <chr>          <dbl> <int>
1 cascade         8.07 11419
2 pool           11.9   5470

This is the result of your test:


    Welch Two Sample t-test

data:  weight_g by unittype
t = -15.984, df = 4581.5, p-value < 2.2e-16
alternative hypothesis: true difference in means between group cascade and group pool is not equal to 0
95 percent confidence interval:
 -4.254626 -3.324975
sample estimates:
mean in group cascade    mean in group pool 
             8.069594             11.859394

How would you write about the results of this test?

--- title: "Problem set 2" editor: source published-title: "Due date" date: 2026-04-23 date-modified: last-modified categories: [problem set] --- # Set up Refer back to the directions for Problem set 1 in the "Set up" section for formatting and completion guidelines. ```{r} #| echo: false #| message: false library(tidyverse) library(lterdatasampler) ``` # Problems ## 1. Spread vs precision a. Mathematically, how do SD and SE differ (in their formulas)? b. Conceptually, how do SD and SE differ (i.e. how do they differ in what they represent)? ## 2. Hypothesis writing What are the main characteristics of a: a. biological, b. statistical, and c. mathematical hypothesis? ## 3. Hypothesis testing ### a. Pickleweed You read in a paper that mean pickleweed (_Salicornia virginica_) stem diameter in a slough in San Diego is 0.43 cm. You think that the pickleweed at Devereux Slough has a different height. - What are your null and alternative hypotheses in biological _and_ statistical terms? - What test would you choose to test your hypothesis? Justify your answer. ### b. Oak growth rates You think that growth rate (measured in cm/week) of oak trees grown in the greenhouses will increase after a few weeks of nitrogen addition to the soil in which they are growing. You measure growth rate for two weeks before adding nitrogen, then measure growth rate for two weeks after adding nitrogen. - What are your null and alternative hypotheses in biological _and_ statistical terms? - What test would you choose to test your hypotheses? Justify your answer. ## 4. Interpreting p-values ### a. Mean monthly air temperature You're comparing _mean_ daily air temperature between January and March at Toolik Field Station. This is some summary information about the data: ```{r} #| echo: false temp_df <- arc_weather |> mutate(month = month(date, label = TRUE), year = year(date)) |> filter(year == 2018 & month %in% c("Jan", "Mar")) |> rename(mean_daily_airtemp = mean_airtemp) temp_df |> group_by(month) |> summarize(mean_monthly_airtemp = mean(mean_daily_airtemp), n = length(mean_daily_airtemp)) |> ungroup() ``` And this is the result of your test: ```{r} #| echo: false t.test( mean_daily_airtemp ~ month, data = temp_df, var.equal = FALSE ) ``` - What kind of t-test is this? - What _must_ be true to run this type of t-test? - Write a statement summarizing the results of this test in 1-2 sentences. ### b. Crab carapace size You're comparing the carapace size (measured in mm) of crabs between Narragansett Bay National Estuarine Research Reserve and Cape Cod. This is some summary information about the data: ```{r} #| echo: false crab_df <- pie_crab |> filter(site %in% c("CC", "NB")) crab_df |> group_by(name) |> summarize(mean_size = mean(size), n = length(size)) |> ungroup() ``` This is the result of your test: ```{r} #| echo: false t.test( size ~ name, data = crab_df, var.equal = TRUE ) ``` - What kind of t-test is this? - What _must_ be true to run this type of t-test? - Write a statement summarizing the results of this test in 1-2 sentences. ### c. Trout weights You're comparing the weights of cutthroat trout (measured in g) between cascade vs pool habitats. This is some summary information about the data: ```{r} #| echo: false trout_df <- and_vertebrates |> filter(species == "Cutthroat trout" & unittype %in% c("C", "P")) |> mutate(unittype = case_when( unittype == "C" ~ "cascade", unittype == "P" ~ "pool" )) trout_df |> group_by(unittype) |> summarize(mean_weight = mean(weight_g, na.rm = TRUE), n = length(weight_g)) |> ungroup() ``` This is the result of your test: ```{r} #| echo: false t.test( weight_g ~ unittype, data = trout_df, var.equal = FALSE ) ``` - How would you write about the results of this test?