HW 06: Foundations for Inference

Conceptual and mathematical foundations for conducting statistical inference.

Overview

Setup

TipBackground

Wildlife biologists are studying head lengths of brushtail possums. Prior research shows that adult head lengths are approximately Normally distributed with

\[ \mu = 92.6 \text{ mm}, \qquad \sigma = 3.6 \text{ mm}. \]

A new field team collects data from different regions in Australia and New Zealand and wants to determine whether their samples are consistent with the historical population.

Throughout this assignment, you will help analyze their findings.

Submission instructions

  • Use the template. provided.
  • Show all your work.
  • There is no raw data to analyze, but you will need to use R functions and math notation wrapped in dollar signs like this: \(P(Z > 1)\)
  • Compile to PDF and submit to Canvas(Gradescope) by the due date

Part I — Individual Possums

1. Standardizing Measurements

  1. Compute the z-score for a possum with head length 95.4 mm.
  2. Compute the z-score for a possum with head length 85.8 mm.
  3. Which possum is more unusual? Justify using the magnitude of the z-score
  4. Interpret the z-score for the 85.8 mm possum in complete context

2. Probability for Individual Possums

  1. Write the probability statement for the chance that a randomly selected possum has head length less than 85 mm.
  2. Convert this probability to standard normal form.
  3. Compute the probability using R.
  4. Would a possum measuring 85 mm be considered unusually small? Justify numerically.

3. Identifying Unusual Observations

A field researcher reports finding a possum with head length 101 mm.

  1. Compute its z-score.
  2. Compute the probability of observing a possum at least this large.
  3. Based on your result, would you question whether this possum belongs to the same population? Explain.

Part II — Sampling Distributions

A new research team measures n = 40 possums from a study location in Eastern Australia.

4. Distribution of Sample Means

  1. What is the mean of the sampling distribution of \(\bar{x}\) when \(n=40\)?
  2. What is the standard deviation of \(\bar{x}\)?
  3. How does this standard deviation compare to the population standard deviation? Explain why.

5. Probability for a Sample Mean

The sample of 40 possums has an average head length of 90.5 mm.

  1. Write the probability statement for observing a sample mean this small or smaller.
  2. Convert to standard normal form.
  3. Compute the probability.
  4. Does this provide strong evidence that this region’s possums differ from the historical population? Explain.

6. Comparing Two Field Sites

Another team samples 40 possums at a site in New Zealand (they are considered an invasive species there) and finds a mean head length of 94.1 mm.

  1. Compute the probability of observing a sample mean this large or larger.
  2. Which field site (90.5 mm or 94.1 mm) shows stronger evidence of being different from the historical population? Explain

Part III — Percentiles and Central Ranges

Refer to Section 3.4.2 of the course packet for example code using qnorm.

7. Percentiles for Individual Possums

  1. Find the 90th percentile for individual head lengths.
  2. Interpret this value in context.

8. Percentiles for Sample Means

  1. Find the 90th percentile for sample mean at the original Australia site when \(n=40\).
  2. Explain why this percentile is closer to the mean than in Question 7.

9. Middle 95%

  1. Find the interval bounding the middle 95% of individual possum head lengths.
  2. Find the interval bounding the middle 95% of sample means when \(n=40\).
  3. Explain, in plain language, why these two intervals differ in width.

Reflection

In 3–4 sentences, explain how the Central Limit Theorem helps researchers distinguish between unusual individual possums, and unusual regions of possums.