---
title: "Foundations for Inferencet"
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# Part I — Individual Possums

## 1. Standardizing Measurements

a) Compute the z-score for a possum with head length 95.4 mm.

```{r}

```

b) Compute the z-score for a possum with head length 85.8 mm.
```{r}

```

c) Which possum is more unusual? Justify using the magnitude of the z-score

d) Interpret the z-score for the 85.8 mm possum in complete context


## 2. Probability for Individual Possums

a) Write the probability statement for the chance that a randomly selected possum has head length less than 85 mm.

$P(X ....)$ (replace the `...` with your answer)

b) Convert this probability to standard normal form.

$P(Z ....)$ (replace the `...` with your answer)

c) Compute the probability using R.
```{r}

```

d) 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.

a) Compute its z-score.
```{r}

```


b) Compute the probability of observing a possum at least this large.
```{r}

```


c) 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

a) What is the mean of the sampling distribution of $\bar{x}$ when $n=40$?

b) What is the standard deviation of $\bar{x}$? 
```{r}

```


c) 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.

a) Write the probability statement for observing a sample mean this small or smaller.  

$P(\bar{X} ....)$ (replace the `...` with your answer)

b) Convert to standard normal form.

$P(Z ....)$ (replace the `...` with your answer)

c) Compute the probability.
```{r}

```

d) 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.

a) Compute the probability of observing a sample mean this large or larger.
```{r}

```

b) 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

a) Find the 90th percentile for individual head lengths.  
```{r}

```

b) Interpret this value in context.


## 8. Percentiles for Sample Means

a) Find the 90th percentile for sample mean at the original Australia site when $n=40$.  
```{r}

```

b) Explain why this percentile is closer to the mean than in Question 7.


## 9. Middle 95%

a) Find the interval bounding the middle 95% of individual possum head lengths.  
```{r}

```

b) Find the interval bounding the middle 95% of sample means when $n=40$.  
```{r}

```

c) 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.
