Lecture 04: Probability and Inference

Author

Bill Perry

Lecture 4: Probability and Statistical Inference

Review of probability distributions
Standard normal distribution and Z-scores
Standard error and confidence intervals
Statistical inference fundamentals
Hypothesis testing principles

Practice Exercise 1: Exploring the Grayling Dataset

Let’s explore the Arctic grayling data from lakes I3 and I8. Use the grayling_df data frame to create basic summary statistics.

# Write your code here to explore the basic structure of the data
# also note plottig a box plot is really useful


# Calculate summary statistics
grayling_summary <- grayling_df %>% 
  group_by(lake) %>%
  summarize(
    mean_length = mean(length_mm, na.rm = TRUE),
    sd_length = sd(length_mm, na.rm = TRUE),
    se_length = sd_length/sqrt(sum(!is.na(length_mm))),
    count = sum(!is.na(length_mm)),
    .groups = "drop")
grayling_summary

# A tibble: 2 × 5
  lake  mean_length sd_length se_length count
  <chr>       <dbl>     <dbl>     <dbl> <int>
1 I3           266.      28.3      3.48    66
2 I8           363.      52.3      5.18   102

Lecture 4: Probability Distributions

Probability Distribution Functions

A probability distribution describes the probability of different outcomes in an experiment
We’ve seen histograms of observed data
Theoretical distributions help us model and understand real-world data
We will focus on a standard normal distribution and a students t distribution

Lecture 4: The Standard Normal Distribution

The standard normal distribution is crucial for understanding statistical inference:

Has mean (μ) = 0 and standard deviation (σ) = 1
Symmetrical bell-shaped curve
Area under the curve = 1 (total probability)
Approximately:
- 68% of data within ±1σ of the mean
- 95% of data within ±2σ of the mean - really 1.96σ
- 99.7% of data within ±3σ of the mean

Z-scores allow us to convert any normal distribution to the standard normal distribution.

Practice Exercise 2: Calculating Z-scores of lake I3

Practice Exercise 2: Calculating Z-scores

Let’s practice converting raw values to Z-scores using the Arctic grayling data.

Z Score = (length - mean) / standard deviation

# Calculate the mean and standard deviation of fish lengths
mean_length <- mean(i3_df$length_mm, na.rm = TRUE)
sd_length <- sd(i3_df$length_mm, na.rm = TRUE)

# Calculate Z-scores for fish lengths
i3_df <- i3_df %>%
  mutate(z_score = (length_mm - mean_length) / sd_length)

# View the first few rows with Z-scores
head(i3_df)

# A tibble: 6 × 6
   site lake  species         length_mm mass_g z_score
  <dbl> <chr> <chr>               <dbl>  <dbl>   <dbl>
1   113 I3    arctic grayling       266    135  0.0139
2   113 I3    arctic grayling       290    185  0.862 
3   113 I3    arctic grayling       262    145 -0.127 
4   113 I3    arctic grayling       275    160  0.332 
5   113 I3    arctic grayling       240    105 -0.905 
6   113 I3    arctic grayling       265    145 -0.0214

Lecture 4: The fish data as a z score

So if we plot this data what does it look like in a standard normal distributon?

Z-score Results

How to get area under 1 STD DEV?

Proportion within 1 standard deviation = sum of absolute values of Z Scores that are less than or equal to 1 divided by the number in the sample…

Remember in a true normal distribution it is 68% within 1 std dev.

should be approximately (varies if distribution is not normal):

68% of data within ±1σ of the mean
95% of data within ±2σ of the mean - really 1.96σ
99.7% of data within ±3σ of the mean

# What proportion of fish are within 1 standard deviation of the mean?
within_1sd <- sum(abs(i3_df$z_score) <= 1, na.rm = TRUE) / sum(!is.na(i3_df$z_score))
cat("Proportion within 1 SD:", round(within_1sd * 100, 1), "%\n")

Proportion within 1 SD: 81.8 %

Lecture 4: Standard normal distribution - Fish Data

You want to know things about this population like

probability of a fish having a certain length (e.g., > 300 mm)
Can solve this by integrating the area under curve
But it is tedious to do every time
Instead
- we can use the standard normal distribution (SND)
- and can use the proportions from the density curve

# A tibble: 1 × 1
  mean_length
        <dbl>
1        266.

Lecture 4: Standard normal distribution properties

Standard Normal Distribution

“benchmark” normal distribution with µ = 0, σ = 1
The Standard Normal Distribution is defined so that:
- ~68% of the curve area within +/- 1 σ of the mean,
- ~95% within +/- 2 σ of the mean,
- ~99.7% within +/- 3 σ of the mean

*remember σ = standard deviation

Lecture 4: Using Z-tables

Areas under curve of Standard Normal Distribution

Have been calculated for a range of sample sizes
Can be looked up in z-table
No need to integrate
Any normally distributed data can be standardized
- transformed into the standard normal distribution
- a value can be looked up in a table

Lecture 4: Z-score Formula

Done by converting original data points to z-scores

Z-scores calculated as:

\(\text{Z = }\frac{x_i-\mu}{\sigma}\)

z = z-score for observation
xi = original observation
µ = mean of data distribution
σ = SD of data distribution

So lets do this for a fish that is 300mm long and guess the probability of catching something larger

z = (300 - 265.61)/28.3 = 1.215194

Lecture 4: Z-score example from table

Done by converting original data points to z-scores

Z-scores calculated as:

\(\text{Z = }\frac{X_i-\mu}{\sigma}\)

z = z-score for observation
xi = original observation
µ = mean of data distribution
σ = SD of data distribution

So lets do this for a fish that is 300mm long and guess the probability of catching something larger

z = (300 - 265.61)/28.3 = 1.22
look up 1.2 on left and 0.02 on top to get 0.8888 in table
Means 88.9% is the area left of the curve and
100 - 88.9 = 11.27% of fish are expected to be longer

At what point do you think its not likely to catch a larger fish - what percentage?

do this the other way using that percent and why?

Lecture 4: Z-score example calculation in r

We can use R to get these values easier…

# For standard normal distribution (mean=0, sd=1):

pnorm(z) # gives cumulative probability (area to the left)
qnorm(p) # gives z-value for a given probability
dnorm(z) # gives probability density

# Examples:
z_value <-  1.22
prob_left <- pnorm(z_value)          # 0.975 (97.5% to the left)
prob_right <- 1 - pnorm(z_value)     # 0.025 (2.5% to the right)
prob_between <- pnorm(2) - pnorm(-2)  # 0.95 (95% between ±1.96)
# To find z-value for a given probability:
z_for_95_percent <- qnorm(0.888)     # 1.96
print(prob_left)

[1] 0.8887676

print(prob_right)

[1] 0.1112324

print(prob_between)

[1] 0.9544997

print(z_for_95_percent)

[1] 1.21596

Lecture 4: We can now use this for fun in the fish

Lets say we are interested in knowing at what point from I3 it is not likely to catch a larger fish?

Maybe we expect 95% of the time to catch a fish that is “common” but the 5% is the unlikely portion….

# Examples:
# What fish length corresponds to the top 5% (unlikely)?
top_5_percent_z <- qnorm(0.95)  # z-score for 95th percentile
unlikely_length <- mean_length + (top_5_percent_z * sd_length)

cat("Only 5% of fish are longer than:", round(unlikely_length, 1), "mm\n")

Only 5% of fish are longer than: 312.2 mm

cat("This corresponds to z-score:", round(top_5_percent_z, 3), "\n")

This corresponds to z-score: 1.645

Lecture 4: What this means

Given that we can: transform data to z-scores from standard normal distribution…

…figure out area under the curve (probability) associated with range of z-scores…

…can therefore figure out probability associated with a range of original data

Lecture 4: So what is next…

We can look at Standard normal distributions and know probability of a value being in a range under the standard normal curve…

Previously we had calculated Standard Error and Confidence Intervals -

Now can assess our confidence that the population mean is within a certain range
Can use t distribution to ask questions like:
- “What is probability of getting sample with mean = ȳ from population with mean = µ?“ (1 sample t-test)
- “What is the probability that two samples came from same population?” (2 sample t-test)

Lecture 4: When Population σ is Unknown

When calculating confidence intervals we usually DON’T know the population σ (standard deviation) or 𝝁 population mean

estimate it from the samples when don’t know the population σ or 𝝁
and when sample size is small < ~30
can’t use the standard normal (z) distribution

Instead, we use Student’s t distribution

Lecture 4: Understanding t-distribution

When sample sizes are small, the t-distribution is more appropriate than the normal distribution.

Similar to normal distribution but with heavier tails
Shape depends on degrees of freedom (df = n-1)
With large df (>30), approaches the normal distribution
Used for:
- Small sample sizes
- When population standard deviation is unknown
- Calculating confidence intervals
- Conducting t-tests

Student’s t-distribution Formula

To calculate CI for sample from “unknown” population:

\(\text{CI} = \bar{y} \pm t \cdot \frac{s}{\sqrt{n}}\)

Where:

ȳ is sample mean
𝑛 is sample size
s is sample standard deviation
t t-value corresponding the probability of the CI
t in t-table for different degrees of freedom (n-1)

Lecture 4: Student’s t-distribution Table

Here is a t-table

Values of t that correspond to probabilities
Probabilities listed along top
Sample dfs are listed in the left-most column
Probabilities are given for one-tailed and two-tailed “questions”

Lecture 4: One-tailed Questions

One-tailed questions: area of distribution left or (right) of a certain value

n=20 (df=19) - 90% of the observations found left
t= 1.328 (10% are outside)

Lecture 4: Two-tailed Questions

Two-tailed questions refer to area between certain values

n= 20 (df=19), 90% of the observations are between
t=-1.729 and t=1.729 (10% are outside)

Lecture 4: t-distribution CI Example

Let’s calculate CIs again:

Use two-sided test

\(\text{CI} = \bar{y} \pm t \cdot \frac{s}{\sqrt{n}}\)

95% CI Sample A: = 272.8 ± 2.306 * (37.81/(9^0.5))
mean = 272.8, N = 20, and s = 37.81 - t is ?
CI = 29.06
The 95% CI is between 243.7 and 301.9
“The 95% CI for the population mean from sample A is 272.8 ± 29.06

Practice Exercise 4: Using the t-distribution

Let’s compare confidence intervals using the normal approximation (z) versus the t-distribution for our fish data. I3 data and 10 fish Mean is 266.7 - sd is 17.12 - se is 5.41

## \(\text{CI} = \bar{y} \pm t \cdot \frac{s}{\sqrt{n}}\)

Practice Exercise 4: Using the t-distribution

Let’s compare confidence intervals using the normal approximation (z) versus the t-distribution for our fish data. I3 data and 10 fish Mean is 266.7 - sd is 17.12 - se is 5.41

## \(\text{CI} = \bar{y} \pm t \cdot \frac{s}{\sqrt{n}}\)

# Display results
cat("Mean:", round(sample_mean, 1), "mm\n")

Mean: 276.3 mm

cat("Standard deviation:", round(sample_sd, 2), "mm\n")

Standard deviation: 26.73 mm

cat("Standard error:", round(sample_se, 2), "mm\n")

Standard error: 8.45 mm

cat("95% CI using z:", round(z_ci_lower, 1), "to", round(z_ci_upper, 1), "mm\n")

95% CI using z: 259.7 to 292.9 mm

cat("95% CI using t:", round(t_ci_lower, 1), "to", round(t_ci_upper, 1), "mm\n")

95% CI using t: 257.2 to 295.4 mm

cat("t critical value:", round(t_crit, 3), "vs z critical value: 1.96\n")

t critical value: 2.262 vs z critical value: 1.96

Lecture 4: Intro to Hypothesis Testing one tailed

Hypothesis testing is a systematic way to evaluate research questions using data.

Key components:

Null hypothesis (H₀): Typically assumes “no effect” or “no difference”
Alternative hypothesis (Hₐ): The claim we’re trying to support
Statistical test: Method for evaluating evidence against H₀
P-value: Probability of observing our results (or more extreme) if H₀ is true
Significance level (α): Threshold for rejecting H₀, typically 0.05

Decision rule: Reject H₀ if p-value < α

lets test if our sample mean of 320 is larger than 270 or not? Essentially we are looking at the confidence intervals!!! But we are only interested if it is larger

Summary of One-Tailed Hypothesis Test:

Sample mean: 320

Hypothesized mean: 285

Sample size: 12

Standard deviation: 42.15

Standard error: 12.168

t-statistic: 2.876

Critical t-value (one-tailed): 1.796

Critical value: 306.85

Decision: Reject Ho (sample mean falls in upper rejection region)

Lecture 4: Intro to Hypothesis Testing one tailed

Hypothesis testing is a systematic way to evaluate research questions using data.

Key components:

Null hypothesis (H₀): Typically assumes “no effect” or “no difference”
Alternative hypothesis (Hₐ): The claim we’re trying to support
Statistical test: Method for evaluating evidence against H₀
P-value: Probability of observing our results (or more extreme) if H₀ is true
Significance level (α): Threshold for rejecting H₀, typically 0.05

Lecture 4: Hypothesis Testing two tailed

Hypothesis testing is a systematic way to evaluate research questions using data.

Key components:

Null hypothesis (H₀): Typically assumes “no effect” or “no difference”
Alternative hypothesis (Hₐ): The claim we’re trying to support
Statistical test: Method for evaluating evidence against H₀
P-value: Probability of observing our results (or more extreme) if H₀ is true
Significance level (α): Threshold for rejecting H₀, typically 0.05

Decision rule: Reject H₀ if p-value < α

lets test if our sample mean of 320 is equal to 270 or not? Essentially we are looking at the confidence intervals!!!

Summary of Hypothesis Test:

Sample mean: 320

Hypothesized mean: 270

Standard error: 12.603

t-statistic: 3.967

Critical t-value (±): 2.306

Critical values: 240.94 to 299.06

Decision: Reject Ho (sample mean falls in rejection region)

Lecture 4: Hypothesis Testing two tailed

Hypothesis testing is a systematic way to evaluate research questions using data.

Key components:

Null hypothesis (H₀): Typically assumes “no effect” or “no difference”
Alternative hypothesis (Hₐ): The claim we’re trying to support
Statistical test: Method for evaluating evidence against H₀
P-value: Probability of observing our results (or more extreme) if H₀ is true
Significance level (α): Threshold for rejecting H₀, typically 0.05

Decision rule: Reject H₀ if p-value < α

lets test if our sample mean of 320 is equal to 270 or not? Essentially we are looking at the confidence intervals!!!

Practice Exercise 5: One-Sample t-Test

Practice Exercise 5: Lets practice a One-Sample t-Test

Let’s perform a one-sample t-test to determine if the mean fish length in Lake I3 differs from 260 mm:

# get only lake I3
i3_df <- grayling_df %>% filter(lake=="I3")

# what is the mean
i3_mean <- mean(i3_df$length_mm, na.rm=TRUE)
cat("Mean:", round(i3_mean, 1), "mm\n")

Mean: 265.6 mm

# Perform a one-sample t-test
t_test_result <- t.test(i3_df$length_mm, mu = 260)

# View the test results
t_test_result


    One Sample t-test

data:  i3_df$length_mm
t = 1.6091, df = 65, p-value = 0.1124
alternative hypothesis: true mean is not equal to 260
95 percent confidence interval:
 258.6481 272.5640
sample estimates:
mean of x 
 265.6061

Interpret this test result by answering these questions:

What was the null hypothesis?
What was the alternative hypothesis?
What does the p-value tell us?
Should we reject or fail to reject the null hypothesis at α = 0.05?
What is the practical interpretation of this result for fish biologists?

Practice Exercise 6: Formulating Hypotheses

For the following research questions about Arctic grayling, write the null and alternative hypotheses:

Are fish in Lake I8 longer than fish in Lake I3?

# Let's test one of these hypotheses: Are fish in Lake I8 longer than fish in Lake I3?

# Perform an independent t-test
t_test_result <- t.test(length_mm ~ lake, data = grayling_df, 
                       alternative = "less")  # H₀: μ_I3 ≥ μ_I8, H₁: μ_I3 < μ_I8

# Display the results
t_test_result


    Welch Two Sample t-test

data:  length_mm by lake
t = -15.532, df = 161.63, p-value < 2.2e-16
alternative hypothesis: true difference in means between group I3 and group I8 is less than 0
95 percent confidence interval:
      -Inf -86.66138
sample estimates:
mean in group I3 mean in group I8 
        265.6061         362.5980

Based on this t-test, what can we conclude about the difference in fish length between the two lakes?

Lecture 4: Understanding P-values

A p-value is the probability of observing the sample result (or something more extreme) if the null hypothesis is true.

Common interpretations:

- p < 0.05: Strong evidence against H₀
- 0.05 ≤ p < 0.10: Moderate evidence against H₀
- p ≥ 0.10: Insufficient evidence against H₀

Common misinterpretations:

- p-value is NOT the probability that H₀ is true
- p-value is NOT the probability that results occurred by chance
- Statistical significance ≠ practical significance
the smaller the p value does not necessarily mean much… use < 0.05 even if is is 10^-16

Lecture 4: Type I and Type II Errors

When making decisions based on hypothesis tests, two types of errors can occur:

Type I Error (False Positive)

- Rejecting H₀ when it’s actually true
- Probability = α (significance level)
- “Finding an effect that isn’t real”

Type II Error (False Negative)

- Failing to reject H₀ when it’s actually false
- Probability = β - “Missing an effect that is real”

Statistical Power = 1 - β

- Probability of correctly rejecting a false H₀
- Increases with:
- - Larger sample size
- - Larger effect size
- - Lower variability
- - Higher α level

Lecture 4: Type I and Type II Errors

What does it mean…
- Black curve (Null Distribution): distribution of test statistics when Ho is true
- Green curve (Alternative Distribution): distribution when Ha is true (is an effect)
Red shaded area (Type I Error): probability of rejecting Ho when it’s actually tru
- area under the null distribution (black curve) to the right α (p=0.05)
Blue shaded area (Type II Error): probability of failing to reject Ho when alternative is actually true
- area under alternative distribution (green curve) left of α (depends on effect size, sample size, etc.)

The Key Insight fundamental trade-off in hypothesis testing:

as α value moves left or right, change balance between Type I and Type II errors
Moving left reduces Type II errors increases Type I errors, and vice versa
power (1 - β) area under the green curve RIGHT of dashed line
the probability of correctly detecting a real effect.

Practice Exercise 7: Interpreting Errors and Power

Practice Exercise 6: Interpreting P-values and Errors

Given the following scenarios, identify whether a Type I or Type II error might have occurred:

A researcher concludes that a new fishing regulation increased grayling size, when in fact it had no effect.
A study fails to detect a real decline in grayling population due to warming water, concluding there was no effect.
Let’s calculate the power of our t-test to detect a 30 mm difference in length between lakes:

# Calculate power for detecting a 30 mm difference
# First determine parameters
lake_I3 <- grayling_df %>% filter(lake == "I3")
lake_I8 <- grayling_df %>% filter(lake == "I8") 

n1 <- nrow(lake_I3)
n2 <- nrow(lake_I8)
sd_pooled <- sqrt((var(lake_I3$length_mm) * (n1-1) + 
                  var(lake_I8$length_mm) * (n2-1)) / 
                  (n1 + n2 - 2))

# Calculate power
effect_size <- 30 / sd_pooled  # Cohen's d
df <- n1 + n2 - 2
alpha <- 0.05
power <- power.t.test(n = min(n1, n2), 
                     delta = effect_size,
                     sd = 1,  # Using standardized effect size
                     sig.level = alpha,
                     type = "two.sample",
                     alternative = "two.sided")

# Display results
power


     Two-sample t test power calculation 

              n = 66
          delta = 0.6741298
             sd = 1
      sig.level = 0.05
          power = 0.9702076
    alternative = two.sided

NOTE: n is number in *each* group

Lecture 4: Summary

Key concepts covered:

Probability distributions model random phenomena
- Normal distribution is especially important
- Z-scores standardize measurements
Standard error measures precision of estimates
- Decreases with larger sample sizes
- Used to construct confidence intervals
Confidence intervals express uncertainty
- Provide plausible range for parameters
- 95% CI: mean ± 1.96 × SE
Hypothesis testing evaluates claims
- Null vs. alternative hypotheses
- P-values quantify evidence against H₀
- Consider both statistical and practical significance

Other Formats

Lecture 4: Probability and Statistical Inference

Practice Exercise 1: Exploring the Grayling Dataset

Lecture 4: Probability Distributions

Probability Distribution Functions

Lecture 4: The Standard Normal Distribution

Practice Exercise 2: Calculating Z-scores of lake I3

Lecture 4: The fish data as a z score

Z-score Results

How to get area under 1 STD DEV?

Lecture 4: Standard normal distribution - Fish Data

Lecture 4: Standard normal distribution properties

Lecture 4: Using Z-tables

Lecture 4: Z-score Formula

\(\text{Z = }\frac{x_i-\mu}{\sigma}\)

Lecture 4: Z-score example from table

\(\text{Z = }\frac{X_i-\mu}{\sigma}\)

Lecture 4: Z-score example calculation in r

Lecture 4: We can now use this for fun in the fish

Lecture 4: What this means

Lecture 4: So what is next…

Lecture 4: When Population σ is Unknown

When calculating confidence intervals we usually DON’T know the population σ (standard deviation) or 𝝁 population mean

Lecture 4: Understanding t-distribution

When sample sizes are small, the t-distribution is more appropriate than the normal distribution.

Student’s t-distribution Formula

To calculate CI for sample from “unknown” population:

\(\text{CI} = \bar{y} \pm t \cdot \frac{s}{\sqrt{n}}\)

Lecture 4: Student’s t-distribution Table

Here is a t-table

Lecture 4: One-tailed Questions

One-tailed questions: area of distribution left or (right) of a certain value

Lecture 4: Two-tailed Questions

Lecture 4: t-distribution CI Example

\(\text{CI} = \bar{y} \pm t \cdot \frac{s}{\sqrt{n}}\)

Practice Exercise 4: Using the t-distribution

Lecture 4: Intro to Hypothesis Testing one tailed

Lecture 4: Intro to Hypothesis Testing one tailed

Lecture 4: Hypothesis Testing two tailed

Lecture 4: Hypothesis Testing two tailed

Practice Exercise 5: One-Sample t-Test

Practice Exercise 6: Formulating Hypotheses

Lecture 4: Understanding P-values

Lecture 4: Type I and Type II Errors

Lecture 4: Type I and Type II Errors

Practice Exercise 7: Interpreting Errors and Power

Lecture 4: Summary