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

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.

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 under curve
• But it is tedious to do every time
• Instead
  • we can use the standard normal distribution (SND)

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

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

The standard error of the mean (SEM) tells us how precise our sample mean is as an estimate of the population mean.

Standard Error Formula: \[ SE_{\bar{Y}} = \frac{s}{\sqrt{n}} \]

Where:

• \(s\) is the sample standard deviation
• \(n\) is the sample size

Key properties:

• SEM decreases as sample size increases
• SEM is used to construct confidence intervals
• SEM measures the precision of the sample mean

Notice: - \(s_{\bar{y}}\) depends on - sample s (standard deviation) - sample n - (\(s_{\bar{y}} = \frac{s}{\sqrt{n}}\))

How and why? - Decreases with sample n - number - increases with sample s - standard deviation

• Large sample, low s = greater confidence in estimate of \(\mu\)

The standard error of the mean (SEM) tells us how precise our sample mean is as an estimate of the population mean.

Standard Error Formula: \[ SE_{\bar{Y}} = \frac{s}{\sqrt{n}} \]

Where:

• \(s\) is the sample standard deviation
• \(n\) is the sample size

Key properties:

• SEM decreases as sample size increases
• SEM is used to construct confidence intervals
• SEM measures the precision of the sample mean

A confidence interval is a range of values that is likely to contain the true population parameter.

95% Confidence Interval Formula: \[\text{95% CI} = \bar{y} \pm z \cdot \frac{\sigma}{\sqrt{n}}\]

Where:

• ȳ is the sample mean
• 𝑛 is the sample size
• σ is the population standard deviation
• z is the z-value corresponding the probability of the CI

A confidence interval is a range of values that is likely to contain the true population parameter.

Interpretation: If we were to take many samples and calculate the 95% CI for each, about 95% of these intervals would contain the true population mean.

Common misinterpretation: “There is a 95% probability that the true mean is in this interval.”

• Interpret 95% CI to mean:
  • Range of values that contains µ (population mean) with 95% probability
• More correctly:
  • If we took 100 samples from population
  • calculate a CI from each
  • 95 of the 100 CIs will contain the true population mean - µ

Lets compare what the two plots look like near each other

In the more typical case DON’T know the population σ

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

Instead, we use Student’s 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

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”

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)

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)

Let’s calculate CIs again:

Use two-sided test

• 95% CI Sample A: = 272.8 ± 2.262 * (37.81/(9^0.5)) = 1.650788
• The 95% CI is between 244.3 and 301.3
• “The 95% CI for the population mean from sample A is 272.8 ± 28.5”

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

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