Sampling bias occurs when a sample does not accurately represent the population, causing some members to be overrepresented or underrepresented

Result:

• Skewed or invalid results

• Cannot generalize findings to the population

1. Self-Selection Bias (Voluntary Response Bias)

People with strong opinions volunteer to participate

Example: Online polls where only motivated people respond

2. Nonresponse Bias

People who refuse to participate differ systematically from those who do

Example: Busy people less likely to complete long surveys

3. Undercoverage Bias

Some groups in the population are inadequately represented

Example: Online surveys miss people without internet access

4. Convenience Sampling Bias

Sampling only easily accessible individuals

Example: Surveying only students in one classroom

5. Survivorship Bias

Only studying "survivors" or successful cases

Example: Only interviewing successful entrepreneurs

Measures the average squared deviation from the mean (spread of data)

Population Variance:

\[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{N} \]

Sample Variance:

\[ s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1} \]

Note: Divide by \( n-1 \) for sample (Bessel's correction)

Square root of variance; measures typical distance from the mean

\[ \sigma = \sqrt{\sigma^2} \quad \text{or} \quad s = \sqrt{s^2} \]

A data value that is significantly different (much larger or smaller) from other values in the dataset

Rule:

Any data value more than 3 standard deviations from the mean is an outlier

\[ \text{Outlier if: } x < \mu - 3\sigma \text{ or } x > \mu + 3\sigma \]

Steps:

1. Find Q1 (25th percentile) and Q3 (75th percentile)

2. Calculate IQR = Q3 - Q1

3. Check for outliers:

\[ \text{Lower outliers: } x < Q1 - 1.5 \times IQR \]

\[ \text{Upper outliers: } x > Q3 + 1.5 \times IQR \]

Data: 10, 12, 14, 15, 16, 18, 50. Mean = 19.3, SD = 14.4

No outliers using 3-sigma rule (though 50 appears unusual)

Removing outliers typically affects:

Mean:

• Most affected by outliers

• Will move toward center of remaining data

Median:

• Resistant to outliers (less affected)

• May change slightly or not at all

Standard Deviation:

• Usually decreases (data less spread out)

• Indicates more consistent data

Range:

• Always decreases

Data: 10, 12, 14, 15, 16, 18, 100

Effect: Mean decreased significantly, SD decreased dramatically, median barely changed

A range of values that likely contains the true population parameter with a specified level of confidence

\[ \text{CI} = \bar{x} \pm z^* \cdot \frac{\sigma}{\sqrt{n}} \]

where:

• \( \bar{x} \) = sample mean

• \( z^* \) = critical value (z-score for confidence level)

• \( \sigma \) = population standard deviation

• \( n \) = sample size

• \( \frac{\sigma}{\sqrt{n}} \) = standard error

Sample: n = 100, \( \bar{x} = 75 \), σ = 10. Find 95% CI.

95% CI: (73.04, 76.96)

\[ \text{CI} = \hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \]

where:

• \( \hat{p} = \frac{x}{n} \) = sample proportion

• \( x \) = number of successes

• \( n \) = sample size

• \( z^* \) = critical value

• Random sample

• \( n\hat{p} \geq 10 \) and \( n(1-\hat{p}) \geq 10 \)

• Sample size < 10% of population

Survey: 200 people, 120 support a policy. Find 95% CI for proportion.

95% CI: (53.2%, 66.8%)

For a 95% confidence interval (a, b):

✓ CORRECT:

"We are 95% confident that the true population mean lies between a and b"

"If we repeated this process many times, about 95% of intervals would contain the true mean"

✗ INCORRECT:

"There is a 95% probability the true mean is between a and b" (probability is wrong - the interval either contains μ or it doesn't)

"95% of the data falls in this interval" (CI is about parameter, not data)

• Higher confidence level → Wider interval (more certainty, less precision)

• Lower confidence level → Narrower interval (less certainty, more precision)

• Larger sample size → Narrower interval (more precision)

1. Control Group

Receives no treatment or standard treatment (baseline for comparison)

2. Treatment Group

Receives the experimental treatment

3. Random Assignment

Randomly assign subjects to groups to eliminate bias

4. Replication

Use enough subjects to detect effects (larger sample = more reliable)

5. Blinding

• Single-blind: Subjects don't know which group they're in

• Double-blind: Neither subjects nor researchers know (reduces bias)

Observational Study:

Observe subjects without intervention; can show association but NOT causation

Experiment:

Researcher imposes treatment; CAN establish causation with proper design

Simulations help determine if observed results could have occurred by chance

Steps:

1. State null hypothesis (no effect/difference)

2. Run many simulations assuming null hypothesis is true

3. Compare observed result to simulation distribution

4. Calculate p-value (proportion of simulations as extreme as observed)

5. Make conclusion

P-value:

Probability of getting results as extreme as observed, assuming no real effect

• Small p-value (< 0.05): Results unlikely due to chance → Statistically significant

• Large p-value (> 0.05): Results could easily occur by chance → Not significant

A coin is flipped 100 times, getting 60 heads. Is the coin fair?

Conclusion: Result is statistically significant (p < 0.05); coin may be biased