Inference for quantitative data

Purpose of t-score

Estimating the population mean (μ) by considering a single sample mean (x̄)

Using s as an estimate for σ

Causes the % of intervals that actually succeed in capturing μ to be less than our stated CI, so we use t-score instead

→ smaller sample = more stretched tails

→ larger sample = closer to normal distribution

Degree of freedom

The number of variables in the final calculation (amount of logically independent values)

Calculated as n - 1

Conditions for one sample t-interval for mean

1. n >/= 30 (must be approx normal and symmetric) - it’s okay if under 30 if there’s no skewness or outliers

2. Parent population must be approx normal

   → If there are outliers, perform an analysis twice (with and without them) even if the sample is large

One sample t-interval for mean steps

1. Define parameter (let … be …)

2. Identify the procedure

3. Check conditions

4. Calculate the CI (x̄ ± t* (s / √n))

5. Interpret the interval in context

For a smaller (more precise) interval

Either decrease the df or increase the sample size

For an increased df

We must accept a wider CI or increase the sample size

→ we ALWAYS want a bigger sample size

Confidence interval for a difference of two population means

1. Define parameter (μ₁ and μ₂)

2. Identify procedure and check conditions (if parent pop is approx normal, CLT is applied)

3. Calc the CI ((x̄₁ - x̄₂) ± t* × SE(x̄₁ - x̄₂) → DF is hard to find so we can use (n₁ + n₂ - 2) assuming that the variance is similar but n₁ ≉ n₂

4. Conclusion

T-test for a difference in population means

1. Define parameters

2. Identify procedure and check conditions

3. Calc test stat and find p-value

4. Conclusion (include possible error and consequence)

Paired t-method

A one sample t-test for the means of pairwise differences where n = the amount of pairs

Uses formula for one sample t-interval (improvement/change is the parameters)

Used to test if there is a difference in two points of time or 2 groups with similar characteristics (ex. before and after study)

Steps for paired t-method

1. Define parameters

2. Procedure and checks

3. Mechanics (CI or p-value)

4. Conclusion