Single-samples t-tests

When are single samples t-tests used? (1)

• When comparing a sample mean/given value to a population mean/constant, when you don’t know the population standard deviation

Kurtosis (3)

• Platykurtic: flatter peaks, shorter, thinner tails and fewer extreme outliers (t-distribution)

• Mesokurtic: normal

• Leptokurtic: taller peaks, fatter tails, and more extreme outliers

Relationship between sample size and t-distribution?

• As sample size increases → t-distribution approaches normal distribution (mesokurtic) from more platykurtic distribution

Required info (4)

• Known population mean/constant

• N = sample size

• X̅ = sample mean

• ∑(X−X̅)² = sum of squares (if not provided with estimated population standard deviation)

Step 1 (1)

• Estimate population standard deviation (by just calculating the sample standard deviation)

Step 2 (1)

• Estimate standard error (estimated population standard deviation / square root of sample size)

Step 3 (1)

• Calculate t-obtained ( (sample mean - population mean) / standard error )

What does the t-obtained indicate? (1)

• How many standard error units the sample mean is away from the population mean

Step 4 (1, 1→2, 1)

• Compare t-obtained to t-crit

• Find t-crit, using:

  • df = N-1

  • Significance (usually alpha = .05)

• Is t-obtained greater than t-crit.? If so, then p < .05 → likely statistically significant

Assumptions (2)

• Normality

• Independence of observations

Capital S vs lowercase s?

• Capital when actual population level value, lowercase when estimated population level value

What is Bessel’s correction?

• N-1 in s_x (estimated population standard deviation) formula: corrects for samples underestimate standard deviation of populations