## Lesson 11: The t-distribution for Small Sample Inference

October 1, 2024

Review:

- Estimation with Confidence Intervals
- Inference for proportions
- Quiz.

Presentation:

- What if the best you can do is a sample size of n<30?
- t-distribution
- For larger samples, n ≥ 30 use Z-table
- For small samples, n < 30 use t-distribution
- Only impacts calculation of the test statistic
- margin of error for confidence intervals
- p-value for significance testing (don’t worry, we cover these topics later)

**t distribution critical values**- print the t distribution table for the next exam
- use the 95% confidence or “0.025” column

- select the row corresponding to df = n – 1
- df = degrees of freedom
- n = number of records in sample data

- find the t* critical value
- row = df = n – 1
- column = .025 (bottom label) or 95% confidence level (top label)
- e.g., if n = 13
- df = n – 1 = 12
- go to the row where df = 12 and move to the .025 column
- you should find
**t* = 2.18**

- print the t distribution table for the next exam
- t-statistic formulas
- for
**confidence intervals**- x-bar ± t * s/√n
- instead of the Z equivalent: x-bar ± z * σ/√n

- for
**Video Unit 26**- Examples:
- Identify appropriate t distribution critical values
- n=10 with 95% confidence, t* = 2.26
- n=20 with 95% confidence, t* = 2.09

- A sample of size n = 25 produced the sample mean, x-bar = 36.0 and standard deviation, s = 9.0. Construct a 95% confidence interval to estimate the population mean.
- n = 25, x-bar = 36.0, s = 9.0
- df = 24, t* = 2.064
- m = 2.064*(9/(√25)) = 3.7152
- Estimate for μ = 36.0 ± 3.7152
- 95% confidence interval: [32.2848, 39.7152]
- previous lesson interval with Z=1.96 and n=400: [35.118, 36.882]
*sample size can make a big difference*

- Identify appropriate t distribution critical values

Activity:

**Problem 1.**Use the t distribution table, assuming 95% confidence, to find the value of**t***for each of the following sample sizes.- n = 12
- n = 18
- n = 24
- n = 28

**Problem 2.**The weights (in lbs) from a random sample of n=16 four-year-old children who took part in a study on childhood obesity are provided below. Estimate the mean weight of four-year-old children using a 95% confidence interval. The sample standard deviation equals 4.2.- {37.1, 26.7, 36.1, 36.2, 40.3, 43.9, 36.2, 40.7, 42.5, 34.8, 37.9, 34.5, 31.1, 36.4, 35.7, 33.4}

**Problem 3.**After purchasing a new fuel-efficient vehicle, the owner calculates miles per gallon (mpg) the first n=12 times he fills the fuel tank. His calculations are provided below. Estimate vehicle fuel efficiency using a 95% confidence interval. You will need to calculate standard deviation.- {42, 43, 37, 36, 34, 45, 48, 43, 38, 42, 43, 46}

*These examples and problems are from the Against All Odds video series and Introduction to the Practice of Statistics (6th Ed.) by Moore et al.*