Tech Tutorial: Foundations

Tech Tutorial — Foundations. This unit works with probability distributions directly — no data file. You give a distribution its parameters and read off probabilities and percentiles. Pick one tool and follow its tab.

To keep things concrete we imagine the population behind the class survey: study hours across all intro-stats students are roughly Normal, mean 13, SD 4, and a fraction 0.44 of all students live on campus.

What you’ll do

  1. Normal probabilities and percentiles.
  2. Binomial probabilities.

1. Normal distribution

Take study hours to be \(X \sim N(\mu = 13,\ \sigma = 4)\). Find:

  • (a) the probability a student studies more than 20 hours, \(P(X > 20)\);
  • (b) the 90th percentile of study hours.
  1. Open the Normal Distribution calculator (mean and SD are pre-filled).
  2. For (a), set the boundary to 20 and select the upper tail — read the shaded probability.
  3. For (b), switch to inverse / find-x mode and enter 0.90 to get the percentile.
  1. Install the distrACTION module (Jamovi library), then distrACTION → Normal Distribution.
  2. Enter Mean = 13, SD = 4.
  3. Use Compute probability with \(x_1 = 20\) and the upper tail for (a); use Compute quantile with \(p = 0.90\) for (b).

(Jamovi screenshots to be added.)

# (a) P(X > 20) for X ~ N(13, 4)
pnorm(20, mean = 13, sd = 4, lower.tail = FALSE)
[1] 0.04005916
# (b) 90th percentile
qnorm(0.90, mean = 13, sd = 4)
[1] 18.12621

2. Binomial distribution

Suppose 44% of all students live on campus. In a random sample of \(n = 12\) students, let \(X\) be the number living on campus, so \(X \sim \text{Binomial}(n = 12,\ p = 0.44)\). Find:

  • (a) \(P(X = 6)\) — exactly six;
  • (b) \(P(X \ge 6)\) — six or more.
  1. Open the Binomial Distribution calculator.
  2. Set n = 12 and p = 0.44.
  3. For (a) read the bar at k = 6; for (b) select the ≥ 6 (upper) region.
  1. distrACTION → Binomial Distribution (same module as above).
  2. Enter n = 12, p = 0.44.
  3. Compute probability with \(x = 6\) for (a), and the \(P(X \ge x)\) option for (b).

(Jamovi screenshots to be added.)

# (a) P(X = 6) for X ~ Binomial(12, 0.44)
dbinom(6, size = 12, prob = 0.44)
[1] 0.2067836
# (b) P(X >= 6) = 1 - P(X <= 5)
pbinom(5, size = 12, prob = 0.44, lower.tail = FALSE)
[1] 0.4448014

Check yourself

The OpenIntro IMS and OpenIntro Statistics materials cover the probability foundations; the R functions above (pnorm, qnorm, dbinom, pbinom) are the core toolkit. See the normal tutorial for how the normal model underlies later inference.