Statistics 1

In-Class Exercise 9

Yu-You Liou

Shih Chien University

2026-09-16

Question 1

Areas Under the Standard Normal Curve

Use pnorm() and qnorm() in R.

1. P(z<1.28)
2. P(z>-0.75)
3. P(-1.50<z<2.10)
4. The value of z that has an area of 0.95 to its left

Question 2

Applications of the Normal Distribution

IQ scores are normally distributed with \mu=100 and \sigma=15.

1. Find P(X>120).
2. Find P(90<X<115).
3. Verify part 1 by first converting to a z score with z=(X-\mu)/\sigma and then using pnorm().

Question 3

Working Backward

For the same IQ distribution:

1. Find the score that separates the top 5% from the rest.
2. Find the two scores that bound the middle 90%.
3. Draw the normal curve with ggplot() and dnorm(), and shade the middle 90% with geom_area().