UI STAT 251 Fall 2010 Quiz 2 NAME:
UI #:
Class: 9:30 / 10:30 (circle yours)
1. Explain the 68-95-99.7 rule in your own words.
For a normally distributed population, 68% of the observations will lie within 1 sd away from the mean, 95% of the
observations will lie within 2 sd away from the mean, and 99.7% of the observations will lie within 3 sd away from the
mean.
2. The height of women is known to be normally distributed with mean 67 inches and standard deviation 2.5 inches.
a) What percent of women are shorter than 65 inches?
> z=(65-67)/2.5
>z
[1] -0.8
> pnorm(z)
[1] 0.2118554
b) What percent of women are between 68 and 69 inches tall?
> z1=(69-67)/2.5
> z2=(68-67)/2.5
> pnorm(z1)
[1] 0.7881446
> pnorm(z2)
[1] 0.6554217
> pnorm(z1)-pnorm(z2)
[1] 0.1327229
c) What are the values of the Q1 and Q3 for the height of women?
> 67+qnorm(c(.25,.75))*2.5
[1] 65.31378 68.68622
3. Cereal boxes are labeled 12 oz. The boxes are filled by a machine. The amount the machine fills is normally
distributed with mean 12.2 oz and standard deviation 0.15 oz.
a) What is the probability a box of cereal is underfilled?
> z=(12-12.2)/0.15
> pnorm(z)
[1] 0.09121122
b) A consumer group wants to the company to change the mean amount of cereal the machine fills so that only 5%
of boxes are underfilled. What do we need to change the mean to while keeping the same standard deviation?
New mean = 12 + 1.64*.15 = 12.246 oz
c) Another approach to control the machine fills so that only 5% of boxes are underfilled while keeping the same
mean at 12.2 oz is to change to standard deviation. What should the new standard deviation be changed to?
New sd = (12 – 12.2)/-1.64 = 0.122 oz